# QNT/351 Descriptive Statistics – Real Estate Data Part 1

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MERRILL LYNCH recently completed a study of online investment portfolios for a sample of clients. For the 70 participants in the study, organize these data into a frequency distribution. (See Exercise 43 and LO2-3.)

LEARNING OBJECTIVES When you have completed this chapter, you will be able to:

LO2-1 Summarize qualitative variables with frequency and relative frequency tables.

LO2-2 Display a frequency table using a bar or pie chart.

LO2-3 Summarize quantitative variables with frequency and relative frequency distributions.

LO2-4 Display a frequency distribution using a histogram or frequency polygon.

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. INTRODUCTION

The highly competitive automobile retailing industry in the United States has changed dramatically in recent years. These changes spurred events such as the

bankruptcies of General Motors and Chrysler in 2009.

elimination of well-known brands like Pontiac and Saturn.

closing of over 1,500 local dealerships.

collapse of consumer credit availability.

consolidation of dealership groups.

Traditionally, a local family owned and operated the community dealership, which might have included one or two manufacturers or brands, like Pontiac and GMC Trucks or Chrysler and the popular Jeep line. Recently, however, skillfully managed and well-financed companies have been acquiring local dealerships throughout the country. As these groups acquire the local dealerships, they often bring standardized selling practices, common software and hardware technology platforms, and management reporting techniques. The goal of these new organizations is to provide an improved buying experience for the consumer, while increasing profitability. Megadealerships often employ over 10,000 people, generate several billion dollars in annual sales, own more than 50 franchises, and are traded on the New York Stock Exchange or NASDAQ. Today, the largest megadealership is AutoNation (ticker symbol AN). Others include Penske Auto Group (PAG and second largest), Asbury Automotive Group (ABG), and Hendrick Auto Group (which is privately held).

The Applewood Auto Group is an ownership group that includes four dealerships. The group sells a wide range of vehicles, including the inexpensive but popular Korean brands Kia and Hyundai, BMW and Volvo sedans and luxury SUVs, and a full line of Ford and Chevrolet cars and trucks.

Ms. Kathryn Ball is a member of the senior management team at Applewood Auto Group, which has its corporate offices adjacent to Kane Motors. She is responsible for tracking and analyzing vehicle sales and the profitability of those vehicles. Kathryn would like to summarize the profit earned on the vehicles sold with tables, charts, and graphs that she would review monthly. She wants to know the profit per vehicle sold, as well as the lowest and highest amount of profit. She is also interested in describing the demographics of the buyers. What are their ages? How many vehicles have they previously purchased from one of the Applewood dealerships? What type of vehicle did they purchase?

The Applewood Auto Group operates four dealerships:

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Tionesta Ford Lincoln Mercury sells the Ford, Lincoln, and Mercury cars and trucks.

Olean Automotive Inc. has the Nissan franchise as well as the General Motors brands of Chevrolet, Cadillac, and GMC Trucks.

Sheffield Motors Inc. sells Buick, GMC trucks, Hyundai, and Kia.

Kane Motors offers the Chrysler, Dodge, and Jeep line as well as BMW and Volvo.

Every month, Ms. Ball collects data from each of the four dealerships and enters them into an Excel spreadsheet. Last month the Applewood Auto Group sold 180 vehicles at the four dealerships. A copy of the first few observations appears to the left. The variables collected include:

Age—the age of the buyer at the time of the purchase.

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Profit—the amount earned by the dealership on the sale of each vehicle.

Location—the dealership where the vehicle was purchased.

Vehicle type—SUV, sedan, compact, hybrid, or truck.

Previous—the number of vehicles previously purchased at any of the four Apple-wood dealerships by the consumer.

The entire data set is available at the McGraw-Hill website (www.mhhe.com/lind16e) and in Appendix A.4 at the end of the text.

LO2-1

Summarize qualitative variables with frequency and relative frequency tables.

CONSTRUCTING FREQUENCY TABLES Recall from Chapter 1 that techniques used to describe a set of data are called descriptive statistics. Descriptive statistics organize data to show the general pattern of the data, to identify where values tend to concentrate, and to expose extreme or unusual data values. The first technique we discuss is a frequency table.

FREQUENCY TABLE A grouping of qualitative data into mutually exclusive and collectively exhaustive classes showing the number of observations in each class.

In Chapter 1, we distinguished between qualitative and quantitative variables. To review, a qualitative variable is nonnumeric, that is, it can only be classified into distinct categories. Examples of qualitative data include political affiliation (Republican, Democrat, Independent), state of birth (Alabama, . . . , Wyoming), and method of payment for a purchase at Barnes & Noble (cash, check, debit, or credit). On the other hand, quantitative variables are numerical in nature. Examples of quantitative data relating to college students include the price of their textbooks, their age, and the number of credit hours they are registered for this semester.

In the Applewood Auto Group data set, there are five variables for each vehicle sale: age of the buyer, amount of profit, dealer that made the sale, type of vehicle sold, and number of previous purchases by the buyer. The dealer and the type of vehicle are qualitative variables. The amount of profit, the age of the buyer, and the number of previous purchases are quantitative variables.

Suppose Ms. Ball wanted to summarize last month’s sales by location. To summarize this qualitative data, we classify the vehicles sold last month according to their location: Tionesta, Olean, Sheffield, or Kane. We use location to develop a frequency table with four mutually exclusive (distinctive) classes. This means that a particular vehicle cannot belong to more than one class. Each vehicle is uniquely classified into one of the four mutually exclusive locations. In addition, the frequency table must be collectively exhaustive. This means that every vehicle is accounted for in the table. This frequency table is shown in Table 2–1. The number of observations, representing the sales at each location, is called the class frequency. So the class frequency for vehicles sold at the Kane location is 52.

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TABLE 2–1 Frequency Table for Vehicles Sold Last Month at Applewood Auto Group by Location

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CHART 2–1 Number of Vehicles Sold by Location

We explain the details of constructing a pie chart using the information in Table 2–3, which shows the frequency and percent of cars sold by the Applewood Auto Group for each vehicle type.

TABLE 2–3 Vehicle Sales by Type at Applewood Auto Group

The first step to develop a pie chart is to mark the percentages 0, 5, 10, 15, and so on evenly around the circumference of a circle (see Chart 2–2). To plot the 40% of total sales represented by sedans, draw a line from the center of the circle to 0 and another line from the center of the circle to 40%. The area in this “slice” represents the number of sedans sold as a percentage of the total sales. Next, add the SUV’s percentage of total sales, 30%, to the sedan’s percentage of total sales, 40%. The result is 70%. Draw a line from the center of the circle to 70%, so the area between 40 and 70 shows the sales of SUVs as a percentage of total sales. Continuing, add the 15% of total sales for compact vehicles, which gives us a total of 85%. Draw a line from the center of the circle to 85, so the “slice” between 70% and 85% represents the number of compact vehicles sold as a percentage of the total sales. The remaining 10% for truck sales and 5% for hybrid sales are added to the chart using the same method.

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Chart 2–2 Pie Chart of Vehicles by Type

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. Because each slice of the pie represents the relative frequency of each vehicle type as a percentage of the total sales, we can easily compare them:

The largest percentage of sales is for sedans.

Sedans and SUVs together account for 70% of vehicle sales.

Hybrids account for 5% of vehicle sales, in spite of being on the market for only a few years.

We can use Excel software to quickly count the number of cars for each vehicle type and create the frequency table, bar chart, and pie chart below. The Excel tool is called a Pivot Table. The instructions to produce these descriptive statistics and charts are given in Appendix C.

Pie and bar charts both serve to illustrate frequency and relative frequency tables. When is a pie chart preferred to a bar chart? In most cases, pie charts are used to show and compare the relative differences in the percentage of observations for each value or class of a qualitative variable. Bar charts are preferred when the goal is to compare the number or frequency of observations for each value or class of a qualitative variable. The following Example/Solution shows another application of bar and pie charts.

E X A M P L E

SkiLodges.com is test marketing its new website and is interested in how easy its website design is to navigate. It randomly selected 200 regular Internet users and asked them to perform a search task on the website. Each person was asked to rate the relative ease of navigation as poor, good, excellent, or awesome. The results are shown in the following table:

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. 1. What type of measurement scale is used for ease of navigation?

2. Draw a bar chart for the survey results.

3. Draw a pie chart for the survey results.

S O L U T I O N

The data are measured on an ordinal scale. That is, the scale is ranked in relative ease when moving from “poor” to “awesome.” Also, the interval between each rating is unknown so it is impossible, for example, to conclude that a rating of good is twice the value of a poor rating.

We can use a bar chart to graph the data. The vertical scale shows the relative frequency and the horizontal scale shows the values of the ease-of-navigation variable.

A pie chart can also be used to graph these data. The pie chart emphasizes that more than half of the respondents rate the relative ease of using the website awesome.

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2–1

The answers are in Appendix E.

DeCenzo Specialty Food and Beverage Company has been serving a cola drink with an additional flavoring, Cola-Plus, that is very popular among its customers. The company is interested in customer preferences for Cola-Plus versus Coca-Cola, Pepsi, and a lemon-lime beverage. They ask 100 randomly sampled customers to take a taste test and select the beverage they prefer most. The results are shown in the following table:

(a) Is the data qualitative or quantitative? Why?

(b) What is the table called? What does it show?

(c) Develop a bar chart to depict the information.

(d) Develop a pie chart using the relative frequencies.

E X E R C I S E S

The answers to the odd-numbered exercises are at the end of the book in Appendix D.

1. A pie chart shows the relative market share of cola products. The “slice” for Pepsi-Cola has a central angle of 90 degrees. What is its market share?

2. In a marketing study, 100 consumers were asked to select the best digital music player from the iPod, the iRiver, and the Magic Star MP3. To summarize the consumer responses with a frequency table, how many classes would the frequency table have?

3. A total of 1,000 residents in Minnesota were asked which season they preferred. One hundred liked winter best, 300 liked spring, 400 liked summer, and 200 liked fall. Develop a frequency table and a relative frequency table to summarize this information.

4. Two thousand frequent business travelers are asked which midwestern city they prefer: Indianapolis, Saint Louis, Chicago, or Milwaukee. One hundred liked Indianapolis best, 450 liked Saint Louis, 1,300 liked Chicago, and the remainder preferred Milwaukee. Develop a frequency table and a relative frequency table to summarize this information.

5. Wellstone Inc. produces and markets replacement covers for cell phones in five different colors: bright white, metallic black, magnetic lime, tangerine orange, and fusion red. To estimate the demand for each

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color, the company set up a kiosk in the Mall of America for several hours and asked randomly selected people which cover color was their favorite. The results follow:

a. What is the table called?

b. Draw a bar chart for the table.

c. Draw a pie chart.

d. If Wellstone Inc. plans to produce 1 million cell phone covers, how many of each color should it produce?

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. 6. A small business consultant is investigating the performance of several companies. The fourth-quarter sales for last year (in thousands of dollars) for the selected companies were:

The consultant wants to include a chart in his report comparing the sales of the six companies. Use a bar chart to compare the fourth-quarter sales of these corporations and write a brief report summarizing the bar chart.

LO2-3

Summarize quantitative variables with frequency and relative frequency distributions.

CONSTRUCTING FREQUENCY DISTRIBUTIONS In Chapter 1 and earlier in this chapter, we distinguished between qualitative and quantitative data. In the previous section, using the Applewood Automotive Group data, we summarized two qualitative variables: the location of the sale and the type of vehicle sold. We created frequency and relative frequency tables and depicted the results in bar and pie charts.

The Applewood Auto Group data also includes several quantitative variables: the age of the buyer, the profit earned on the sale of the vehicle, and the number of previous purchases. Suppose Ms. Ball wants to summarize last month’s sales by profit earned. We can describe profit using a frequency distribution.

FREQUENCY DISTRIBUTION A grouping of quantitative data into mutually exclusive and collectively exhaustive classes showing the number of observations in each class.

How do we develop a frequency distribution? The following example shows the steps to construct a frequency distribution. Remember, our goal is to construct tables, charts, and graphs that will quickly reveal the concentration, extreme values, and shape of the data.

E X A M P L E

We return to the situation where Ms. Kathryn Ball of the Applewood Auto Group wants to summarize the quantitative variable profit with a frequency distribution and display the distribution with charts and graphs. With this information, Ms. Ball can easily answer the following questions: What is the typical profit on each sale? What is the largest or maximum profit on any sale? What is the smallest or minimum profit on any sale? Around what value do the profits tend to cluster?

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S O L U T I O N

To begin, we need the profits for each of the 180 vehicle sales, which are listed in Table 2–4. This information is called raw or ungrouped data because it is simply a listing of the individual, observed profits. It is possible to search the list and find the smallest or minimum profit ($294) and the largest or maximum profit ($3,292),

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. but that is about all. It is difficult to determine a typical profit or to visualize where the profits tend to cluster. The raw data are more easily interpreted if we summarize the data with a frequency distribution as follows.

TABLE 2–4 Profit on Vehicles Sold Last Month by the Applewood Auto Group

Step 1: Decide on the number of classes. A useful recipe to determine the number of classes (k) is the “2 to the k rule.” This guide suggests you select the smallest number (k) for the number of classes such

that 2k (in words, 2 raised to the power of k) is greater than the number of observations (n). In the Applewood Auto Group example, there were 180 vehicles sold. So n 5 180. If we try k 5 = 7, which

means we would use 7 classes, 27 = 128, which is less than 180. Hence, 7 is too few classes. If we

let k = 8, then 28 = 256, which is greater than 180. So the recommended number of classes is 8.

Step 2: Determine the class interval. Generally, the class interval is the same for all classes. The classes all taken together must cover at least the distance from the minimum value in the data up to the maximum value. Expressing these words in a formula:

where i is the class interval, and k is the number of classes. For the Applewood Auto Group, the minimum value is $294 and the maximum value is $3,292. If

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we need 8 classes, the interval should be:

In practice, this interval size is usually rounded up to some convenient number, such as a multiple of 10 or 100. The value of $400 is a reasonable choice.

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. Step 3: Set the individual class limits. State clear class limits so you can put each observation into only one category. This means you must avoid overlapping or unclear class limits. For example, classes such as “$1,300–$1,400” and “$1,400–$1,500” should not be used because it is not clear whether the value $1,400 is in the first or second class. In this text, we will generally use the format $1,300 up to $1,400 and $1,400 up to $1,500 and so on. With this format, it is clear that $1,399 goes into the first class and $1,400 in the second. Because we always round the class interval up to get a convenient class size, we cover a larger than necessary range. For example, using 8 classes with an interval of $400 in the Applewood Auto Group example results in a range of 8($400) = $3,200. The actual range is $2,998, found by ($3,292 − $294). Comparing that value to $3,200, we have an excess of $202. Because we need to cover only the range (Maximum – Minimum), it is natural to put approximately equal amounts of the excess in each of the two tails. Of course, we should also select convenient class limits. A guideline is to make the lower limit of the first class a multiple of the class interval. Sometimes this is not possible, but the lower limit should at least be rounded. So here are the classes we could use for these data.

Step 4: Tally the vehicle profit into the classes. To begin, the profit from the sale of the first vehicle in Table 2–4 is $1,387. It is tallied in the $1,000 up to $1,400 class. The second profit in the first row of Table 2–4 is $2,148. It is tallied in the $1,800 up to $2,200 class. The other profits are tallied in a similar manner. When all the profits are tallied, the table would appear as:

Step 5: Count the number of items in each class. The number of observations in each class is called the class frequency. In the $200 up to $600 class there are 8 observations, and in the $600 up to $1,000

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Page 28class there are 11 observations. Therefore, the class frequency in

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. the first class is 8 and the class frequency in the second class is 11. There are a total of 180 observations in the entire set of data. So the sum of all the frequencies should be equal to 180.

TABLE 2–5 Frequency Distribution of Profit for Vehicles Sold Last Month at Applewood Auto Group

Now that we have organized the data into a frequency distribution (see Table 2–5), we can summarize the profits of the vehicles for the Applewood Auto Group. Observe the following:

1. The profits from vehicle sales range between $200 and $3,400.

2. The vehicle profits are classified using a class interval of $400. The class interval is determined by subtracting consecutive lower or upper class limits. For example, the lower limit of the first class is $200, and the lower limit of the second class is $600. The difference is the class interval of $400.

3. The profits are concentrated between $1,000 and $3,000. The profit on 157 vehicles, or 87%, was within this range.

4. For each class, we can determine the typical profit or class midpoint. It is halfway between the lower or upper limits of two consecutive classes. It is computed by adding the lower or upper limits of consecutive classes and dividing by 2. Referring to Table 2–5, the lower class limit of the first class is $200, and the next class limit is $600. The class midpoint is $400, found by ($600 1 $200)y2. The midpoint best represents, or is typical of, the profits of the vehicles in that class. Applewood sold 8 vehicles with a typical profit of $400.

5. The largest concentration, or highest frequency, of vehicles sold is in the $1,800 up to $2,200 class. There are 45 vehicles in this class. The class midpoint is $2,000. So we say that the typical profit in the class with the highest frequency is $2,000.

By presenting this information to Ms. Ball, we give her a clear picture of the distribution of the vehicle profits for last month.

We admit that arranging the information on profits into a frequency distribution does result in the loss of some detailed information. That is, by organizing the data into a frequency distribution, we cannot pinpoint the exact profit on any vehicle, such as $1,387, $2,148, or $2,201. Further, we cannot tell that the actual minimum profit for any vehicle sold is $294 or that the maximum profit was $3,292. However, the lower limit of the first class and the upper limit of the last class convey essentially the same meaning. Likely, Ms. Ball will make the same judgment if she knows the smallest profit is about $200 that she will if she knows

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the exact profit is $292. The advantages of summarizing the 180 profits into a more understandable and organized form more than offset this disadvantage.

In 1788, James Madison, John Jay, and Alexander Hamilton anonymously published a series of essays entitled The Federalist. These Federalist papers were an attempt to convince the people of New York that they should ratify the Constitution. In the course of history, the authorship of most of these papers became known, but 12 remained contested. Through the use of statistical analysis, and particularly the study of the frequency of the use of various words, we can now conclude that James Madison is the likely author of the 12 papers. In fact, the statistical evidence that Madison is the author is overwhelming.

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. When we summarize raw data with frequency distributions, equal class intervals are preferred. However, in certain situations unequal class intervals may be necessary to avoid a large number of classes with very small frequencies. Such is the case in Table 2–6. The U.S. Internal Revenue Service uses unequal-sized class intervals for adjusted gross income on individual tax returns to summarize the number of

individual tax returns. If we use our method to find equal class intervals, the 2k rule results in 25 classes, and a class interval of $400,000, assuming $0 and $10,000,000 as the minimum and maximum values for adjusted gross income. Using equal class intervals, the first 13 classes in Table 2–6 would be combined into one class of about 99.9% of all tax returns and 24 classes for the 0.1% of the returns with an adjusted gross income above $400,000. Using equal class intervals does not provide a good understanding of the raw data. In this case, good judgment in the use of unequal class intervals, as demonstrated in Table 2–6, is required to show the distribution of the number of tax returns filed, especially for incomes under $500,000.

TABLE 2–6 Adjusted Gross Income for Individuals Filing Income Tax Returns

2–2

The commissions earned for the first quarter of last year by the 11 members of the sales staff at Master Chemical Company are:

$1,650 $1,475 $1,510 $1,670 $1,595 $1,760 $1,540 $1,495 $1,590 $1,625 $1,510

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(a) What are the values such as $1,650 and $1,475 called?

(b) Using $1,400 up to $1,500 as the first class, $1,500 up to $1,600 as the second class, and so forth, organize the quarterly commissions into a frequency distribution.

(c) What are the numbers in the right column of your frequency distribution called?

(d) Describe the distribution of quarterly commissions, based on the frequency distribution. What is the largest concentration of commissions earned? What is the smallest, and the largest? What is the typical amount earned?

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. Relative Frequency Distribution

It may be desirable, as we did earlier with qualitative data, to convert class frequencies to relative class frequencies to show the proportion of the total number of observations in each class. In our vehicle profits, we may want to know what percentage of the vehicle profits are in the $1,000 up to $1,400 class. In another study, we may want to know what percentage of the employees used 5 up to 10 personal leave days last year. To convert a frequency distribution to a relative frequency distribution, each of the class frequencies is divided by the total number of observations. From the distribution of vehicle profits, Table 2–5, the relative frequency for the $1,000 up to $1,400 class is 0.128, found by dividing 23 by 180. That is, profit on 12.8% of the vehicles sold is between $1,000 and $1,400. The relative frequencies for the remaining classes are shown in Table 2–7.

TABLE 2–7 Relative Frequency Distribution of Profit for Vehicles Sold Last Month at Applewood Auto Group

As we mentioned in Chapter 1, there are many software packages that perform statistical calculations. Throughout this text, we will show the output from Microsoft Excel, MegaStat (a Microsoft Excel add-in), and Minitab (a statistical software package).

On page 22 of this chapter, we used the Pivot Table tool in Excel to create a frequency table. We use the same Excel tool to create frequency and relative frequency distributions for the profit variable in the Applewood Auto Group data. The necessary steps for the table to the left are given in the Software Commands section in Appendix C.

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2–3

Barry Bonds of the San Francisco Giants established a new single-season Major League Baseball home run record by hitting 73 home runs during the 2001 season. Listed below is the sorted distance of each of the 73 home runs.

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. (a) For this data, show that seven classes would be used to create a frequency distribution using

the 2k rule.

(b) Show that a class interval of 30 would summarize the data in seven classes.

(c) Construct frequency and relative frequency distributions for the data with seven classes and a class interval of 30. Start the first class with a lower limit of 300.

(d) How many home runs traveled a distance of 360 up to 390 feet?

(e) What percentage of the home runs traveled a distance of 360 up to 390 feet?

(f) What percentage of the home runs traveled a distance of 390 feet or more?

E X E R C I S E S

This icon indicates that the data are available at the text website: www.mhhe.com/lind16e. You will be able to download the data directly into Excel or Minitab from this site.

7. A set of data consists of 38 observations. How many classes would you recommend for the frequency distribution?

8. A set of data consists of 45 observations between $0 and $29. What size would you recommend for the class interval?

9. A set of data consists of 230 observations between $235 and $567. What class interval would you recommend?

10. A set of data contains 53 observations. The minimum value is 42 and the maximum value is 129. The data are to be organized into a frequency distribution.

a. How many classes would you suggest?

b. What would you suggest as the lower limit of the first class?

For DATA FILE, please visit www.mhhe.com/lind16e

11. Wachesaw Manufacturing Inc. produced the following number of units in the last 16 days.

The information is to be organized into a frequency distribution.

a. How many classes would you recommend?

b. What class interval would you suggest?

c. What lower limit would you recommend for the first class?

d. Organize the information into a frequency distribution and determine the relative frequency distribution.

e. Comment on the shape of the distribution.

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For DATA FILE, please visit www.mhhe.com/lind16e

12. The Quick Change Oil Company has a number of outlets in the metropolitan Seattle area. The daily number of oil changes at the Oak Street outlet in the past 20 days are:

The data are to be organized into a frequency distribution.

a. How many classes would you recommend?

b. What class interval would you suggest?

c. What lower limit would you recommend for the first class?

d. Organize the number of oil changes into a frequency distribution.

e. Comment on the shape of the frequency distribution. Also determine the relative frequency distribution.

For DATA FILE, please visit www.mhhe.com/lind16e

13. The manager of the BiLo Supermarket in Mt. Pleasant, Rhode Island, gathered the following information on the number of times a customer visits the store during a month. The responses of 51 customers were:

a. Starting with 0 as the lower limit of the first class and using a class interval of 3, organize the data into a frequency distribution.

b. Describe the distribution. Where do the data tend to cluster?

c. Convert the distribution to a relative frequency distribution.

For DATA FILE, please visit www.mhhe.com/lind16e

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. 14. The food services division of Cedar River Amusement Park Inc. is studying the amount that families who visit the amusement park spend per day on food and drink. A sample of 40 families who visited the park yesterday revealed they spent the following amounts:

a. Organize the data into a frequency distribution, using seven classes and 15 as the lower limit of the first class. What class interval did you select?

b. Where do the data tend to cluster?

c. Describe the distribution.

d. Determine the relative frequency distribution.

LO2-4

Display a frequency distribution using a histogram or frequency polygon.

GRAPHIC PRESENTATION OF A FREQUENCY DISTRIBUTION Sales managers, stock analysts, hospital administrators, and other busy executives often need a quick picture of the distributions of sales, stock prices, or hospital costs. These distributions can often be depicted by the use of charts and graphs. Three charts that will help portray a frequency distribution graphically are the histogram, the frequency polygon, and the cumulative frequency polygon.

Histogram A histogram for a frequency distribution based on quantitative data is similar to the bar chart showing the distribution of qualitative data. The classes are marked on the horizontal axis and the class frequencies on the vertical axis. The class frequencies are represented by the heights of the bars. However, there is one important difference based on the nature of the data. Quantitative data are usually measured using scales that are continuous, not discrete. Therefore, the horizontal axis represents all possible values, and the bars are drawn adjacent to each other to show the continuous nature of the data.

HISTOGRAM A graph in which the classes are marked on the horizontal axis and the class frequencies on the vertical axis. The class frequencies are represented by the heights of the bars, and the bars are drawn adjacent to each other.

E X A M P L E

Below is the frequency distribution of the profits on vehicle sales last month at the Applewood Auto Group.

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. Construct a histogram. What observations can you reach based on the information presented in the histogram?

S O L U T I O N

The class frequencies are scaled along the vertical axis (Y-axis) and either the class limits or the class midpoints along the horizontal axis. To illustrate the construction of the histogram, the first three classes are shown in Chart 2–3.

Chart 2–3 Construction of a Histogram

From Chart 2–3 we note the profit on eight vehicles was $200 up to $600. Therefore, the height of the column for that class is 8. There are 11 vehicle sales where the profit was $600 up to $1,000. So, logically, the height of that column is 11. The height of the bar represents the number of observations in the class.

This procedure is continued for all classes. The complete histogram is shown in Chart 2–4. Note that there is no space between the bars. This is a feature of the histogram. Why is this so? Because the variable plotted on the horizontal axis is a continuous variable. In a bar chart, the scale of measurement is often nominal and the vertical bars are separated. This is an important distinction between the histogram and the bar chart.

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Chart 2–4 Histogram of the Profit on 180 Vehicles Sold at the Applewood Auto Group

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. We can make the following statements using Chart 2–4. They are the same as the observations based on Table 2–5 on page 28.

1. The profits from vehicle sales range between $200 and $3,400.

2. The vehicle profits are classified using a class interval of $400. The class interval is determined by subtracting consecutive lower or upper class limits. For example, the lower limit of the first class is $200, and the lower limit of the second class is $600. The difference is the class interval or $400.

3. The profits are concentrated between $1,000 and $3,000. The profit on 157 vehicles, or 87%, was within this range.

4. For each class, we can determine the typical profit or class midpoint. It is halfway between the lower or upper limits of two consecutive classes. It is computed by adding the lower or upper limits of consecutive classes and dividing by 2. Referring to Chart 2–4, the lower class limit of the first class is $200, and the next class limit is $600. The class midpoint is $400, found by ($600 1 $200)y2. The midpoint best represents, or is typical of, the profits of the vehicles in that class. Applewood sold 8 vehicles with a typical profit of $400.

5. The largest concentration, or highest frequency of vehicles sold, is in the $1,800 up to $2,200 class. There are 45 vehicles in this class. The class midpoint is $2,000. So we say that the typical profit in the class with the highest frequency is $2,000.

Thus, the histogram provides an easily interpreted visual representation of a frequency distribution. We should also point out that we would have made the same observations and the shape of the histogram would have been the same had we used a relative frequency distribution instead of the actual frequencies. That is, if we had used the relative frequencies of Table 2–7, we would have had a histogram of the same shape as Chart 2–4. The only difference is that the vertical axis would have been reported in percentage of vehicles instead of the number of vehicles. The Excel commands to create Chart 2–4 are given in Appendix C.

Florence Nightingale is known as the founder of the nursing profession. However, she also saved many lives by using statistical analysis. When she encountered an unsanitary condition or an undersup-plied hospital, she improved the conditions and then used statistical data to document the improvement. Thus, she was able to convince others of the need for medical reform, particularly in the area of sanitation. She developed original graphs to demonstrate that, during the Crimean War, more soldiers died from unsanitary conditions than were killed in combat.

Frequency Polygon A frequency polygon also shows the shape of a distribution and is similar to a histogram. It consists of line segments connecting the points formed by the intersections of the class midpoints and the class frequencies. The construction of a frequency polygon is illustrated in Chart 2–5 on the next page. We use the profits from the cars sold last month at the Applewood Auto Group. The midpoint of each class is scaled on the X-axis and the class frequencies on the Y-axis. Recall that the class midpoint is the value at the center of a class and

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represents the typical values in that class. The class frequency is the number of observations in a particular class. The profit earned on the vehicles sold last month by the Applewood Auto Group is repeated below.

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Chart 2–5 Frequency Polygon of Profit on 180 Vehicles Sold at Applewood Auto Group

As noted previously, the $200 up to $600 class is represented by the midpoint $400. To construct a frequency polygon, move horizontally on the graph to the midpoint, $400, and then vertically to 8, the class frequency, and place a dot. The x and the y values of this point are called the coordinates. The coordinates of the next point are x = 800 and y = 11. The process is continued for all classes. Then the points are connected in order. That is, the point representing the lowest class is joined to the one representing the second class and so on. Note in Chart 2–5 that, to complete the frequency polygon, midpoints of $0 and $3,600 are added to the X-axis to “anchor” the polygon at zero frequencies. These two values, $0 and $3,600, were derived by subtracting the class interval of $400 from the lowest midpoint ($400) and by adding $400 to the highest midpoint ($3,200) in the frequency distribution.

Both the histogram and the frequency polygon allow us to get a quick picture of the main characteristics of the data (highs, lows, points of concentration, etc.). Although the two representations are similar in purpose, the histogram has the advantage of depicting each class as a rectangle, with the height of the rectangular bar representing the number in each class. The frequency polygon, in turn, has an advantage over the histogram. It allows us to compare directly two or more frequency distributions. Suppose Ms. Ball wants to compare the profit per vehicle sold at Applewood Auto Group with a similar auto group, Fowler Auto in Grayling, Michigan. To do this, two frequency polygons are constructed, one on top of the other, as in Chart 2–6. Two things are clear from the chart:

The typical vehicle profit is larger at Fowler Motors—about $2,000 for Applewood and about $2,400 for Fowler.

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Chart 2–6 Distribution of Profit at Applewood Auto Group and Fowler Motors

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PRINTED BY: [email protected]email.phoenix.edu. Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. There is less dispersion in the profits at Fowler Motors than at Applewood. The lower limit

of the first class for Applewood is $0 and the upper limit is $3,600. For Fowler Motors, the lower limit is $800 and the upper limit is the same: $3,600.

The total number of cars sold at the two dealerships is about the same, so a direct comparison is possible. If the difference in the total number of cars sold is large, then converting the frequencies to relative frequencies and then plotting the two distributions would allow a clearer comparison.

2–4

The annual imports of a selected group of electronic suppliers are shown in the following frequency distribution.

(a) Portray the imports as a histogram.

(b) Portray the imports as a relative frequency polygon.

(c) Summarize the important facets of the distribution (such as classes with the highest and lowest frequencies).

E X E R C I S E S

15. Molly’s Candle Shop has several retail stores in the coastal areas of North and South Carolina. Many of Molly’s customers ask her to ship their purchases. The following chart shows the number of packages shipped per day for the last 100 days.

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a. What is this chart called?

b. What is the total number of frequencies?

c. What is the class interval?

d. What is the class frequency of the 10 up to 15 class?

e. What is the relative frequency of the 10 up to 15 class?

f. What is the midpoint of the 10 up to 15 class?

g. On how many days were there 25 or more packages shipped?

16. The following chart shows the number of patients admitted daily to Memorial Hospital through the emergency room.

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. a. What is the midpoint of the 2 up to 4 class?

b. How many days were 2 up to 4 patients admitted?

c. What is the class interval?

d. What is this chart called?

17. The following frequency distribution reports the number of frequent flier miles, reported in thousands, for employees of Brumley Statistical Consulting Inc. during the most recent quarter.

a. How many employees were studied?

b. What is the midpoint of the first class?

c. Construct a histogram.

d. A frequency polygon is to be drawn. What are the coordinates of the plot for the first class?

e. Construct a frequency polygon.

f. Interpret the frequent flier miles accumulated using the two charts.

18. Ecommerce.com, a large Internet retailer, is studying the lead time (elapsed time between when an order is placed and when it is filled) for a sample of recent orders. The lead times are reported in days.

a. How many orders were studied?

b. What is the midpoint of the first class?

c. What are the coordinates of the first class for a frequency polygon?

d. Draw a histogram.

e. Draw a frequency polygon.

f. Interpret the lead times using the two charts.

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Cumulative Frequency Distributions Consider once again the distribution of the profits on vehicles sold by the Apple-wood Auto Group. Suppose we were interested in the number of vehicles that sold for a profit of less than $1,400 or the profit earned on the lowest-selling 40% of the vehicles. These values can be approximated by developing a cumulative frequency distribution and portraying it graphically in a cumulative frequency polygon.

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E X A M P L E

The frequency distribution of the profits earned at Applewood Auto Group is repeated from Table 2–5.

Construct a cumulative frequency polygon to answer the following two questions. Seventy-five percent of the vehicles sold earned a profit of less than what amount? Sixty of the vehicles earned a profit of less than what amount?

S O L U T I O N

As the names imply, a cumulative frequency distribution and a cumulative frequency polygon require cumulative frequencies. To construct a cumulative frequency distribution, refer to the preceding table and note that there were eight vehicles in which the profit earned was less than $600. Those 8 vehicles, plus the 11 in the next higher class, for a total of 19, earned a profit of less than $1,000. The cumulative frequency for the next higher class is 42, found by 8 1 11 1 23. This process is continued for all the classes. All the vehicles earned a profit of less than $3,400. (See Table 2–8.)

TABLE 2–8 Cumulative Frequency Distribution for Profit on Vehicles Sold Last Month at Applewood Auto Group

To plot a cumulative frequency distribution, scale the upper limit of each class along the X-axis and the corresponding cumulative frequencies along the Y-axis. To provide additional information, you can label

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the vertical axis on the left in units and the vertical axis on the right in percent. In the Applewood Auto Group,

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. the vertical axis on the left is labeled from 0 to 180 and on the right from 0 to 100%. The value of 50% corresponds to 90 vehicles.

To begin, the first plot is at x = 200 and y = 0. None of the vehicles sold for a profit of less than $200. The profit on 8 vehicles was less than $600, so the next plot is at x = 600 and y = 8. Continuing, the next plot is x = 1,000 and y = 19. There were 19 vehicles that sold for a profit of less than $1,000. The rest of the points are plotted and then the dots connected to form Chart 2–7.

Chart 2–7 Cumulative Frequency Polygon for Profit on Vehicles Sold Last Month at Applewood Auto Group

Using Chart 2–7 to find the amount of profit on 75% of the cars sold, draw a horizontal line from the 75% mark on the right-hand vertical axis over to the polygon, then drop down to the X-axis and read the amount of profit. The value on the X-axis is about $2,300, so we estimate that 75% of the vehicles sold earned a profit of $2,300 or less for the Applewood group.

To find the highest profit earned on 60 of the 180 vehicles, we use Chart 2–7 to locate the value of 60 on the left-hand vertical axis. Next, we draw a horizontal line from the value of 60 to the polygon and then drop down to the X-axis and read the profit. It is about $1,600, so we estimate that 60 of the vehicles sold for a profit of less than $1,600. We can also make estimates of the percentage of vehicles that sold for less than a particular amount. To explain, suppose we want to estimate the percentage of vehicles that sold for a profit of less than $2,000. We begin by locating the value of $2,000 on the X-axis, move vertically to the polygon, and then horizontally to the vertical axis on the right. The value is about 56%, so we conclude 56% of the

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vehicles sold for a profit of less than $2,000.

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2–5

A sample of the hourly wages of 15 employees at Home Depot in Brunswick, Georgia, was organized into the following table.

(a) What is the table called?

(b) Develop a cumulative frequency distribution and portray the distribution in a cumulative frequency polygon.

(c) On the basis of the cumulative frequency polygon, how many employees earn less than $11 per hour?

E X E R C I S E S

19. The following cumulative frequency polygon shows the hourly wages of a sample of certified welders in the Atlanta, Georgia, area.

a. How many welders were studied?

b. What is the class interval?

c. About how many welders earn less than $10.00 per hour?

d. About 75% of the welders make less than what amount?

e. Ten of the welders studied made less than what amount?

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f. What percent of the welders make less than $20.00 per hour?

20. The following cumulative frequency polygon shows the selling price ($000) of houses sold in the Billings, Montana, area.

a. How many homes were studied?

b. What is the class interval?

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. c. One hundred homes sold for less than what amount?

d. About 75% of the homes sold for less than what amount?

e. Estimate the number of homes in the $150,000 up to $200,000 class.

f. About how many homes sold for less than $225,000?

21. The frequency distribution representing the number of frequent flier miles accumulated by employees at Brumley Statistical Consulting Inc. is repeated from Exercise 17.

a. How many employees accumulated less than 3,000 miles?

b. Convert the frequency distribution to a cumulative frequency distribution.

c. Portray the cumulative distribution in the form of a cumulative frequency polygon.

d. Based on the cumulative frequency polygon, about 75% of the employees accumulated how many miles or less?

22. The frequency distribution of order lead time at Ecommerce.com from Exercise 18 is repeated below.

a. How many orders were filled in less than 10 days? In less than 15 days?

b. Convert the frequency distribution to a cumulative frequency distribution.

c. Develop a cumulative frequency polygon.

d. About 60% of the orders were filled in less than how many days?

C H A P T E R S U M M A R Y

I. A frequency table is a grouping of qualitative data into mutually exclusive and collectively exhaustive classes showing the number of observations in each class.

II. A relative frequency table shows the fraction of the number of frequencies in each class.

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III. A bar chart is a graphic representation of a frequency table.

IV. A pie chart shows the proportion each distinct class represents of the total number of observations.

V. A frequency distribution is a grouping of data into mutually exclusive and collectively exhaustive classes showing the number of observations in each class.

A. The steps in constructing a frequency distribution are

1. Decide on the number of classes.

2. Determine the class interval.

3. Set the individual class limits.

4. Tally the raw data into classes.

5. Count the number of tallies in each class.

B. The class frequency is the number of observations in each class.

C. The class interval is the difference between the limits of two consecutive classes.

D. The class midpoint is halfway between the limits of consecutive classes.

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. VI. A relative frequency distribution shows the percent of observations in each class.

VII. There are three methods for graphically portraying a frequency distribution.

A. A histogram portrays the number of frequencies in each class in the form of a rectangle.

B. A frequency polygon consists of line segments connecting the points formed by the intersection of the class midpoint and the class frequency.

C. A cumulative frequency distribution shows the number or percent of observations less than a given value.

C H A P T E R E X E R C I S E S

23. Describe the similarities and differences of qualitative and quantitative variables. Be sure to include the following:

a. What level of measurement is required for each variable type?

b. Can both types be used to describe both samples and populations?

24. Describe the similarities and differences between a frequency table and a frequency distribution. Be sure to include which requires qualitative data and which requires quantitative data.

25. Alexandra Damonte will be building a new resort in Myrtle Beach, South Carolina. She must decide how to design the resort based on the type of activities that the resort will offer to its customers. A recent poll of 300 potential customers showed the following results about customers’ preferences for planned resort activities:

a. What is the table called?

b. Draw a bar chart to portray the survey results.

c. Draw a pie chart for the survey results.

d. If you are preparing to present the results to Ms. Damonte as part of a report, which graph would you prefer to show? Why?

For DATA FILE, please visit www.mhhe.com/lind16e

26. Speedy Swift is a package delivery service that serves the greater Atlanta, Georgia, metropolitan area. To maintain customer loyalty, one of Speedy Swift’s performance objectives is on-time delivery. To monitor its performance, each delivery is measured on the following scale: early (package delivered before the promised time), on-time (package delivered within 5 minutes of the promised time), late (package delivered more than 5 minutes past the promised time), lost (package never delivered). Speedy Swift’s objective is to deliver 99% of all packages either early or on-time. Another objective is to never lose a

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package. Speedy collected the following data for last month’s performance:

a. What scale is used to measure delivery performance? What kind of variable is delivery performance?

b. Construct a frequency table for delivery performance for last month.

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. c. Construct a relative frequency table for delivery performance last month.

d. Construct a bar chart of the frequency table for delivery performance for last month.

e. Construct a pie chart of on-time delivery performance for last month.

f. Analyze the data summaries and write an evaluation of last month’s delivery performance as it relates to Speedy Swift’s performance objectives. Write a general recommendation for further analysis.

27. A data set consists of 83 observations. How many classes would you recommend for a frequency distribution?

28. A data set consists of 145 observations that range from 56 to 490. What size class interval would you recommend?

For DATA FILE, please visit www.mhhe.com/lind16e

29. The following is the number of minutes to commute from home to work for a group of 25 automobile executives.

a. How many classes would you recommend?

b. What class interval would you suggest?

c. What would you recommend as the lower limit of the first class?

d. Organize the data into a frequency distribution.

e. Comment on the shape of the frequency distribution.

For DATA FILE, please visit www.mhhe.com/lind16e

30. The following data give the weekly amounts spent on groceries for a sample of 45 households.

a. How many classes would you recommend?

b. What class interval would you suggest?

c. What would you recommend as the lower limit of the first class?

d. Organize the data into a frequency distribution.

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For DATA FILE, please visit www.mhhe.com/lind16e

31. A social scientist is studying the use of iPods by college students. A sample of 45 students revealed they played the following number of songs yesterday.

Organize the above information into a frequency distribution.

a. How many classes would you suggest?

b. What is the most suitable class interval?

c. What is the lower limit of the initial class?

d. Create the frequency distribution.

e. Describe the profile of the distribution.

For DATA FILE, please visit www.mhhe.com/lind16e

32. David Wise handles his own investment portfolio, and has done so for many years. Listed below is the holding time (recorded to the nearest whole year) between purchase and sale for his collection of 36 stocks.

a. How many classes would you propose?

b. What class interval would you suggest?

c. What quantity would you use for the lower limit of the initial class?

d. Using your responses to parts (a), (b), and (c), create a frequency distribution.

e. Identify the appearance of the frequency distribution.

For DATA FILE, please visit www.mhhe.com/lind16e

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Page 18

MERRILL LYNCH recently completed a study of online investment portfolios for a sample of clients. For the 70 participants in the study, organize these data into a frequency distribution. (See Exercise 43 and LO2-3.)

LEARNING OBJECTIVES When you have completed this chapter, you will be able to:

LO2-1 Summarize qualitative variables with frequency and relative frequency tables.

LO2-2 Display a frequency table using a bar or pie chart.

LO2-3 Summarize quantitative variables with frequency and relative frequency distributions.

LO2-4 Display a frequency distribution using a histogram or frequency polygon.

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. INTRODUCTION

The highly competitive automobile retailing industry in the United States has changed dramatically in recent years. These changes spurred events such as the

bankruptcies of General Motors and Chrysler in 2009.

elimination of well-known brands like Pontiac and Saturn.

closing of over 1,500 local dealerships.

collapse of consumer credit availability.

consolidation of dealership groups.

Traditionally, a local family owned and operated the community dealership, which might have included one or two manufacturers or brands, like Pontiac and GMC Trucks or Chrysler and the popular Jeep line. Recently, however, skillfully managed and well-financed companies have been acquiring local dealerships throughout the country. As these groups acquire the local dealerships, they often bring standardized selling practices, common software and hardware technology platforms, and management reporting techniques. The goal of these new organizations is to provide an improved buying experience for the consumer, while increasing profitability. Megadealerships often employ over 10,000 people, generate several billion dollars in annual sales, own more than 50 franchises, and are traded on the New York Stock Exchange or NASDAQ. Today, the largest megadealership is AutoNation (ticker symbol AN). Others include Penske Auto Group (PAG and second largest), Asbury Automotive Group (ABG), and Hendrick Auto Group (which is privately held).

The Applewood Auto Group is an ownership group that includes four dealerships. The group sells a wide range of vehicles, including the inexpensive but popular Korean brands Kia and Hyundai, BMW and Volvo sedans and luxury SUVs, and a full line of Ford and Chevrolet cars and trucks.

Ms. Kathryn Ball is a member of the senior management team at Applewood Auto Group, which has its corporate offices adjacent to Kane Motors. She is responsible for tracking and analyzing vehicle sales and the profitability of those vehicles. Kathryn would like to summarize the profit earned on the vehicles sold with tables, charts, and graphs that she would review monthly. She wants to know the profit per vehicle sold, as well as the lowest and highest amount of profit. She is also interested in describing the demographics of the buyers. What are their ages? How many vehicles have they previously purchased from one of the Applewood dealerships? What type of vehicle did they purchase?

The Applewood Auto Group operates four dealerships:

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Tionesta Ford Lincoln Mercury sells the Ford, Lincoln, and Mercury cars and trucks.

Olean Automotive Inc. has the Nissan franchise as well as the General Motors brands of Chevrolet, Cadillac, and GMC Trucks.

Sheffield Motors Inc. sells Buick, GMC trucks, Hyundai, and Kia.

Kane Motors offers the Chrysler, Dodge, and Jeep line as well as BMW and Volvo.

Every month, Ms. Ball collects data from each of the four dealerships and enters them into an Excel spreadsheet. Last month the Applewood Auto Group sold 180 vehicles at the four dealerships. A copy of the first few observations appears to the left. The variables collected include:

Age—the age of the buyer at the time of the purchase.

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Profit—the amount earned by the dealership on the sale of each vehicle.

Location—the dealership where the vehicle was purchased.

Vehicle type—SUV, sedan, compact, hybrid, or truck.

Previous—the number of vehicles previously purchased at any of the four Apple-wood dealerships by the consumer.

The entire data set is available at the McGraw-Hill website (www.mhhe.com/lind16e) and in Appendix A.4 at the end of the text.

LO2-1

Summarize qualitative variables with frequency and relative frequency tables.

CONSTRUCTING FREQUENCY TABLES Recall from Chapter 1 that techniques used to describe a set of data are called descriptive statistics. Descriptive statistics organize data to show the general pattern of the data, to identify where values tend to concentrate, and to expose extreme or unusual data values. The first technique we discuss is a frequency table.

FREQUENCY TABLE A grouping of qualitative data into mutually exclusive and collectively exhaustive classes showing the number of observations in each class.

In Chapter 1, we distinguished between qualitative and quantitative variables. To review, a qualitative variable is nonnumeric, that is, it can only be classified into distinct categories. Examples of qualitative data include political affiliation (Republican, Democrat, Independent), state of birth (Alabama, . . . , Wyoming), and method of payment for a purchase at Barnes & Noble (cash, check, debit, or credit). On the other hand, quantitative variables are numerical in nature. Examples of quantitative data relating to college students include the price of their textbooks, their age, and the number of credit hours they are registered for this semester.

In the Applewood Auto Group data set, there are five variables for each vehicle sale: age of the buyer, amount of profit, dealer that made the sale, type of vehicle sold, and number of previous purchases by the buyer. The dealer and the type of vehicle are qualitative variables. The amount of profit, the age of the buyer, and the number of previous purchases are quantitative variables.

Suppose Ms. Ball wanted to summarize last month’s sales by location. To summarize this qualitative data, we classify the vehicles sold last month according to their location: Tionesta, Olean, Sheffield, or Kane. We use location to develop a frequency table with four mutually exclusive (distinctive) classes. This means that a particular vehicle cannot belong to more than one class. Each vehicle is uniquely classified into one of the four mutually exclusive locations. In addition, the frequency table must be collectively exhaustive. This means that every vehicle is accounted for in the table. This frequency table is shown in Table 2–1. The number of observations, representing the sales at each location, is called the class frequency. So the class frequency for vehicles sold at the Kane location is 52.

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TABLE 2–1 Frequency Table for Vehicles Sold Last Month at Applewood Auto Group by Location

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CHART 2–1 Number of Vehicles Sold by Location

We explain the details of constructing a pie chart using the information in Table 2–3, which shows the frequency and percent of cars sold by the Applewood Auto Group for each vehicle type.

TABLE 2–3 Vehicle Sales by Type at Applewood Auto Group

The first step to develop a pie chart is to mark the percentages 0, 5, 10, 15, and so on evenly around the circumference of a circle (see Chart 2–2). To plot the 40% of total sales represented by sedans, draw a line from the center of the circle to 0 and another line from the center of the circle to 40%. The area in this “slice” represents the number of sedans sold as a percentage of the total sales. Next, add the SUV’s percentage of total sales, 30%, to the sedan’s percentage of total sales, 40%. The result is 70%. Draw a line from the center of the circle to 70%, so the area between 40 and 70 shows the sales of SUVs as a percentage of total sales. Continuing, add the 15% of total sales for compact vehicles, which gives us a total of 85%. Draw a line from the center of the circle to 85, so the “slice” between 70% and 85% represents the number of compact vehicles sold as a percentage of the total sales. The remaining 10% for truck sales and 5% for hybrid sales are added to the chart using the same method.

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Chart 2–2 Pie Chart of Vehicles by Type

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. Because each slice of the pie represents the relative frequency of each vehicle type as a percentage of the total sales, we can easily compare them:

The largest percentage of sales is for sedans.

Sedans and SUVs together account for 70% of vehicle sales.

Hybrids account for 5% of vehicle sales, in spite of being on the market for only a few years.

We can use Excel software to quickly count the number of cars for each vehicle type and create the frequency table, bar chart, and pie chart below. The Excel tool is called a Pivot Table. The instructions to produce these descriptive statistics and charts are given in Appendix C.

Pie and bar charts both serve to illustrate frequency and relative frequency tables. When is a pie chart preferred to a bar chart? In most cases, pie charts are used to show and compare the relative differences in the percentage of observations for each value or class of a qualitative variable. Bar charts are preferred when the goal is to compare the number or frequency of observations for each value or class of a qualitative variable. The following Example/Solution shows another application of bar and pie charts.

E X A M P L E

SkiLodges.com is test marketing its new website and is interested in how easy its website design is to navigate. It randomly selected 200 regular Internet users and asked them to perform a search task on the website. Each person was asked to rate the relative ease of navigation as poor, good, excellent, or awesome. The results are shown in the following table:

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. 1. What type of measurement scale is used for ease of navigation?

2. Draw a bar chart for the survey results.

3. Draw a pie chart for the survey results.

S O L U T I O N

The data are measured on an ordinal scale. That is, the scale is ranked in relative ease when moving from “poor” to “awesome.” Also, the interval between each rating is unknown so it is impossible, for example, to conclude that a rating of good is twice the value of a poor rating.

We can use a bar chart to graph the data. The vertical scale shows the relative frequency and the horizontal scale shows the values of the ease-of-navigation variable.

A pie chart can also be used to graph these data. The pie chart emphasizes that more than half of the respondents rate the relative ease of using the website awesome.

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2–1

The answers are in Appendix E.

DeCenzo Specialty Food and Beverage Company has been serving a cola drink with an additional flavoring, Cola-Plus, that is very popular among its customers. The company is interested in customer preferences for Cola-Plus versus Coca-Cola, Pepsi, and a lemon-lime beverage. They ask 100 randomly sampled customers to take a taste test and select the beverage they prefer most. The results are shown in the following table:

(a) Is the data qualitative or quantitative? Why?

(b) What is the table called? What does it show?

(c) Develop a bar chart to depict the information.

(d) Develop a pie chart using the relative frequencies.

E X E R C I S E S

The answers to the odd-numbered exercises are at the end of the book in Appendix D.

1. A pie chart shows the relative market share of cola products. The “slice” for Pepsi-Cola has a central angle of 90 degrees. What is its market share?

2. In a marketing study, 100 consumers were asked to select the best digital music player from the iPod, the iRiver, and the Magic Star MP3. To summarize the consumer responses with a frequency table, how many classes would the frequency table have?

3. A total of 1,000 residents in Minnesota were asked which season they preferred. One hundred liked winter best, 300 liked spring, 400 liked summer, and 200 liked fall. Develop a frequency table and a relative frequency table to summarize this information.

4. Two thousand frequent business travelers are asked which midwestern city they prefer: Indianapolis, Saint Louis, Chicago, or Milwaukee. One hundred liked Indianapolis best, 450 liked Saint Louis, 1,300 liked Chicago, and the remainder preferred Milwaukee. Develop a frequency table and a relative frequency table to summarize this information.

5. Wellstone Inc. produces and markets replacement covers for cell phones in five different colors: bright white, metallic black, magnetic lime, tangerine orange, and fusion red. To estimate the demand for each

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color, the company set up a kiosk in the Mall of America for several hours and asked randomly selected people which cover color was their favorite. The results follow:

a. What is the table called?

b. Draw a bar chart for the table.

c. Draw a pie chart.

d. If Wellstone Inc. plans to produce 1 million cell phone covers, how many of each color should it produce?

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. 6. A small business consultant is investigating the performance of several companies. The fourth-quarter sales for last year (in thousands of dollars) for the selected companies were:

The consultant wants to include a chart in his report comparing the sales of the six companies. Use a bar chart to compare the fourth-quarter sales of these corporations and write a brief report summarizing the bar chart.

LO2-3

Summarize quantitative variables with frequency and relative frequency distributions.

CONSTRUCTING FREQUENCY DISTRIBUTIONS In Chapter 1 and earlier in this chapter, we distinguished between qualitative and quantitative data. In the previous section, using the Applewood Automotive Group data, we summarized two qualitative variables: the location of the sale and the type of vehicle sold. We created frequency and relative frequency tables and depicted the results in bar and pie charts.

The Applewood Auto Group data also includes several quantitative variables: the age of the buyer, the profit earned on the sale of the vehicle, and the number of previous purchases. Suppose Ms. Ball wants to summarize last month’s sales by profit earned. We can describe profit using a frequency distribution.

FREQUENCY DISTRIBUTION A grouping of quantitative data into mutually exclusive and collectively exhaustive classes showing the number of observations in each class.

How do we develop a frequency distribution? The following example shows the steps to construct a frequency distribution. Remember, our goal is to construct tables, charts, and graphs that will quickly reveal the concentration, extreme values, and shape of the data.

E X A M P L E

We return to the situation where Ms. Kathryn Ball of the Applewood Auto Group wants to summarize the quantitative variable profit with a frequency distribution and display the distribution with charts and graphs. With this information, Ms. Ball can easily answer the following questions: What is the typical profit on each sale? What is the largest or maximum profit on any sale? What is the smallest or minimum profit on any sale? Around what value do the profits tend to cluster?

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S O L U T I O N

To begin, we need the profits for each of the 180 vehicle sales, which are listed in Table 2–4. This information is called raw or ungrouped data because it is simply a listing of the individual, observed profits. It is possible to search the list and find the smallest or minimum profit ($294) and the largest or maximum profit ($3,292),

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. but that is about all. It is difficult to determine a typical profit or to visualize where the profits tend to cluster. The raw data are more easily interpreted if we summarize the data with a frequency distribution as follows.

TABLE 2–4 Profit on Vehicles Sold Last Month by the Applewood Auto Group

Step 1: Decide on the number of classes. A useful recipe to determine the number of classes (k) is the “2 to the k rule.” This guide suggests you select the smallest number (k) for the number of classes such

that 2k (in words, 2 raised to the power of k) is greater than the number of observations (n). In the Applewood Auto Group example, there were 180 vehicles sold. So n 5 180. If we try k 5 = 7, which

means we would use 7 classes, 27 = 128, which is less than 180. Hence, 7 is too few classes. If we

let k = 8, then 28 = 256, which is greater than 180. So the recommended number of classes is 8.

Step 2: Determine the class interval. Generally, the class interval is the same for all classes. The classes all taken together must cover at least the distance from the minimum value in the data up to the maximum value. Expressing these words in a formula:

where i is the class interval, and k is the number of classes. For the Applewood Auto Group, the minimum value is $294 and the maximum value is $3,292. If

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we need 8 classes, the interval should be:

In practice, this interval size is usually rounded up to some convenient number, such as a multiple of 10 or 100. The value of $400 is a reasonable choice.

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. Step 3: Set the individual class limits. State clear class limits so you can put each observation into only one category. This means you must avoid overlapping or unclear class limits. For example, classes such as “$1,300–$1,400” and “$1,400–$1,500” should not be used because it is not clear whether the value $1,400 is in the first or second class. In this text, we will generally use the format $1,300 up to $1,400 and $1,400 up to $1,500 and so on. With this format, it is clear that $1,399 goes into the first class and $1,400 in the second. Because we always round the class interval up to get a convenient class size, we cover a larger than necessary range. For example, using 8 classes with an interval of $400 in the Applewood Auto Group example results in a range of 8($400) = $3,200. The actual range is $2,998, found by ($3,292 − $294). Comparing that value to $3,200, we have an excess of $202. Because we need to cover only the range (Maximum – Minimum), it is natural to put approximately equal amounts of the excess in each of the two tails. Of course, we should also select convenient class limits. A guideline is to make the lower limit of the first class a multiple of the class interval. Sometimes this is not possible, but the lower limit should at least be rounded. So here are the classes we could use for these data.

Step 4: Tally the vehicle profit into the classes. To begin, the profit from the sale of the first vehicle in Table 2–4 is $1,387. It is tallied in the $1,000 up to $1,400 class. The second profit in the first row of Table 2–4 is $2,148. It is tallied in the $1,800 up to $2,200 class. The other profits are tallied in a similar manner. When all the profits are tallied, the table would appear as:

Step 5: Count the number of items in each class. The number of observations in each class is called the class frequency. In the $200 up to $600 class there are 8 observations, and in the $600 up to $1,000

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Page 28class there are 11 observations. Therefore, the class frequency in

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. the first class is 8 and the class frequency in the second class is 11. There are a total of 180 observations in the entire set of data. So the sum of all the frequencies should be equal to 180.

TABLE 2–5 Frequency Distribution of Profit for Vehicles Sold Last Month at Applewood Auto Group

Now that we have organized the data into a frequency distribution (see Table 2–5), we can summarize the profits of the vehicles for the Applewood Auto Group. Observe the following:

1. The profits from vehicle sales range between $200 and $3,400.

2. The vehicle profits are classified using a class interval of $400. The class interval is determined by subtracting consecutive lower or upper class limits. For example, the lower limit of the first class is $200, and the lower limit of the second class is $600. The difference is the class interval of $400.

3. The profits are concentrated between $1,000 and $3,000. The profit on 157 vehicles, or 87%, was within this range.

4. For each class, we can determine the typical profit or class midpoint. It is halfway between the lower or upper limits of two consecutive classes. It is computed by adding the lower or upper limits of consecutive classes and dividing by 2. Referring to Table 2–5, the lower class limit of the first class is $200, and the next class limit is $600. The class midpoint is $400, found by ($600 1 $200)y2. The midpoint best represents, or is typical of, the profits of the vehicles in that class. Applewood sold 8 vehicles with a typical profit of $400.

5. The largest concentration, or highest frequency, of vehicles sold is in the $1,800 up to $2,200 class. There are 45 vehicles in this class. The class midpoint is $2,000. So we say that the typical profit in the class with the highest frequency is $2,000.

By presenting this information to Ms. Ball, we give her a clear picture of the distribution of the vehicle profits for last month.

We admit that arranging the information on profits into a frequency distribution does result in the loss of some detailed information. That is, by organizing the data into a frequency distribution, we cannot pinpoint the exact profit on any vehicle, such as $1,387, $2,148, or $2,201. Further, we cannot tell that the actual minimum profit for any vehicle sold is $294 or that the maximum profit was $3,292. However, the lower limit of the first class and the upper limit of the last class convey essentially the same meaning. Likely, Ms. Ball will make the same judgment if she knows the smallest profit is about $200 that she will if she knows

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the exact profit is $292. The advantages of summarizing the 180 profits into a more understandable and organized form more than offset this disadvantage.

In 1788, James Madison, John Jay, and Alexander Hamilton anonymously published a series of essays entitled The Federalist. These Federalist papers were an attempt to convince the people of New York that they should ratify the Constitution. In the course of history, the authorship of most of these papers became known, but 12 remained contested. Through the use of statistical analysis, and particularly the study of the frequency of the use of various words, we can now conclude that James Madison is the likely author of the 12 papers. In fact, the statistical evidence that Madison is the author is overwhelming.

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. When we summarize raw data with frequency distributions, equal class intervals are preferred. However, in certain situations unequal class intervals may be necessary to avoid a large number of classes with very small frequencies. Such is the case in Table 2–6. The U.S. Internal Revenue Service uses unequal-sized class intervals for adjusted gross income on individual tax returns to summarize the number of

individual tax returns. If we use our method to find equal class intervals, the 2k rule results in 25 classes, and a class interval of $400,000, assuming $0 and $10,000,000 as the minimum and maximum values for adjusted gross income. Using equal class intervals, the first 13 classes in Table 2–6 would be combined into one class of about 99.9% of all tax returns and 24 classes for the 0.1% of the returns with an adjusted gross income above $400,000. Using equal class intervals does not provide a good understanding of the raw data. In this case, good judgment in the use of unequal class intervals, as demonstrated in Table 2–6, is required to show the distribution of the number of tax returns filed, especially for incomes under $500,000.

TABLE 2–6 Adjusted Gross Income for Individuals Filing Income Tax Returns

2–2

The commissions earned for the first quarter of last year by the 11 members of the sales staff at Master Chemical Company are:

$1,650 $1,475 $1,510 $1,670 $1,595 $1,760 $1,540 $1,495 $1,590 $1,625 $1,510

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(a) What are the values such as $1,650 and $1,475 called?

(b) Using $1,400 up to $1,500 as the first class, $1,500 up to $1,600 as the second class, and so forth, organize the quarterly commissions into a frequency distribution.

(c) What are the numbers in the right column of your frequency distribution called?

(d) Describe the distribution of quarterly commissions, based on the frequency distribution. What is the largest concentration of commissions earned? What is the smallest, and the largest? What is the typical amount earned?

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. Relative Frequency Distribution

It may be desirable, as we did earlier with qualitative data, to convert class frequencies to relative class frequencies to show the proportion of the total number of observations in each class. In our vehicle profits, we may want to know what percentage of the vehicle profits are in the $1,000 up to $1,400 class. In another study, we may want to know what percentage of the employees used 5 up to 10 personal leave days last year. To convert a frequency distribution to a relative frequency distribution, each of the class frequencies is divided by the total number of observations. From the distribution of vehicle profits, Table 2–5, the relative frequency for the $1,000 up to $1,400 class is 0.128, found by dividing 23 by 180. That is, profit on 12.8% of the vehicles sold is between $1,000 and $1,400. The relative frequencies for the remaining classes are shown in Table 2–7.

TABLE 2–7 Relative Frequency Distribution of Profit for Vehicles Sold Last Month at Applewood Auto Group

As we mentioned in Chapter 1, there are many software packages that perform statistical calculations. Throughout this text, we will show the output from Microsoft Excel, MegaStat (a Microsoft Excel add-in), and Minitab (a statistical software package).

On page 22 of this chapter, we used the Pivot Table tool in Excel to create a frequency table. We use the same Excel tool to create frequency and relative frequency distributions for the profit variable in the Applewood Auto Group data. The necessary steps for the table to the left are given in the Software Commands section in Appendix C.

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2–3

Barry Bonds of the San Francisco Giants established a new single-season Major League Baseball home run record by hitting 73 home runs during the 2001 season. Listed below is the sorted distance of each of the 73 home runs.

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. (a) For this data, show that seven classes would be used to create a frequency distribution using

the 2k rule.

(b) Show that a class interval of 30 would summarize the data in seven classes.

(c) Construct frequency and relative frequency distributions for the data with seven classes and a class interval of 30. Start the first class with a lower limit of 300.

(d) How many home runs traveled a distance of 360 up to 390 feet?

(e) What percentage of the home runs traveled a distance of 360 up to 390 feet?

(f) What percentage of the home runs traveled a distance of 390 feet or more?

E X E R C I S E S

This icon indicates that the data are available at the text website: www.mhhe.com/lind16e. You will be able to download the data directly into Excel or Minitab from this site.

7. A set of data consists of 38 observations. How many classes would you recommend for the frequency distribution?

8. A set of data consists of 45 observations between $0 and $29. What size would you recommend for the class interval?

9. A set of data consists of 230 observations between $235 and $567. What class interval would you recommend?

10. A set of data contains 53 observations. The minimum value is 42 and the maximum value is 129. The data are to be organized into a frequency distribution.

a. How many classes would you suggest?

b. What would you suggest as the lower limit of the first class?

For DATA FILE, please visit www.mhhe.com/lind16e

11. Wachesaw Manufacturing Inc. produced the following number of units in the last 16 days.

The information is to be organized into a frequency distribution.

a. How many classes would you recommend?

b. What class interval would you suggest?

c. What lower limit would you recommend for the first class?

d. Organize the information into a frequency distribution and determine the relative frequency distribution.

e. Comment on the shape of the distribution.

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For DATA FILE, please visit www.mhhe.com/lind16e

12. The Quick Change Oil Company has a number of outlets in the metropolitan Seattle area. The daily number of oil changes at the Oak Street outlet in the past 20 days are:

The data are to be organized into a frequency distribution.

a. How many classes would you recommend?

b. What class interval would you suggest?

c. What lower limit would you recommend for the first class?

d. Organize the number of oil changes into a frequency distribution.

e. Comment on the shape of the frequency distribution. Also determine the relative frequency distribution.

For DATA FILE, please visit www.mhhe.com/lind16e

13. The manager of the BiLo Supermarket in Mt. Pleasant, Rhode Island, gathered the following information on the number of times a customer visits the store during a month. The responses of 51 customers were:

a. Starting with 0 as the lower limit of the first class and using a class interval of 3, organize the data into a frequency distribution.

b. Describe the distribution. Where do the data tend to cluster?

c. Convert the distribution to a relative frequency distribution.

For DATA FILE, please visit www.mhhe.com/lind16e

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. 14. The food services division of Cedar River Amusement Park Inc. is studying the amount that families who visit the amusement park spend per day on food and drink. A sample of 40 families who visited the park yesterday revealed they spent the following amounts:

a. Organize the data into a frequency distribution, using seven classes and 15 as the lower limit of the first class. What class interval did you select?

b. Where do the data tend to cluster?

c. Describe the distribution.

d. Determine the relative frequency distribution.

LO2-4

Display a frequency distribution using a histogram or frequency polygon.

GRAPHIC PRESENTATION OF A FREQUENCY DISTRIBUTION Sales managers, stock analysts, hospital administrators, and other busy executives often need a quick picture of the distributions of sales, stock prices, or hospital costs. These distributions can often be depicted by the use of charts and graphs. Three charts that will help portray a frequency distribution graphically are the histogram, the frequency polygon, and the cumulative frequency polygon.

Histogram A histogram for a frequency distribution based on quantitative data is similar to the bar chart showing the distribution of qualitative data. The classes are marked on the horizontal axis and the class frequencies on the vertical axis. The class frequencies are represented by the heights of the bars. However, there is one important difference based on the nature of the data. Quantitative data are usually measured using scales that are continuous, not discrete. Therefore, the horizontal axis represents all possible values, and the bars are drawn adjacent to each other to show the continuous nature of the data.

HISTOGRAM A graph in which the classes are marked on the horizontal axis and the class frequencies on the vertical axis. The class frequencies are represented by the heights of the bars, and the bars are drawn adjacent to each other.

E X A M P L E

Below is the frequency distribution of the profits on vehicle sales last month at the Applewood Auto Group.

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. Construct a histogram. What observations can you reach based on the information presented in the histogram?

S O L U T I O N

The class frequencies are scaled along the vertical axis (Y-axis) and either the class limits or the class midpoints along the horizontal axis. To illustrate the construction of the histogram, the first three classes are shown in Chart 2–3.

Chart 2–3 Construction of a Histogram

From Chart 2–3 we note the profit on eight vehicles was $200 up to $600. Therefore, the height of the column for that class is 8. There are 11 vehicle sales where the profit was $600 up to $1,000. So, logically, the height of that column is 11. The height of the bar represents the number of observations in the class.

This procedure is continued for all classes. The complete histogram is shown in Chart 2–4. Note that there is no space between the bars. This is a feature of the histogram. Why is this so? Because the variable plotted on the horizontal axis is a continuous variable. In a bar chart, the scale of measurement is often nominal and the vertical bars are separated. This is an important distinction between the histogram and the bar chart.

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Chart 2–4 Histogram of the Profit on 180 Vehicles Sold at the Applewood Auto Group

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. We can make the following statements using Chart 2–4. They are the same as the observations based on Table 2–5 on page 28.

1. The profits from vehicle sales range between $200 and $3,400.

2. The vehicle profits are classified using a class interval of $400. The class interval is determined by subtracting consecutive lower or upper class limits. For example, the lower limit of the first class is $200, and the lower limit of the second class is $600. The difference is the class interval or $400.

4. For each class, we can determine the typical profit or class midpoint. It is halfway between the lower or upper limits of two consecutive classes. It is computed by adding the lower or upper limits of consecutive classes and dividing by 2. Referring to Chart 2–4, the lower class limit of the first class is $200, and the next class limit is $600. The class midpoint is $400, found by ($600 1 $200)y2. The midpoint best represents, or is typical of, the profits of the vehicles in that class. Applewood sold 8 vehicles with a typical profit of $400.

5. The largest concentration, or highest frequency of vehicles sold, is in the $1,800 up to $2,200 class. There are 45 vehicles in this class. The class midpoint is $2,000. So we say that the typical profit in the class with the highest frequency is $2,000.

Thus, the histogram provides an easily interpreted visual representation of a frequency distribution. We should also point out that we would have made the same observations and the shape of the histogram would have been the same had we used a relative frequency distribution instead of the actual frequencies. That is, if we had used the relative frequencies of Table 2–7, we would have had a histogram of the same shape as Chart 2–4. The only difference is that the vertical axis would have been reported in percentage of vehicles instead of the number of vehicles. The Excel commands to create Chart 2–4 are given in Appendix C.

Florence Nightingale is known as the founder of the nursing profession. However, she also saved many lives by using statistical analysis. When she encountered an unsanitary condition or an undersup-plied hospital, she improved the conditions and then used statistical data to document the improvement. Thus, she was able to convince others of the need for medical reform, particularly in the area of sanitation. She developed original graphs to demonstrate that, during the Crimean War, more soldiers died from unsanitary conditions than were killed in combat.

Frequency Polygon A frequency polygon also shows the shape of a distribution and is similar to a histogram. It consists of line segments connecting the points formed by the intersections of the class midpoints and the class frequencies. The construction of a frequency polygon is illustrated in Chart 2–5 on the next page. We use the profits from the cars sold last month at the Applewood Auto Group. The midpoint of each class is scaled on the X-axis and the class frequencies on the Y-axis. Recall that the class midpoint is the value at the center of a class and

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represents the typical values in that class. The class frequency is the number of observations in a particular class. The profit earned on the vehicles sold last month by the Applewood Auto Group is repeated below.

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Chart 2–5 Frequency Polygon of Profit on 180 Vehicles Sold at Applewood Auto Group

As noted previously, the $200 up to $600 class is represented by the midpoint $400. To construct a frequency polygon, move horizontally on the graph to the midpoint, $400, and then vertically to 8, the class frequency, and place a dot. The x and the y values of this point are called the coordinates. The coordinates of the next point are x = 800 and y = 11. The process is continued for all classes. Then the points are connected in order. That is, the point representing the lowest class is joined to the one representing the second class and so on. Note in Chart 2–5 that, to complete the frequency polygon, midpoints of $0 and $3,600 are added to the X-axis to “anchor” the polygon at zero frequencies. These two values, $0 and $3,600, were derived by subtracting the class interval of $400 from the lowest midpoint ($400) and by adding $400 to the highest midpoint ($3,200) in the frequency distribution.

Both the histogram and the frequency polygon allow us to get a quick picture of the main characteristics of the data (highs, lows, points of concentration, etc.). Although the two representations are similar in purpose, the histogram has the advantage of depicting each class as a rectangle, with the height of the rectangular bar representing the number in each class. The frequency polygon, in turn, has an advantage over the histogram. It allows us to compare directly two or more frequency distributions. Suppose Ms. Ball wants to compare the profit per vehicle sold at Applewood Auto Group with a similar auto group, Fowler Auto in Grayling, Michigan. To do this, two frequency polygons are constructed, one on top of the other, as in Chart 2–6. Two things are clear from the chart:

The typical vehicle profit is larger at Fowler Motors—about $2,000 for Applewood and about $2,400 for Fowler.

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Chart 2–6 Distribution of Profit at Applewood Auto Group and Fowler Motors

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. There is less dispersion in the profits at Fowler Motors than at Applewood. The lower limit

of the first class for Applewood is $0 and the upper limit is $3,600. For Fowler Motors, the lower limit is $800 and the upper limit is the same: $3,600.

The total number of cars sold at the two dealerships is about the same, so a direct comparison is possible. If the difference in the total number of cars sold is large, then converting the frequencies to relative frequencies and then plotting the two distributions would allow a clearer comparison.

2–4

The annual imports of a selected group of electronic suppliers are shown in the following frequency distribution.

(a) Portray the imports as a histogram.

(b) Portray the imports as a relative frequency polygon.

(c) Summarize the important facets of the distribution (such as classes with the highest and lowest frequencies).

E X E R C I S E S

15. Molly’s Candle Shop has several retail stores in the coastal areas of North and South Carolina. Many of Molly’s customers ask her to ship their purchases. The following chart shows the number of packages shipped per day for the last 100 days.

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a. What is this chart called?

b. What is the total number of frequencies?

c. What is the class interval?

d. What is the class frequency of the 10 up to 15 class?

e. What is the relative frequency of the 10 up to 15 class?

f. What is the midpoint of the 10 up to 15 class?

g. On how many days were there 25 or more packages shipped?

16. The following chart shows the number of patients admitted daily to Memorial Hospital through the emergency room.

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. a. What is the midpoint of the 2 up to 4 class?

b. How many days were 2 up to 4 patients admitted?

c. What is the class interval?

d. What is this chart called?

17. The following frequency distribution reports the number of frequent flier miles, reported in thousands, for employees of Brumley Statistical Consulting Inc. during the most recent quarter.

a. How many employees were studied?

b. What is the midpoint of the first class?

c. Construct a histogram.

d. A frequency polygon is to be drawn. What are the coordinates of the plot for the first class?

e. Construct a frequency polygon.

f. Interpret the frequent flier miles accumulated using the two charts.

18. Ecommerce.com, a large Internet retailer, is studying the lead time (elapsed time between when an order is placed and when it is filled) for a sample of recent orders. The lead times are reported in days.

a. How many orders were studied?

b. What is the midpoint of the first class?

c. What are the coordinates of the first class for a frequency polygon?

d. Draw a histogram.

e. Draw a frequency polygon.

f. Interpret the lead times using the two charts.

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Cumulative Frequency Distributions Consider once again the distribution of the profits on vehicles sold by the Apple-wood Auto Group. Suppose we were interested in the number of vehicles that sold for a profit of less than $1,400 or the profit earned on the lowest-selling 40% of the vehicles. These values can be approximated by developing a cumulative frequency distribution and portraying it graphically in a cumulative frequency polygon.

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E X A M P L E

The frequency distribution of the profits earned at Applewood Auto Group is repeated from Table 2–5.

Construct a cumulative frequency polygon to answer the following two questions. Seventy-five percent of the vehicles sold earned a profit of less than what amount? Sixty of the vehicles earned a profit of less than what amount?

S O L U T I O N

As the names imply, a cumulative frequency distribution and a cumulative frequency polygon require cumulative frequencies. To construct a cumulative frequency distribution, refer to the preceding table and note that there were eight vehicles in which the profit earned was less than $600. Those 8 vehicles, plus the 11 in the next higher class, for a total of 19, earned a profit of less than $1,000. The cumulative frequency for the next higher class is 42, found by 8 1 11 1 23. This process is continued for all the classes. All the vehicles earned a profit of less than $3,400. (See Table 2–8.)

TABLE 2–8 Cumulative Frequency Distribution for Profit on Vehicles Sold Last Month at Applewood Auto Group

To plot a cumulative frequency distribution, scale the upper limit of each class along the X-axis and the corresponding cumulative frequencies along the Y-axis. To provide additional information, you can label

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the vertical axis on the left in units and the vertical axis on the right in percent. In the Applewood Auto Group,

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. the vertical axis on the left is labeled from 0 to 180 and on the right from 0 to 100%. The value of 50% corresponds to 90 vehicles.

To begin, the first plot is at x = 200 and y = 0. None of the vehicles sold for a profit of less than $200. The profit on 8 vehicles was less than $600, so the next plot is at x = 600 and y = 8. Continuing, the next plot is x = 1,000 and y = 19. There were 19 vehicles that sold for a profit of less than $1,000. The rest of the points are plotted and then the dots connected to form Chart 2–7.

Chart 2–7 Cumulative Frequency Polygon for Profit on Vehicles Sold Last Month at Applewood Auto Group

Using Chart 2–7 to find the amount of profit on 75% of the cars sold, draw a horizontal line from the 75% mark on the right-hand vertical axis over to the polygon, then drop down to the X-axis and read the amount of profit. The value on the X-axis is about $2,300, so we estimate that 75% of the vehicles sold earned a profit of $2,300 or less for the Applewood group.

To find the highest profit earned on 60 of the 180 vehicles, we use Chart 2–7 to locate the value of 60 on the left-hand vertical axis. Next, we draw a horizontal line from the value of 60 to the polygon and then drop down to the X-axis and read the profit. It is about $1,600, so we estimate that 60 of the vehicles sold for a profit of less than $1,600. We can also make estimates of the percentage of vehicles that sold for less than a particular amount. To explain, suppose we want to estimate the percentage of vehicles that sold for a profit of less than $2,000. We begin by locating the value of $2,000 on the X-axis, move vertically to the polygon, and then horizontally to the vertical axis on the right. The value is about 56%, so we conclude 56% of the

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vehicles sold for a profit of less than $2,000.

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2–5

A sample of the hourly wages of 15 employees at Home Depot in Brunswick, Georgia, was organized into the following table.

(a) What is the table called?

(b) Develop a cumulative frequency distribution and portray the distribution in a cumulative frequency polygon.

(c) On the basis of the cumulative frequency polygon, how many employees earn less than $11 per hour?

E X E R C I S E S

19. The following cumulative frequency polygon shows the hourly wages of a sample of certified welders in the Atlanta, Georgia, area.

a. How many welders were studied?

b. What is the class interval?

c. About how many welders earn less than $10.00 per hour?

d. About 75% of the welders make less than what amount?

e. Ten of the welders studied made less than what amount?

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f. What percent of the welders make less than $20.00 per hour?

20. The following cumulative frequency polygon shows the selling price ($000) of houses sold in the Billings, Montana, area.

a. How many homes were studied?

b. What is the class interval?

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. c. One hundred homes sold for less than what amount?

d. About 75% of the homes sold for less than what amount?

e. Estimate the number of homes in the $150,000 up to $200,000 class.

f. About how many homes sold for less than $225,000?

21. The frequency distribution representing the number of frequent flier miles accumulated by employees at Brumley Statistical Consulting Inc. is repeated from Exercise 17.

a. How many employees accumulated less than 3,000 miles?

b. Convert the frequency distribution to a cumulative frequency distribution.

c. Portray the cumulative distribution in the form of a cumulative frequency polygon.

d. Based on the cumulative frequency polygon, about 75% of the employees accumulated how many miles or less?

22. The frequency distribution of order lead time at Ecommerce.com from Exercise 18 is repeated below.

a. How many orders were filled in less than 10 days? In less than 15 days?

b. Convert the frequency distribution to a cumulative frequency distribution.

c. Develop a cumulative frequency polygon.

d. About 60% of the orders were filled in less than how many days?

C H A P T E R S U M M A R Y

I. A frequency table is a grouping of qualitative data into mutually exclusive and collectively exhaustive classes showing the number of observations in each class.

II. A relative frequency table shows the fraction of the number of frequencies in each class.

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III. A bar chart is a graphic representation of a frequency table.

IV. A pie chart shows the proportion each distinct class represents of the total number of observations.

V. A frequency distribution is a grouping of data into mutually exclusive and collectively exhaustive classes showing the number of observations in each class.

A. The steps in constructing a frequency distribution are

1. Decide on the number of classes.

2. Determine the class interval.

3. Set the individual class limits.

4. Tally the raw data into classes.

5. Count the number of tallies in each class.

B. The class frequency is the number of observations in each class.

C. The class interval is the difference between the limits of two consecutive classes.

D. The class midpoint is halfway between the limits of consecutive classes.

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. VI. A relative frequency distribution shows the percent of observations in each class.

VII. There are three methods for graphically portraying a frequency distribution.

A. A histogram portrays the number of frequencies in each class in the form of a rectangle.

B. A frequency polygon consists of line segments connecting the points formed by the intersection of the class midpoint and the class frequency.

C. A cumulative frequency distribution shows the number or percent of observations less than a given value.

C H A P T E R E X E R C I S E S

23. Describe the similarities and differences of qualitative and quantitative variables. Be sure to include the following:

a. What level of measurement is required for each variable type?

b. Can both types be used to describe both samples and populations?

24. Describe the similarities and differences between a frequency table and a frequency distribution. Be sure to include which requires qualitative data and which requires quantitative data.

25. Alexandra Damonte will be building a new resort in Myrtle Beach, South Carolina. She must decide how to design the resort based on the type of activities that the resort will offer to its customers. A recent poll of 300 potential customers showed the following results about customers’ preferences for planned resort activities:

a. What is the table called?

b. Draw a bar chart to portray the survey results.

c. Draw a pie chart for the survey results.

d. If you are preparing to present the results to Ms. Damonte as part of a report, which graph would you prefer to show? Why?

For DATA FILE, please visit www.mhhe.com/lind16e

26. Speedy Swift is a package delivery service that serves the greater Atlanta, Georgia, metropolitan area. To maintain customer loyalty, one of Speedy Swift’s performance objectives is on-time delivery. To monitor its performance, each delivery is measured on the following scale: early (package delivered before the promised time), on-time (package delivered within 5 minutes of the promised time), late (package delivered more than 5 minutes past the promised time), lost (package never delivered). Speedy Swift’s objective is to deliver 99% of all packages either early or on-time. Another objective is to never lose a

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package. Speedy collected the following data for last month’s performance:

a. What scale is used to measure delivery performance? What kind of variable is delivery performance?

b. Construct a frequency table for delivery performance for last month.

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. c. Construct a relative frequency table for delivery performance last month.

d. Construct a bar chart of the frequency table for delivery performance for last month.

e. Construct a pie chart of on-time delivery performance for last month.

f. Analyze the data summaries and write an evaluation of last month’s delivery performance as it relates to Speedy Swift’s performance objectives. Write a general recommendation for further analysis.

27. A data set consists of 83 observations. How many classes would you recommend for a frequency distribution?

28. A data set consists of 145 observations that range from 56 to 490. What size class interval would you recommend?

For DATA FILE, please visit www.mhhe.com/lind16e

29. The following is the number of minutes to commute from home to work for a group of 25 automobile executives.

a. How many classes would you recommend?

b. What class interval would you suggest?

c. What would you recommend as the lower limit of the first class?

d. Organize the data into a frequency distribution.

e. Comment on the shape of the frequency distribution.

For DATA FILE, please visit www.mhhe.com/lind16e

30. The following data give the weekly amounts spent on groceries for a sample of 45 households.

a. How many classes would you recommend?

b. What class interval would you suggest?

c. What would you recommend as the lower limit of the first class?

d. Organize the data into a frequency distribution.

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For DATA FILE, please visit www.mhhe.com/lind16e

31. A social scientist is studying the use of iPods by college students. A sample of 45 students revealed they played the following number of songs yesterday.

Organize the above information into a frequency distribution.

a. How many classes would you suggest?

b. What is the most suitable class interval?

c. What is the lower limit of the initial class?

d. Create the frequency distribution.

e. Describe the profile of the distribution.

For DATA FILE, please visit www.mhhe.com/lind16e

32. David Wise handles his own investment portfolio, and has done so for many years. Listed below is the holding time (recorded to the nearest whole year) between purchase and sale for his collection of 36 stocks.

a. How many classes would you propose?

b. What class interval would you suggest?

c. What quantity would you use for the lower limit of the initial class?

d. Using your responses to parts (a), (b), and (c), create a frequency distribution.

e. Identify the appearance of the frequency distribution.

For DATA FILE, please visit www.mhhe.com/lind16e

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MERRILL LYNCH recently completed a study of online investment portfolios for a sample of clients. For the 70 participants in the study, organize these data into a frequency distribution. (See Exercise 43 and LO2-3.)

LEARNING OBJECTIVES When you have completed this chapter, you will be able to:

LO2-1 Summarize qualitative variables with frequency and relative frequency tables.

LO2-2 Display a frequency table using a bar or pie chart.

LO2-3 Summarize quantitative variables with frequency and relative frequency distributions.

LO2-4 Display a frequency distribution using a histogram or frequency polygon.

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. INTRODUCTION

The highly competitive automobile retailing industry in the United States has changed dramatically in recent years. These changes spurred events such as the

bankruptcies of General Motors and Chrysler in 2009.

elimination of well-known brands like Pontiac and Saturn.

closing of over 1,500 local dealerships.

collapse of consumer credit availability.

consolidation of dealership groups.

Traditionally, a local family owned and operated the community dealership, which might have included one or two manufacturers or brands, like Pontiac and GMC Trucks or Chrysler and the popular Jeep line. Recently, however, skillfully managed and well-financed companies have been acquiring local dealerships throughout the country. As these groups acquire the local dealerships, they often bring standardized selling practices, common software and hardware technology platforms, and management reporting techniques. The goal of these new organizations is to provide an improved buying experience for the consumer, while increasing profitability. Megadealerships often employ over 10,000 people, generate several billion dollars in annual sales, own more than 50 franchises, and are traded on the New York Stock Exchange or NASDAQ. Today, the largest megadealership is AutoNation (ticker symbol AN). Others include Penske Auto Group (PAG and second largest), Asbury Automotive Group (ABG), and Hendrick Auto Group (which is privately held).

The Applewood Auto Group is an ownership group that includes four dealerships. The group sells a wide range of vehicles, including the inexpensive but popular Korean brands Kia and Hyundai, BMW and Volvo sedans and luxury SUVs, and a full line of Ford and Chevrolet cars and trucks.

Ms. Kathryn Ball is a member of the senior management team at Applewood Auto Group, which has its corporate offices adjacent to Kane Motors. She is responsible for tracking and analyzing vehicle sales and the profitability of those vehicles. Kathryn would like to summarize the profit earned on the vehicles sold with tables, charts, and graphs that she would review monthly. She wants to know the profit per vehicle sold, as well as the lowest and highest amount of profit. She is also interested in describing the demographics of the buyers. What are their ages? How many vehicles have they previously purchased from one of the Applewood dealerships? What type of vehicle did they purchase?

The Applewood Auto Group operates four dealerships:

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Tionesta Ford Lincoln Mercury sells the Ford, Lincoln, and Mercury cars and trucks.

Olean Automotive Inc. has the Nissan franchise as well as the General Motors brands of Chevrolet, Cadillac, and GMC Trucks.

Sheffield Motors Inc. sells Buick, GMC trucks, Hyundai, and Kia.

Kane Motors offers the Chrysler, Dodge, and Jeep line as well as BMW and Volvo.

Every month, Ms. Ball collects data from each of the four dealerships and enters them into an Excel spreadsheet. Last month the Applewood Auto Group sold 180 vehicles at the four dealerships. A copy of the first few observations appears to the left. The variables collected include:

Age—the age of the buyer at the time of the purchase.

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Profit—the amount earned by the dealership on the sale of each vehicle.

Location—the dealership where the vehicle was purchased.

Vehicle type—SUV, sedan, compact, hybrid, or truck.

Previous—the number of vehicles previously purchased at any of the four Apple-wood dealerships by the consumer.

The entire data set is available at the McGraw-Hill website (www.mhhe.com/lind16e) and in Appendix A.4 at the end of the text.

LO2-1

Summarize qualitative variables with frequency and relative frequency tables.

CONSTRUCTING FREQUENCY TABLES Recall from Chapter 1 that techniques used to describe a set of data are called descriptive statistics. Descriptive statistics organize data to show the general pattern of the data, to identify where values tend to concentrate, and to expose extreme or unusual data values. The first technique we discuss is a frequency table.

FREQUENCY TABLE A grouping of qualitative data into mutually exclusive and collectively exhaustive classes showing the number of observations in each class.

In Chapter 1, we distinguished between qualitative and quantitative variables. To review, a qualitative variable is nonnumeric, that is, it can only be classified into distinct categories. Examples of qualitative data include political affiliation (Republican, Democrat, Independent), state of birth (Alabama, . . . , Wyoming), and method of payment for a purchase at Barnes & Noble (cash, check, debit, or credit). On the other hand, quantitative variables are numerical in nature. Examples of quantitative data relating to college students include the price of their textbooks, their age, and the number of credit hours they are registered for this semester.

In the Applewood Auto Group data set, there are five variables for each vehicle sale: age of the buyer, amount of profit, dealer that made the sale, type of vehicle sold, and number of previous purchases by the buyer. The dealer and the type of vehicle are qualitative variables. The amount of profit, the age of the buyer, and the number of previous purchases are quantitative variables.

Suppose Ms. Ball wanted to summarize last month’s sales by location. To summarize this qualitative data, we classify the vehicles sold last month according to their location: Tionesta, Olean, Sheffield, or Kane. We use location to develop a frequency table with four mutually exclusive (distinctive) classes. This means that a particular vehicle cannot belong to more than one class. Each vehicle is uniquely classified into one of the four mutually exclusive locations. In addition, the frequency table must be collectively exhaustive. This means that every vehicle is accounted for in the table. This frequency table is shown in Table 2–1. The number of observations, representing the sales at each location, is called the class frequency. So the class frequency for vehicles sold at the Kane location is 52.

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TABLE 2–1 Frequency Table for Vehicles Sold Last Month at Applewood Auto Group by Location

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CHART 2–1 Number of Vehicles Sold by Location

We explain the details of constructing a pie chart using the information in Table 2–3, which shows the frequency and percent of cars sold by the Applewood Auto Group for each vehicle type.

TABLE 2–3 Vehicle Sales by Type at Applewood Auto Group

The first step to develop a pie chart is to mark the percentages 0, 5, 10, 15, and so on evenly around the circumference of a circle (see Chart 2–2). To plot the 40% of total sales represented by sedans, draw a line from the center of the circle to 0 and another line from the center of the circle to 40%. The area in this “slice” represents the number of sedans sold as a percentage of the total sales. Next, add the SUV’s percentage of total sales, 30%, to the sedan’s percentage of total sales, 40%. The result is 70%. Draw a line from the center of the circle to 70%, so the area between 40 and 70 shows the sales of SUVs as a percentage of total sales. Continuing, add the 15% of total sales for compact vehicles, which gives us a total of 85%. Draw a line from the center of the circle to 85, so the “slice” between 70% and 85% represents the number of compact vehicles sold as a percentage of the total sales. The remaining 10% for truck sales and 5% for hybrid sales are added to the chart using the same method.

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Chart 2–2 Pie Chart of Vehicles by Type

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Page 23

PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. Because each slice of the pie represents the relative frequency of each vehicle type as a percentage of the total sales, we can easily compare them:

The largest percentage of sales is for sedans.

Sedans and SUVs together account for 70% of vehicle sales.

Hybrids account for 5% of vehicle sales, in spite of being on the market for only a few years.

We can use Excel software to quickly count the number of cars for each vehicle type and create the frequency table, bar chart, and pie chart below. The Excel tool is called a Pivot Table. The instructions to produce these descriptive statistics and charts are given in Appendix C.

Pie and bar charts both serve to illustrate frequency and relative frequency tables. When is a pie chart preferred to a bar chart? In most cases, pie charts are used to show and compare the relative differences in the percentage of observations for each value or class of a qualitative variable. Bar charts are preferred when the goal is to compare the number or frequency of observations for each value or class of a qualitative variable. The following Example/Solution shows another application of bar and pie charts.

E X A M P L E

SkiLodges.com is test marketing its new website and is interested in how easy its website design is to navigate. It randomly selected 200 regular Internet users and asked them to perform a search task on the website. Each person was asked to rate the relative ease of navigation as poor, good, excellent, or awesome. The results are shown in the following table:

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. 1. What type of measurement scale is used for ease of navigation?

2. Draw a bar chart for the survey results.

3. Draw a pie chart for the survey results.

S O L U T I O N

The data are measured on an ordinal scale. That is, the scale is ranked in relative ease when moving from “poor” to “awesome.” Also, the interval between each rating is unknown so it is impossible, for example, to conclude that a rating of good is twice the value of a poor rating.

We can use a bar chart to graph the data. The vertical scale shows the relative frequency and the horizontal scale shows the values of the ease-of-navigation variable.

A pie chart can also be used to graph these data. The pie chart emphasizes that more than half of the respondents rate the relative ease of using the website awesome.

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2–1

The answers are in Appendix E.

DeCenzo Specialty Food and Beverage Company has been serving a cola drink with an additional flavoring, Cola-Plus, that is very popular among its customers. The company is interested in customer preferences for Cola-Plus versus Coca-Cola, Pepsi, and a lemon-lime beverage. They ask 100 randomly sampled customers to take a taste test and select the beverage they prefer most. The results are shown in the following table:

(a) Is the data qualitative or quantitative? Why?

(b) What is the table called? What does it show?

(c) Develop a bar chart to depict the information.

(d) Develop a pie chart using the relative frequencies.

E X E R C I S E S

The answers to the odd-numbered exercises are at the end of the book in Appendix D.

1. A pie chart shows the relative market share of cola products. The “slice” for Pepsi-Cola has a central angle of 90 degrees. What is its market share?

2. In a marketing study, 100 consumers were asked to select the best digital music player from the iPod, the iRiver, and the Magic Star MP3. To summarize the consumer responses with a frequency table, how many classes would the frequency table have?

3. A total of 1,000 residents in Minnesota were asked which season they preferred. One hundred liked winter best, 300 liked spring, 400 liked summer, and 200 liked fall. Develop a frequency table and a relative frequency table to summarize this information.

4. Two thousand frequent business travelers are asked which midwestern city they prefer: Indianapolis, Saint Louis, Chicago, or Milwaukee. One hundred liked Indianapolis best, 450 liked Saint Louis, 1,300 liked Chicago, and the remainder preferred Milwaukee. Develop a frequency table and a relative frequency table to summarize this information.

5. Wellstone Inc. produces and markets replacement covers for cell phones in five different colors: bright white, metallic black, magnetic lime, tangerine orange, and fusion red. To estimate the demand for each

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color, the company set up a kiosk in the Mall of America for several hours and asked randomly selected people which cover color was their favorite. The results follow:

a. What is the table called?

b. Draw a bar chart for the table.

c. Draw a pie chart.

d. If Wellstone Inc. plans to produce 1 million cell phone covers, how many of each color should it produce?

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. 6. A small business consultant is investigating the performance of several companies. The fourth-quarter sales for last year (in thousands of dollars) for the selected companies were:

The consultant wants to include a chart in his report comparing the sales of the six companies. Use a bar chart to compare the fourth-quarter sales of these corporations and write a brief report summarizing the bar chart.

LO2-3

Summarize quantitative variables with frequency and relative frequency distributions.

CONSTRUCTING FREQUENCY DISTRIBUTIONS In Chapter 1 and earlier in this chapter, we distinguished between qualitative and quantitative data. In the previous section, using the Applewood Automotive Group data, we summarized two qualitative variables: the location of the sale and the type of vehicle sold. We created frequency and relative frequency tables and depicted the results in bar and pie charts.

The Applewood Auto Group data also includes several quantitative variables: the age of the buyer, the profit earned on the sale of the vehicle, and the number of previous purchases. Suppose Ms. Ball wants to summarize last month’s sales by profit earned. We can describe profit using a frequency distribution.

FREQUENCY DISTRIBUTION A grouping of quantitative data into mutually exclusive and collectively exhaustive classes showing the number of observations in each class.

How do we develop a frequency distribution? The following example shows the steps to construct a frequency distribution. Remember, our goal is to construct tables, charts, and graphs that will quickly reveal the concentration, extreme values, and shape of the data.

E X A M P L E

We return to the situation where Ms. Kathryn Ball of the Applewood Auto Group wants to summarize the quantitative variable profit with a frequency distribution and display the distribution with charts and graphs. With this information, Ms. Ball can easily answer the following questions: What is the typical profit on each sale? What is the largest or maximum profit on any sale? What is the smallest or minimum profit on any sale? Around what value do the profits tend to cluster?

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S O L U T I O N

To begin, we need the profits for each of the 180 vehicle sales, which are listed in Table 2–4. This information is called raw or ungrouped data because it is simply a listing of the individual, observed profits. It is possible to search the list and find the smallest or minimum profit ($294) and the largest or maximum profit ($3,292),

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. but that is about all. It is difficult to determine a typical profit or to visualize where the profits tend to cluster. The raw data are more easily interpreted if we summarize the data with a frequency distribution as follows.

TABLE 2–4 Profit on Vehicles Sold Last Month by the Applewood Auto Group

Step 1: Decide on the number of classes. A useful recipe to determine the number of classes (k) is the “2 to the k rule.” This guide suggests you select the smallest number (k) for the number of classes such

that 2k (in words, 2 raised to the power of k) is greater than the number of observations (n). In the Applewood Auto Group example, there were 180 vehicles sold. So n 5 180. If we try k 5 = 7, which

means we would use 7 classes, 27 = 128, which is less than 180. Hence, 7 is too few classes. If we

let k = 8, then 28 = 256, which is greater than 180. So the recommended number of classes is 8.

Step 2: Determine the class interval. Generally, the class interval is the same for all classes. The classes all taken together must cover at least the distance from the minimum value in the data up to the maximum value. Expressing these words in a formula:

where i is the class interval, and k is the number of classes. For the Applewood Auto Group, the minimum value is $294 and the maximum value is $3,292. If

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we need 8 classes, the interval should be:

In practice, this interval size is usually rounded up to some convenient number, such as a multiple of 10 or 100. The value of $400 is a reasonable choice.

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. Step 3: Set the individual class limits. State clear class limits so you can put each observation into only one category. This means you must avoid overlapping or unclear class limits. For example, classes such as “$1,300–$1,400” and “$1,400–$1,500” should not be used because it is not clear whether the value $1,400 is in the first or second class. In this text, we will generally use the format $1,300 up to $1,400 and $1,400 up to $1,500 and so on. With this format, it is clear that $1,399 goes into the first class and $1,400 in the second. Because we always round the class interval up to get a convenient class size, we cover a larger than necessary range. For example, using 8 classes with an interval of $400 in the Applewood Auto Group example results in a range of 8($400) = $3,200. The actual range is $2,998, found by ($3,292 − $294). Comparing that value to $3,200, we have an excess of $202. Because we need to cover only the range (Maximum – Minimum), it is natural to put approximately equal amounts of the excess in each of the two tails. Of course, we should also select convenient class limits. A guideline is to make the lower limit of the first class a multiple of the class interval. Sometimes this is not possible, but the lower limit should at least be rounded. So here are the classes we could use for these data.

Step 4: Tally the vehicle profit into the classes. To begin, the profit from the sale of the first vehicle in Table 2–4 is $1,387. It is tallied in the $1,000 up to $1,400 class. The second profit in the first row of Table 2–4 is $2,148. It is tallied in the $1,800 up to $2,200 class. The other profits are tallied in a similar manner. When all the profits are tallied, the table would appear as:

Step 5: Count the number of items in each class. The number of observations in each class is called the class frequency. In the $200 up to $600 class there are 8 observations, and in the $600 up to $1,000

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Page 28class there are 11 observations. Therefore, the class frequency in

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. the first class is 8 and the class frequency in the second class is 11. There are a total of 180 observations in the entire set of data. So the sum of all the frequencies should be equal to 180.

TABLE 2–5 Frequency Distribution of Profit for Vehicles Sold Last Month at Applewood Auto Group

Now that we have organized the data into a frequency distribution (see Table 2–5), we can summarize the profits of the vehicles for the Applewood Auto Group. Observe the following:

1. The profits from vehicle sales range between $200 and $3,400.

2. The vehicle profits are classified using a class interval of $400. The class interval is determined by subtracting consecutive lower or upper class limits. For example, the lower limit of the first class is $200, and the lower limit of the second class is $600. The difference is the class interval of $400.

4. For each class, we can determine the typical profit or class midpoint. It is halfway between the lower or upper limits of two consecutive classes. It is computed by adding the lower or upper limits of consecutive classes and dividing by 2. Referring to Table 2–5, the lower class limit of the first class is $200, and the next class limit is $600. The class midpoint is $400, found by ($600 1 $200)y2. The midpoint best represents, or is typical of, the profits of the vehicles in that class. Applewood sold 8 vehicles with a typical profit of $400.

5. The largest concentration, or highest frequency, of vehicles sold is in the $1,800 up to $2,200 class. There are 45 vehicles in this class. The class midpoint is $2,000. So we say that the typical profit in the class with the highest frequency is $2,000.

By presenting this information to Ms. Ball, we give her a clear picture of the distribution of the vehicle profits for last month.

We admit that arranging the information on profits into a frequency distribution does result in the loss of some detailed information. That is, by organizing the data into a frequency distribution, we cannot pinpoint the exact profit on any vehicle, such as $1,387, $2,148, or $2,201. Further, we cannot tell that the actual minimum profit for any vehicle sold is $294 or that the maximum profit was $3,292. However, the lower limit of the first class and the upper limit of the last class convey essentially the same meaning. Likely, Ms. Ball will make the same judgment if she knows the smallest profit is about $200 that she will if she knows

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the exact profit is $292. The advantages of summarizing the 180 profits into a more understandable and organized form more than offset this disadvantage.

In 1788, James Madison, John Jay, and Alexander Hamilton anonymously published a series of essays entitled The Federalist. These Federalist papers were an attempt to convince the people of New York that they should ratify the Constitution. In the course of history, the authorship of most of these papers became known, but 12 remained contested. Through the use of statistical analysis, and particularly the study of the frequency of the use of various words, we can now conclude that James Madison is the likely author of the 12 papers. In fact, the statistical evidence that Madison is the author is overwhelming.

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. When we summarize raw data with frequency distributions, equal class intervals are preferred. However, in certain situations unequal class intervals may be necessary to avoid a large number of classes with very small frequencies. Such is the case in Table 2–6. The U.S. Internal Revenue Service uses unequal-sized class intervals for adjusted gross income on individual tax returns to summarize the number of

individual tax returns. If we use our method to find equal class intervals, the 2k rule results in 25 classes, and a class interval of $400,000, assuming $0 and $10,000,000 as the minimum and maximum values for adjusted gross income. Using equal class intervals, the first 13 classes in Table 2–6 would be combined into one class of about 99.9% of all tax returns and 24 classes for the 0.1% of the returns with an adjusted gross income above $400,000. Using equal class intervals does not provide a good understanding of the raw data. In this case, good judgment in the use of unequal class intervals, as demonstrated in Table 2–6, is required to show the distribution of the number of tax returns filed, especially for incomes under $500,000.

TABLE 2–6 Adjusted Gross Income for Individuals Filing Income Tax Returns

2–2

The commissions earned for the first quarter of last year by the 11 members of the sales staff at Master Chemical Company are:

$1,650 $1,475 $1,510 $1,670 $1,595 $1,760 $1,540 $1,495 $1,590 $1,625 $1,510

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(a) What are the values such as $1,650 and $1,475 called?

(b) Using $1,400 up to $1,500 as the first class, $1,500 up to $1,600 as the second class, and so forth, organize the quarterly commissions into a frequency distribution.

(c) What are the numbers in the right column of your frequency distribution called?

(d) Describe the distribution of quarterly commissions, based on the frequency distribution. What is the largest concentration of commissions earned? What is the smallest, and the largest? What is the typical amount earned?

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. Relative Frequency Distribution

It may be desirable, as we did earlier with qualitative data, to convert class frequencies to relative class frequencies to show the proportion of the total number of observations in each class. In our vehicle profits, we may want to know what percentage of the vehicle profits are in the $1,000 up to $1,400 class. In another study, we may want to know what percentage of the employees used 5 up to 10 personal leave days last year. To convert a frequency distribution to a relative frequency distribution, each of the class frequencies is divided by the total number of observations. From the distribution of vehicle profits, Table 2–5, the relative frequency for the $1,000 up to $1,400 class is 0.128, found by dividing 23 by 180. That is, profit on 12.8% of the vehicles sold is between $1,000 and $1,400. The relative frequencies for the remaining classes are shown in Table 2–7.

TABLE 2–7 Relative Frequency Distribution of Profit for Vehicles Sold Last Month at Applewood Auto Group

As we mentioned in Chapter 1, there are many software packages that perform statistical calculations. Throughout this text, we will show the output from Microsoft Excel, MegaStat (a Microsoft Excel add-in), and Minitab (a statistical software package).

On page 22 of this chapter, we used the Pivot Table tool in Excel to create a frequency table. We use the same Excel tool to create frequency and relative frequency distributions for the profit variable in the Applewood Auto Group data. The necessary steps for the table to the left are given in the Software Commands section in Appendix C.

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2–3

Barry Bonds of the San Francisco Giants established a new single-season Major League Baseball home run record by hitting 73 home runs during the 2001 season. Listed below is the sorted distance of each of the 73 home runs.

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. (a) For this data, show that seven classes would be used to create a frequency distribution using

the 2k rule.

(b) Show that a class interval of 30 would summarize the data in seven classes.

(c) Construct frequency and relative frequency distributions for the data with seven classes and a class interval of 30. Start the first class with a lower limit of 300.

(d) How many home runs traveled a distance of 360 up to 390 feet?

(e) What percentage of the home runs traveled a distance of 360 up to 390 feet?

(f) What percentage of the home runs traveled a distance of 390 feet or more?

E X E R C I S E S

This icon indicates that the data are available at the text website: www.mhhe.com/lind16e. You will be able to download the data directly into Excel or Minitab from this site.

7. A set of data consists of 38 observations. How many classes would you recommend for the frequency distribution?

8. A set of data consists of 45 observations between $0 and $29. What size would you recommend for the class interval?

9. A set of data consists of 230 observations between $235 and $567. What class interval would you recommend?

10. A set of data contains 53 observations. The minimum value is 42 and the maximum value is 129. The data are to be organized into a frequency distribution.

a. How many classes would you suggest?

b. What would you suggest as the lower limit of the first class?

For DATA FILE, please visit www.mhhe.com/lind16e

11. Wachesaw Manufacturing Inc. produced the following number of units in the last 16 days.

The information is to be organized into a frequency distribution.

a. How many classes would you recommend?

b. What class interval would you suggest?

c. What lower limit would you recommend for the first class?

d. Organize the information into a frequency distribution and determine the relative frequency distribution.

e. Comment on the shape of the distribution.

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For DATA FILE, please visit www.mhhe.com/lind16e

12. The Quick Change Oil Company has a number of outlets in the metropolitan Seattle area. The daily number of oil changes at the Oak Street outlet in the past 20 days are:

The data are to be organized into a frequency distribution.

a. How many classes would you recommend?

b. What class interval would you suggest?

c. What lower limit would you recommend for the first class?

d. Organize the number of oil changes into a frequency distribution.

e. Comment on the shape of the frequency distribution. Also determine the relative frequency distribution.

For DATA FILE, please visit www.mhhe.com/lind16e

13. The manager of the BiLo Supermarket in Mt. Pleasant, Rhode Island, gathered the following information on the number of times a customer visits the store during a month. The responses of 51 customers were:

a. Starting with 0 as the lower limit of the first class and using a class interval of 3, organize the data into a frequency distribution.

b. Describe the distribution. Where do the data tend to cluster?

c. Convert the distribution to a relative frequency distribution.

For DATA FILE, please visit www.mhhe.com/lind16e

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. 14. The food services division of Cedar River Amusement Park Inc. is studying the amount that families who visit the amusement park spend per day on food and drink. A sample of 40 families who visited the park yesterday revealed they spent the following amounts:

a. Organize the data into a frequency distribution, using seven classes and 15 as the lower limit of the first class. What class interval did you select?

b. Where do the data tend to cluster?

c. Describe the distribution.

d. Determine the relative frequency distribution.

LO2-4

Display a frequency distribution using a histogram or frequency polygon.

GRAPHIC PRESENTATION OF A FREQUENCY DISTRIBUTION Sales managers, stock analysts, hospital administrators, and other busy executives often need a quick picture of the distributions of sales, stock prices, or hospital costs. These distributions can often be depicted by the use of charts and graphs. Three charts that will help portray a frequency distribution graphically are the histogram, the frequency polygon, and the cumulative frequency polygon.

Histogram A histogram for a frequency distribution based on quantitative data is similar to the bar chart showing the distribution of qualitative data. The classes are marked on the horizontal axis and the class frequencies on the vertical axis. The class frequencies are represented by the heights of the bars. However, there is one important difference based on the nature of the data. Quantitative data are usually measured using scales that are continuous, not discrete. Therefore, the horizontal axis represents all possible values, and the bars are drawn adjacent to each other to show the continuous nature of the data.

HISTOGRAM A graph in which the classes are marked on the horizontal axis and the class frequencies on the vertical axis. The class frequencies are represented by the heights of the bars, and the bars are drawn adjacent to each other.

E X A M P L E

Below is the frequency distribution of the profits on vehicle sales last month at the Applewood Auto Group.

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. Construct a histogram. What observations can you reach based on the information presented in the histogram?

S O L U T I O N

The class frequencies are scaled along the vertical axis (Y-axis) and either the class limits or the class midpoints along the horizontal axis. To illustrate the construction of the histogram, the first three classes are shown in Chart 2–3.

Chart 2–3 Construction of a Histogram

From Chart 2–3 we note the profit on eight vehicles was $200 up to $600. Therefore, the height of the column for that class is 8. There are 11 vehicle sales where the profit was $600 up to $1,000. So, logically, the height of that column is 11. The height of the bar represents the number of observations in the class.

This procedure is continued for all classes. The complete histogram is shown in Chart 2–4. Note that there is no space between the bars. This is a feature of the histogram. Why is this so? Because the variable plotted on the horizontal axis is a continuous variable. In a bar chart, the scale of measurement is often nominal and the vertical bars are separated. This is an important distinction between the histogram and the bar chart.

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Chart 2–4 Histogram of the Profit on 180 Vehicles Sold at the Applewood Auto Group

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. We can make the following statements using Chart 2–4. They are the same as the observations based on Table 2–5 on page 28.

1. The profits from vehicle sales range between $200 and $3,400.

2. The vehicle profits are classified using a class interval of $400. The class interval is determined by subtracting consecutive lower or upper class limits. For example, the lower limit of the first class is $200, and the lower limit of the second class is $600. The difference is the class interval or $400.

4. For each class, we can determine the typical profit or class midpoint. It is halfway between the lower or upper limits of two consecutive classes. It is computed by adding the lower or upper limits of consecutive classes and dividing by 2. Referring to Chart 2–4, the lower class limit of the first class is $200, and the next class limit is $600. The class midpoint is $400, found by ($600 1 $200)y2. The midpoint best represents, or is typical of, the profits of the vehicles in that class. Applewood sold 8 vehicles with a typical profit of $400.

5. The largest concentration, or highest frequency of vehicles sold, is in the $1,800 up to $2,200 class. There are 45 vehicles in this class. The class midpoint is $2,000. So we say that the typical profit in the class with the highest frequency is $2,000.

Thus, the histogram provides an easily interpreted visual representation of a frequency distribution. We should also point out that we would have made the same observations and the shape of the histogram would have been the same had we used a relative frequency distribution instead of the actual frequencies. That is, if we had used the relative frequencies of Table 2–7, we would have had a histogram of the same shape as Chart 2–4. The only difference is that the vertical axis would have been reported in percentage of vehicles instead of the number of vehicles. The Excel commands to create Chart 2–4 are given in Appendix C.

Florence Nightingale is known as the founder of the nursing profession. However, she also saved many lives by using statistical analysis. When she encountered an unsanitary condition or an undersup-plied hospital, she improved the conditions and then used statistical data to document the improvement. Thus, she was able to convince others of the need for medical reform, particularly in the area of sanitation. She developed original graphs to demonstrate that, during the Crimean War, more soldiers died from unsanitary conditions than were killed in combat.

Frequency Polygon A frequency polygon also shows the shape of a distribution and is similar to a histogram. It consists of line segments connecting the points formed by the intersections of the class midpoints and the class frequencies. The construction of a frequency polygon is illustrated in Chart 2–5 on the next page. We use the profits from the cars sold last month at the Applewood Auto Group. The midpoint of each class is scaled on the X-axis and the class frequencies on the Y-axis. Recall that the class midpoint is the value at the center of a class and

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represents the typical values in that class. The class frequency is the number of observations in a particular class. The profit earned on the vehicles sold last month by the Applewood Auto Group is repeated below.

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Chart 2–5 Frequency Polygon of Profit on 180 Vehicles Sold at Applewood Auto Group

As noted previously, the $200 up to $600 class is represented by the midpoint $400. To construct a frequency polygon, move horizontally on the graph to the midpoint, $400, and then vertically to 8, the class frequency, and place a dot. The x and the y values of this point are called the coordinates. The coordinates of the next point are x = 800 and y = 11. The process is continued for all classes. Then the points are connected in order. That is, the point representing the lowest class is joined to the one representing the second class and so on. Note in Chart 2–5 that, to complete the frequency polygon, midpoints of $0 and $3,600 are added to the X-axis to “anchor” the polygon at zero frequencies. These two values, $0 and $3,600, were derived by subtracting the class interval of $400 from the lowest midpoint ($400) and by adding $400 to the highest midpoint ($3,200) in the frequency distribution.

Both the histogram and the frequency polygon allow us to get a quick picture of the main characteristics of the data (highs, lows, points of concentration, etc.). Although the two representations are similar in purpose, the histogram has the advantage of depicting each class as a rectangle, with the height of the rectangular bar representing the number in each class. The frequency polygon, in turn, has an advantage over the histogram. It allows us to compare directly two or more frequency distributions. Suppose Ms. Ball wants to compare the profit per vehicle sold at Applewood Auto Group with a similar auto group, Fowler Auto in Grayling, Michigan. To do this, two frequency polygons are constructed, one on top of the other, as in Chart 2–6. Two things are clear from the chart:

The typical vehicle profit is larger at Fowler Motors—about $2,000 for Applewood and about $2,400 for Fowler.

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Chart 2–6 Distribution of Profit at Applewood Auto Group and Fowler Motors

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. There is less dispersion in the profits at Fowler Motors than at Applewood. The lower limit

of the first class for Applewood is $0 and the upper limit is $3,600. For Fowler Motors, the lower limit is $800 and the upper limit is the same: $3,600.

The total number of cars sold at the two dealerships is about the same, so a direct comparison is possible. If the difference in the total number of cars sold is large, then converting the frequencies to relative frequencies and then plotting the two distributions would allow a clearer comparison.

2–4

The annual imports of a selected group of electronic suppliers are shown in the following frequency distribution.

(a) Portray the imports as a histogram.

(b) Portray the imports as a relative frequency polygon.

(c) Summarize the important facets of the distribution (such as classes with the highest and lowest frequencies).

E X E R C I S E S

15. Molly’s Candle Shop has several retail stores in the coastal areas of North and South Carolina. Many of Molly’s customers ask her to ship their purchases. The following chart shows the number of packages shipped per day for the last 100 days.

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a. What is this chart called?

b. What is the total number of frequencies?

c. What is the class interval?

d. What is the class frequency of the 10 up to 15 class?

e. What is the relative frequency of the 10 up to 15 class?

f. What is the midpoint of the 10 up to 15 class?

g. On how many days were there 25 or more packages shipped?

16. The following chart shows the number of patients admitted daily to Memorial Hospital through the emergency room.

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. a. What is the midpoint of the 2 up to 4 class?

b. How many days were 2 up to 4 patients admitted?

c. What is the class interval?

d. What is this chart called?

17. The following frequency distribution reports the number of frequent flier miles, reported in thousands, for employees of Brumley Statistical Consulting Inc. during the most recent quarter.

a. How many employees were studied?

b. What is the midpoint of the first class?

c. Construct a histogram.

d. A frequency polygon is to be drawn. What are the coordinates of the plot for the first class?

e. Construct a frequency polygon.

f. Interpret the frequent flier miles accumulated using the two charts.

18. Ecommerce.com, a large Internet retailer, is studying the lead time (elapsed time between when an order is placed and when it is filled) for a sample of recent orders. The lead times are reported in days.

a. How many orders were studied?

b. What is the midpoint of the first class?

c. What are the coordinates of the first class for a frequency polygon?

d. Draw a histogram.

e. Draw a frequency polygon.

f. Interpret the lead times using the two charts.

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Cumulative Frequency Distributions Consider once again the distribution of the profits on vehicles sold by the Apple-wood Auto Group. Suppose we were interested in the number of vehicles that sold for a profit of less than $1,400 or the profit earned on the lowest-selling 40% of the vehicles. These values can be approximated by developing a cumulative frequency distribution and portraying it graphically in a cumulative frequency polygon.

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E X A M P L E

The frequency distribution of the profits earned at Applewood Auto Group is repeated from Table 2–5.

Construct a cumulative frequency polygon to answer the following two questions. Seventy-five percent of the vehicles sold earned a profit of less than what amount? Sixty of the vehicles earned a profit of less than what amount?

S O L U T I O N

As the names imply, a cumulative frequency distribution and a cumulative frequency polygon require cumulative frequencies. To construct a cumulative frequency distribution, refer to the preceding table and note that there were eight vehicles in which the profit earned was less than $600. Those 8 vehicles, plus the 11 in the next higher class, for a total of 19, earned a profit of less than $1,000. The cumulative frequency for the next higher class is 42, found by 8 1 11 1 23. This process is continued for all the classes. All the vehicles earned a profit of less than $3,400. (See Table 2–8.)

TABLE 2–8 Cumulative Frequency Distribution for Profit on Vehicles Sold Last Month at Applewood Auto Group

To plot a cumulative frequency distribution, scale the upper limit of each class along the X-axis and the corresponding cumulative frequencies along the Y-axis. To provide additional information, you can label

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the vertical axis on the left in units and the vertical axis on the right in percent. In the Applewood Auto Group,

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. the vertical axis on the left is labeled from 0 to 180 and on the right from 0 to 100%. The value of 50% corresponds to 90 vehicles.

To begin, the first plot is at x = 200 and y = 0. None of the vehicles sold for a profit of less than $200. The profit on 8 vehicles was less than $600, so the next plot is at x = 600 and y = 8. Continuing, the next plot is x = 1,000 and y = 19. There were 19 vehicles that sold for a profit of less than $1,000. The rest of the points are plotted and then the dots connected to form Chart 2–7.

Chart 2–7 Cumulative Frequency Polygon for Profit on Vehicles Sold Last Month at Applewood Auto Group

Using Chart 2–7 to find the amount of profit on 75% of the cars sold, draw a horizontal line from the 75% mark on the right-hand vertical axis over to the polygon, then drop down to the X-axis and read the amount of profit. The value on the X-axis is about $2,300, so we estimate that 75% of the vehicles sold earned a profit of $2,300 or less for the Applewood group.

To find the highest profit earned on 60 of the 180 vehicles, we use Chart 2–7 to locate the value of 60 on the left-hand vertical axis. Next, we draw a horizontal line from the value of 60 to the polygon and then drop down to the X-axis and read the profit. It is about $1,600, so we estimate that 60 of the vehicles sold for a profit of less than $1,600. We can also make estimates of the percentage of vehicles that sold for less than a particular amount. To explain, suppose we want to estimate the percentage of vehicles that sold for a profit of less than $2,000. We begin by locating the value of $2,000 on the X-axis, move vertically to the polygon, and then horizontally to the vertical axis on the right. The value is about 56%, so we conclude 56% of the

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vehicles sold for a profit of less than $2,000.

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2–5

A sample of the hourly wages of 15 employees at Home Depot in Brunswick, Georgia, was organized into the following table.

(a) What is the table called?

(b) Develop a cumulative frequency distribution and portray the distribution in a cumulative frequency polygon.

(c) On the basis of the cumulative frequency polygon, how many employees earn less than $11 per hour?

E X E R C I S E S

19. The following cumulative frequency polygon shows the hourly wages of a sample of certified welders in the Atlanta, Georgia, area.

a. How many welders were studied?

b. What is the class interval?

c. About how many welders earn less than $10.00 per hour?

d. About 75% of the welders make less than what amount?

e. Ten of the welders studied made less than what amount?

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f. What percent of the welders make less than $20.00 per hour?

20. The following cumulative frequency polygon shows the selling price ($000) of houses sold in the Billings, Montana, area.

a. How many homes were studied?

b. What is the class interval?

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. c. One hundred homes sold for less than what amount?

d. About 75% of the homes sold for less than what amount?

e. Estimate the number of homes in the $150,000 up to $200,000 class.

f. About how many homes sold for less than $225,000?

21. The frequency distribution representing the number of frequent flier miles accumulated by employees at Brumley Statistical Consulting Inc. is repeated from Exercise 17.

a. How many employees accumulated less than 3,000 miles?

b. Convert the frequency distribution to a cumulative frequency distribution.

c. Portray the cumulative distribution in the form of a cumulative frequency polygon.

d. Based on the cumulative frequency polygon, about 75% of the employees accumulated how many miles or less?

22. The frequency distribution of order lead time at Ecommerce.com from Exercise 18 is repeated below.

a. How many orders were filled in less than 10 days? In less than 15 days?

b. Convert the frequency distribution to a cumulative frequency distribution.

c. Develop a cumulative frequency polygon.

d. About 60% of the orders were filled in less than how many days?

C H A P T E R S U M M A R Y

I. A frequency table is a grouping of qualitative data into mutually exclusive and collectively exhaustive classes showing the number of observations in each class.

II. A relative frequency table shows the fraction of the number of frequencies in each class.

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III. A bar chart is a graphic representation of a frequency table.

IV. A pie chart shows the proportion each distinct class represents of the total number of observations.

V. A frequency distribution is a grouping of data into mutually exclusive and collectively exhaustive classes showing the number of observations in each class.

A. The steps in constructing a frequency distribution are

1. Decide on the number of classes.

2. Determine the class interval.

3. Set the individual class limits.

4. Tally the raw data into classes.

5. Count the number of tallies in each class.

B. The class frequency is the number of observations in each class.

C. The class interval is the difference between the limits of two consecutive classes.

D. The class midpoint is halfway between the limits of consecutive classes.

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. VI. A relative frequency distribution shows the percent of observations in each class.

VII. There are three methods for graphically portraying a frequency distribution.

A. A histogram portrays the number of frequencies in each class in the form of a rectangle.

B. A frequency polygon consists of line segments connecting the points formed by the intersection of the class midpoint and the class frequency.

C. A cumulative frequency distribution shows the number or percent of observations less than a given value.

C H A P T E R E X E R C I S E S

23. Describe the similarities and differences of qualitative and quantitative variables. Be sure to include the following:

a. What level of measurement is required for each variable type?

b. Can both types be used to describe both samples and populations?

24. Describe the similarities and differences between a frequency table and a frequency distribution. Be sure to include which requires qualitative data and which requires quantitative data.

25. Alexandra Damonte will be building a new resort in Myrtle Beach, South Carolina. She must decide how to design the resort based on the type of activities that the resort will offer to its customers. A recent poll of 300 potential customers showed the following results about customers’ preferences for planned resort activities:

a. What is the table called?

b. Draw a bar chart to portray the survey results.

c. Draw a pie chart for the survey results.

d. If you are preparing to present the results to Ms. Damonte as part of a report, which graph would you prefer to show? Why?

For DATA FILE, please visit www.mhhe.com/lind16e

26. Speedy Swift is a package delivery service that serves the greater Atlanta, Georgia, metropolitan area. To maintain customer loyalty, one of Speedy Swift’s performance objectives is on-time delivery. To monitor its performance, each delivery is measured on the following scale: early (package delivered before the promised time), on-time (package delivered within 5 minutes of the promised time), late (package delivered more than 5 minutes past the promised time), lost (package never delivered). Speedy Swift’s objective is to deliver 99% of all packages either early or on-time. Another objective is to never lose a

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package. Speedy collected the following data for last month’s performance:

a. What scale is used to measure delivery performance? What kind of variable is delivery performance?

b. Construct a frequency table for delivery performance for last month.

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PRINTED BY: [email protected] Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. c. Construct a relative frequency table for delivery performance last month.

d. Construct a bar chart of the frequency table for delivery performance for last month.

e. Construct a pie chart of on-time delivery performance for last month.

f. Analyze the data summaries and write an evaluation of last month’s delivery performance as it relates to Speedy Swift’s performance objectives. Write a general recommendation for further analysis.

27. A data set consists of 83 observations. How many classes would you recommend for a frequency distribution?

28. A data set consists of 145 observations that range from 56 to 490. What size class interval would you recommend?

For DATA FILE, please visit www.mhhe.com/lind16e

29. The following is the number of minutes to commute from home to work for a group of 25 automobile executives.

a. How many classes would you recommend?

b. What class interval would you suggest?

c. What would you recommend as the lower limit of the first class?

d. Organize the data into a frequency distribution.

e. Comment on the shape of the frequency distribution.

For DATA FILE, please visit www.mhhe.com/lind16e

30. The following data give the weekly amounts spent on groceries for a sample of 45 households.

a. How many classes would you recommend?

b. What class interval would you suggest?

c. What would you recommend as the lower limit of the first class?

d. Organize the data into a frequency distribution.

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For DATA FILE, please visit www.mhhe.com/lind16e

31. A social scientist is studying the use of iPods by college students. A sample of 45 students revealed they played the following number of songs yesterday.

Organize the above information into a frequency distribution.

a. How many classes would you suggest?

b. What is the most suitable class interval?

c. What is the lower limit of the initial class?

d. Create the frequency distribution.

e. Describe the profile of the distribution.

For DATA FILE, please visit www.mhhe.com/lind16e

32. David Wise handles his own investment portfolio, and has done so for many years. Listed below is the holding time (recorded to the nearest whole year) between purchase and sale for his collection of 36 stocks.

a. How many classes would you propose?

b. What class interval would you suggest?

c. What quantity would you use for the lower limit of the initial class?

d. Using your responses to parts (a), (b), and (c), create a frequency distribution.

e. Identify the appearance of the frequency distribution.

For DATA FILE, please visit www.mhhe.com/lind16e

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