For this lesson, we will be comparing two group population values

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Practice Problems:    Chapter 10

 

For this lesson, we will be comparing two group population values (or two paired observations for those in the population).   There are some subtle differences in the formulas used for computing the Standard Error used in both our Confidence Intervals of the Difference and in our Hypothesis Tests which are noted in the instructions below.

 

1)  We wish to compare the population proportion of those graduates who pass the Bar Exam on the first try for two small law schools.   The researcher has randomly selected graduates from the previous 4 years graduating classes and asked those selected if they passed the Bar Exam on the first try.   Law School A had 150 out of 200 sampled who indicated that they passed the Bar Exam on the first try.    Law School B had 170 out of 250 sampled who indicated that they passed the Bar Exam on the first try.    We wish to see if there is a difference between the population proportions of those who passed the Bar Exam on the first try for the two law schools (p1 is for School A, p2 is for School B).

 

A)  First, we want to construct a 95% Confidence Interval of the Difference between the two population proportions.    Here we use SE =

We use this SE formula for confidence intervals of the difference between population proportions.

 

i)                    Calculate your SE.   Show all work

ii)                   What is your Z multiplier?

iii)                 Construct your 95% Confidence Interval of the Difference.   Show all work and indicate the endpoints.

iv)                 Is there a difference in the two group population proportions? Why or why not?

B)Now, do a Hypothesis Test to see if there is a difference between the two population proportions.  

i)                    Since we are testing to see if there is a difference, our alternative is that there is a difference and our null is that there is no difference:

Ho:  p1-p2 = 0;   Ha:  p1 - p2 ≠ 0     Given that the null hypothesis is that there is no difference (ie: assuming equal population values), we use the pooled  to calculate our SE

=

 

Calculate and .   Show all work.

ii)                   Calculate our Z test statistic.  Show all work.

iii)                 What is the p-value of our Z test statistic?  Show all work.

iv)                 What is your conclusion and why?

2)  We wish to test whether a program teaching social skills to elementary students has an effect on social skills in third graders.   In one elementary school, we randomly assign three classes out of six classes of third graders to participate in the program.  The three classes who do not participate in the program act as the control group.   We are told that the composition of the six classes is similar in terms of gender composition, IQ, etc.      Every third grader is evaluated by their teacher and given a social skills assessment score at the beginning of the study.    Higher scores mean the student has a higher level of social skills.  At the end of the eight week program all students are then evaluated again by their teacher and given a social skills assessment score.   The “change in the social skills assessment score”(ending score - beginning score) is then compared for the two groups.     This is a pilot program so we have no idea as to whether there will be a difference one way or the other between the two groups (we don’t know if the program will improve scores more than the normal maturation process of being in school for the eight weeks the program is given).   Therefore, we simply test to see if there is a difference in the two group population means of score change.   The group that participated in the program (group 1) averaged a gain of 15 in their social skills assessment score.   The control group (group 2) averaged a gain of 14.is the sample mean change in score,  is the sample variance,   is the sample size

 

 

We can assume that if the program has no effect that both the population means will be equal and that the variability within groups will be the same.  The sample SD for group 1 is not more than twice that for group 2 , therefore, we can use the pooled SD.

df =

the pooled standard deviation, s =

SE =  s

 

A)     Construct a 95% Confidence Interval of the Difference between the two group population means.

i)                    Calculate your SE.  Show all work

ii)                   What is your t multiplier from the t Table?

iii)                 Construct your 95% Confidence Interval of the Difference.   Show all work and indicate the endpoints.

iv)                 Is there a difference in the two group population means?  Why or why not?

B)      Now, do a Hypothesis Test to see if there is a difference between the two population means.

i)                    What are the null and alternative hypotheses?  Use proper statistical notation (use proper symbols for population means)

ii)                   We use the same SE as we used for the Confidence Interval of the Difference in part A.   Calculate your t test statistic.  Show all work.

iii)                 Using the t table in the text.  Indicate the range of the p-value for your t test statistic.

iv)                 What is your conclusion and why?

v)                  Part iv is for a conclusion regarding statistical significance.  Is the difference between the groups of practical significance?  Assume a total possible score of 60.   Look at your CI of the Difference in part A to help answer this question.

3) A certain diet has been effective in past studies.  We want to see if a certain diet is effective for our group.    The duration of the study is 8 weeks. Study participants had their beginning weight measured and then were measured again 8 weeks after starting the diet.   The change in weight (beginning weight - ending weight) for each participant is the response variable.   To simplify our calculations, we will do our analysis on only 4 pairs of observations.   Note the difference between this and the scenario in Question 2.   In Question 2 we were comparing a change in scores between two independent groups and the Hypothesis Test we used was a 2-sample t-test.    In this problem we are comparing the change per person and not comparing between groups.  The Hypothesis Test we use is a Paired t-test.

Below are the observations taken on the four participants.

 

participant

beginning weight

ending weight

difference

(beg wt-end wt)

1

210

200

10

2

150

145

5

3

185

160

25

4

180

184

-4

 

A)     We test to see if the diet was effective.   If the diet was effective, then the difference between the beginning and ending weight would be positive since the ending weight would be less than the beginning weight.

 

i)                     What are the null and alternative hypotheses?   Use proper statistical notation for paired quantitative data (μd.)

ii)                   Find the sample mean difference, , the sample SD for the difference,

Show all work.

remember, n is the number of pairs

iii)                 Calculate the   Show all work.

iv)                 Calculate your t test statistic.  Show all work.

v)                  Using the t table in the text calculate your p-value range.  Remember,

DF = n-1  where n is the number of pairs.

vi)                 What is your conclusion and why?

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For this lesson, we will be comparing two group population values
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