Applications of Statistical Methods

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Question 1 – Investment Portfolio (12 marks)

Consider the daily percent change in the stock price of two companies, A and B, in an investment portfolio. The dataset is called Investment Portfolio.

Answer the following questions manually. Use statistical software or MS Excel for help with the computation of any summary statistics needed for manual computations.

a) Draw a scatterplot of the company A daily percent changes against the company B daily percent changes. Describe the relationship between daily percent changes that you see in this scatterplot.

b) Determine the regression equation to predict the daily percent change in the stock price of company A from the daily percent change in the stock price of company B. Interpret the value of the slope coefficient.

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c) Find the correlation between the percent changes. Does the correlation value support your description of the scatterplot in part a)?

d) Compute the corresponding coefficient of determination and interpret its value. In financial terms, it represents the proportion of non-diversifiable risk in company A.

e) Compute the 95% confidence interval for the slope coefficient. f) Test at the 5% significance level whether the slope coefficient is significantly different

from 1, representing the beta of a highly diversified portfolio. Don’t forget to show your computations.

Questions 2 – Location Analysis (38 marks)

Location analysis is an important decision in operations management of production and service industries. A critical decision for many organizations is where to locate a processing plant, warehouse, retail outlet, etc. A large number of business variables are typically considered in this decision problem.

The management of a large motel/inn chain is aware of the challenges in choosing new motel locations. The chain’s management uses the “operating margin,” which is the ratio of the sum of profit, depreciation, and interest expenses divided by total revenue, to make this type of decision. In general, the higher the “operating margin,” the greater the success of the motel/inn.

The chain’s management has collected data on 100 randomly selected of its current inns. By measuring the “operating margin,” the objective is to predict which sites would likely generate more profit. Below is a description of the different variables considered in this analysis.

Variable Description Location ID Number Location identifier

Operating Margin Operating margin, in percent

Number Number of motels, inns, and hotel rooms within 5 miles

Nearest Number of miles to the closest competitors

Enrollment Number of college and university enrollment (in thousands) in nearby college and universities

Income Average household income (in thousands) of the neighborhood

Distance Distance from downtown

Quality The quality of the service level of the location (1 = bad, 2 = average, 3 = good, 4 = excellent)

High Speed Internet High speed internet availability (1 = no, 2 = yes)

Gym Gym availability (1 = no, 2 = yes)

The dataset is called Location Analysis.

Part 1 (10 marks)

Using Minitab or any other statistical software, run a simple linear regression model to predict Operating Margin based on Distance and answer the following questions:

a) Using an appropriate graph, plot Operating Margin versus Distance and comment on the relationship between these two variables.

b) Write down your estimation of the regression equation for predicting Operating Margin from Distance. Draw the regression line on the plot in part a).

c) Assuming α = 0.01, test whether Distance has statistically significant predictive power in estimating Operating Margin. State the hypotheses, provide a test statistic and p-value, and state your conclusion. Show your calculations.

d) Interpret the values of the regression coefficients (slope and intercept).

Part 2 (6 marks)

Using Minitab or any other statistical software, now perform a multiple linear regression analysis of Operating Margin (response variable) against all the remaining variables as predictors, excluding Location ID Number.

a) Write down the regression equation and provide at least two summary measures of the fit of the model. Based on the summary measures, does the model provide a good fit for the data? Explain.

b) Plot the residuals against the fitted values and comment on whether the usual model conditions are met.

c) The variable Operating Margin New in the dataset corresponds to the Operating Margin variable from which some values have been recorded as missing values. Identify those missing values and explain what they are and why they were recorded as missing.

Part 3 (12 marks)

Using statistical software, run the same multiple linear regression model as in Part 2 above but this time using Operating Margin New as the response variable. Then, answer the following questions:

a) Briefly compare the resulting regression equation and fit with those obtained in Part 2. b) Plot the residuals against the fitted values and comment on whether the model complies

with the usual conditions for multiple linear regression. c) Provide an interpretation for the model intercept and for the regression coefficients

associated with variables Income and Distance. Is an interpretation of the model intercept appropriate in this case? Compare the value of the regression coefficient for Distance with the one obtained in Part 1 above and clearly explain any difference.

d) Do you see any justification for dropping any variable(s) from the model? Explain (hint: multicollinearity; the significance of predictors).

e) Run a final model using Operating Margin New as the response variable and including only the significant predictors (hint: those with a p-value ≤ 5%).

f) Test the overall significance of the final model in part e). Use a 1% significance level and follow all the steps for hypothesis testing indicated in the Instructions section.

Part 4 (10 marks)

Based on your final model in Part 3 above, answer the following questions:

a) Test the marginal contribution of Quality, assuming that the other variables in the model remain constant. Use a 1% significance level, and make sure you follow all the steps for hypothesis testing indicated in the Instructions section. Show the computation of the t- statistics (i.e., the ratio used to compute it).

b) Calculate the 99% prediction interval for the actual operating margin of a new location with the same characteristics as those for Location ID Number 3098 in the data file. Check if the prediction interval includes the actual operating margin associated with Location ID Number 3098 and explain why it does or does not.

c) Calculate the 99% confidence interval for the mean operating margin of a new location with the same characteristics as those for Location ID Number 3098 in the data file. Explain any difference between the size of this interval and the one in part b) above.

Location Analysis

  

Location ID Number


Operating Margin


Number


Nearest


Enrollment


Income


Distance


Quality


High Speed Internet


Gym


Operating Margin New

 

2801


39.3


3591


2.0


18.5


29


7.6


2


1


1


39.3

 

2608


49.5


2726


1.2


19.0


36


9.3


3


2


2


49.5

 

3191


49.0


2890


2.4


20.0


35


2.6


3


1


2


49.0

 

3210


45.1


2172


1.4


7.0


35


9.2


3


2


2


45.1

 

3701


41.9


3517


2.9


6.0


38


9.1


2


2


1


41.9

 

2547


43.5


2910


1.4


12.5


35


1.8


2


2


1


43.5

 

2670


40.3


2848


2.5


19.0


30


8.4


2


1


2


40.3

 

2664


42.8


3003


2.2


19.5


39


12.0


2


1


1


42.8

 

2492


57.3


2601


3.1


18.5


39


11.0


4


2


2


57.3

 

2649


52.1


3140


2.2


23.5


34


3.7


3


2


2


52.1

 

3066


25.1


1613


1.7


21.5


29


4.1


4


2


2

 

2193


54.4


3234


3.2


19.5


39


8.3


4


2


2


54.4

 

2793


47.3


2761


3.4


16.0


39


6.6


3


1


2


47.3

 

3620


57.4


2687


0.9


15.5


42


6.9


4


2


2


57.4

 

2538


38.4


3264


2.7


22.5


29


10.4


2


1


1


38.4

 

3730


56.5


2045


1.5


15.0


38


4.3


4


2


2


56.5

 

3120


35.2


3251


1.7


13.0


35


4.3


1


1


1


35.2

 

2389


39.3


3159


2.7


11.0


34


0.2


2


1


1


39.3

 

3053


47.3


2642


1.8


9.0


30


4.8


3


1


2


47.3

 

3304


44.2


3471


2.2


12.0


39


5.4


3


2


1


44.2

 

3770


54.4


2756


3.2


14.5


39


7.9


4


2


1


54.4

 

3380


51.5


3069


3.3


20.5


37


7.7


3


2


2


51.5

 

2795


46.2


2244


3.6


15.5


36


5.9


3


1


1


46.2

 

2315


49.8


2859


1.7


14.5


39


3.0


3


2


2


49.8

 

2682


41.1


3098


2.3


11.5


34


7.7


2


1


2


41.1

 

2789


51.1


3227


1.9


11.0


37


6.0


3


2


2


51.1

 

2847


52.2


2879


2.1


19.5


36


2.9


3


2


2


52.2

 

2372


49.9


3255


2.2


22.5


36


8.7


3


2


2


49.9

 

3349


39.8


3823


3.6


17.0


38


4.8


2


1


2


39.8

 

3704


43.5


3198


1.8


17.0


36


4.6


2


2


1


43.5

 

2872


45.4


2443


0.1


10.5


36


7.5


3


2


1


45.4

 

2570


58.5


2210


2.7


17.5


35


2.7


4


2


2


58.5

 

2956


46.0


2655


1.1


22.0


34


8.1


3


1


1


46.0

 

3098


81.5


4214


2.4


16.0


34


5.6


1


1


1

 

3256


41.0


3480


1.9


18.0


31


11.0


2


1


2


41.0

 

2486


46.0


2341


2.3


23.0


29


7.4


3


2


2


46.0

 

3466


48.8


2935


3.9


20.0


35


6.1


3


1


2


48.8

 

2912


53.5


3285


2.0


23.5


42


2.9


3


2


2


53.5

 

3200


54.1


2862


2.9


16.5


41


11.8


3


2


1


54.1

 

3208


34.3


3548


2.5


13.5


31


8.3


1


1


2


34.3

 

3799


50.2


3021


1.7


8.5


41


5.5


3


1


2


50.2

 

3450


43.3


3383


2.2


11.5


33


9.5


2


2


1


43.3

 

2403


40.8


2676


3.3


25.0


41


5.1


2


1


1


40.8

 

2555


49.0


2826


4.2


17.0


34


1.2


3


2


2


49.0

 

3056


52.0


3354


2.0


26.5


37


7.2


3


2


2


52.0

 

2331


51.2


3082


1.5


22.0


40


11.2


3


2


2


51.2

 

2806


52.7


3378


3.3


10.0


34


7.6


3


2


2


52.7

 

2802


35.7


3591


1.4


9.5


43


5.0


1


1


1


35.7

 

2501


40.0


3397


1.6


19.5


32


3.1


2


1


1


40.0

 

2636


60.1


2619


2.9


12.0


31


6.8


4


2


2


60.1

 

3016


48.2


2429


2.3


20.0


33


7.6


3


1


2


48.2

 

3581


36.8


3006


2.5


15.5


37


4.2


2


1


1


36.8

 

2395


55.4


2790


1.6


12.0


37


9.1


4


2


1


55.4

 

2365


33.8


2810


2.8


17.5


35


14.4


1


1


1


33.8

 

3152


51.2


2775


2.5


15.0


39


5.6


3


1


2


51.2

 

3071


42.9


3838


2.4


10.0


40


2.9


2


1


1


42.9

 

2698


36.1


2962


0.9


12.5


41


8.6


1


1


1


36.1

 

3026


47.5


3080


1.9


13.5


43


8.1


3


1


2


47.5

 

2785


33.2


2629


1.4


20.0


35


7.2


1


1


1


33.2

 

2973


54.2


2484


2.8


16.0


33


0.6


3


2


1


54.2

 

2785


32.4


3124


1.6


17.0


29


7.3


1


1


1


32.4

 

2834


54.1


2432


2.9


17.5


35


10.3


3


2


1


54.1

 

2146


49.0


1998


3.6


19.0


41


4.9


3


2


2


49.0

 

3625


46.3


2554


2.1


10.0


32


9.9


3


1


1


46.3

 

3328


41.6


2776


2.1


16.0


31


7.3


2


1


2


41.6

 

3358


43.2


2724


1.2


14.5


42


0.3


2


2


1


43.2

 

2230


54.4


3241


2.2


21.5


44


5.1


4


2


2


54.4

 

2662


42.5


2403


1.0


19.0


32


8.2


2


1


1


42.5

 

3736


45.0


3697


2.1


20.5


40


9.0


3


1


1


45.0

 

3227


33.5


3657


1.5


18.0


28


3.5


1


1


1


33.5

 

2718


35.9


2786


1.9


15.5


37


8.5


1


1


1


35.9

 

3675


38.1


3568


2.3


16.5


37


3.7


2


1


1


38.1

 

2960


44.0


2813


3.8


15.0


33


11.8


2


1


1


44.0

 

3405


71.2


3567


2.5


13.5


32


9.1


1


1


2

 

2793


37.8


2980


3.2


12.0


32


6.6


2


1


1


37.8

 

2675


43.7


2915


1.6


13.5


36


5.3


2


2


1


43.7

 

2874


40.2


3670


2.8


17.5


37


9.9


2


1


1


40.2

 

3467


55.5


3203


4.2


8.0


37


2.7


4


2


2


55.5

 

3593


44.9


3003


0.9


15.5


37


10.2


3


2


1


44.9

 

2596


38.8


3330


2.7


13.5


41


4.0


2


1


1


38.8

 

3341


52.6


3013


3.7


20.0


41


14.1


3


2


1


52.6

 

3479


53.0


2844


3.5


13.5


30


10.8


3


2


1


53.0

 

2656


58.0


2382


3.2


12.0


39


1.3


4


2


2


58.0

 

2617


54.2


3018


3.1


20.0


39


8.2


3


2


1


54.2

 

2114


60.5


2932


1.8


19.5


39


6.1


4


2


2


60.5

 

3287


47.6


2751


3.0


10.5


36


9.1


3


2


2


47.6

 

3531


36.9


2923


1.2


14.0


40


7.6


2


1


1


36.9

 

3562


34.5


2730


0.3


17.0


33


4.3


1


1


1


34.5

 

3648


30.8


3622


2.3


15.0


34


7.6


1


1


1


30.8

 

3569


49.0


3759


2.9


19.0


33


10.8


3


2


2


49.0

 

3038


45.5


2691


3.2


13.5


46


5.7


3


2


2


45.5

 

2663


51.4


2640


3.8


19.5


37


7.6


3


2


2


51.4

 

2971


49.6


3359


1.6


10.5


44


2.6


3


2


2


49.6

 

3116


31.9


3422


3.3


15.5


38


12.1


1


1


1


31.9

 

2546


53.4


2772


0.8


14.5


45


11.3


3


2


2


53.4

 

2118


48.8


2825


1.1


13.0


40


9.0


3


1


2


48.8

 

2117


39.2


2996


1.3


15.5


38


8.5


2


1


1


39.2

 

3794


43.4


3597


3.9


15.5


35


13.2


2


1


1


43.4

 

2910


41.0


2846


3.1


20.5


31


8.8


2


1


2


41.0

 

3504


49.6


2922


1.9


19.5


32


4.4


3


1


2


49.6

             

Investment Portfolio

  

Company A


Company B

 

1.31


0.38

 

-0.26


1.27

 

0.88


0.56

 

0.48


0.62

 

-0.49


-0.55

 

0.37


-1.24

 

-0.22


-1.87

 

1.34


1.30

 

0.21


-0.19

 

0.92


0.42

 

0.76


-1.06

 

-0.53


-1.07

 

-0.72


0.58

 

0.06


0.65

 

0.20


0.76

 

0.41


0.85

 

0.26


0.74

 

1.46


0.83

 

0.53


0.83

 

1.28


0.36

 

0.12


0.18

 

-0.52


-1.08

 

-0.73


-0.91

 

1.06


1.87

 

1.33


2.85

 

-0.18


-0.53

 

2.04


1.79

 

1.57


2.73

 

-0.50


-0.76

 

0.91


1.33

 

-0.25


-0.50

 

0.62


1.93

 

-2.04


-0.10

 

0.46


-0.95

 

1.69


0.40

 

0.11


0.18

 

-0.55


-2.59

 

1.13


1.18

 

0.91


-0.73

 

-1.19


-1.71

 

-1.33


-3.11

 

0.89


0.77

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