# Applications of statistical methods | Mathematics homework help

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.

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