Exercise 7

 

1. The Tri-City Office Equipment Corporation sells an imported desk calculator on a franchise basis and performs preventive maintenance and repair service on this calculator. The users of the deck calculators are either training institutions that use a student model, or business firms that employ a commercial model. The data below have been collected from 18 recent calls on users to perform routine preventive maintenance service; for each call, X1 is the number of machines serviced, X2 is the type of calculator model ( S – student, C – commercial ) , and Y is the total number of minutes spent by the service person. An analyst at Tri-City wishes to fit a regression model including both X1 and X2 as independent variables and estimate the effect of calculator model on Y.

 

 

I

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

X1i

7

6

5

1

5

4

7

3

4

2

8

5

2

5

7

1

4

5

X2i

C

S

C

C

C

S

S

C

C

C

C

S

S

C

S

C

C

C

Y

97

86

78

10

75

62

101

39

53

33

118

65

25

71

105

17

49

68

 

Assume that the first-order regression model is appropriate, and let X2 = 1 if student model and 0 if commercial model.

  1. Explain the meaning of all regression coefficients in the model.

  2. Fit the regression model and state the estimated regression function.

  3. Estimate the effect of calculator model on mean service time with a 95 percent

    4. confidence interval. Interpret your interval estimate.

  4. Why would the analyst wish to include the number of calculators variable (X1)

    5. in the model when interest is in estimating the effect of type of calculator

    model on service time?

  5. Obtain the residuals and plot them against X1*X2 . Is there any indication that

    6. an interaction term in the model would be helpful?

  6. Fit the regression model including X1, X2, and their interaction X1* X2 , and

test whether the interaction term can be dropped from the model; control the

Alfa risk at 0.10. State the alternatives, decision rule, and conclusion. If the

interaction term cannot be dropped from the model, describe the nature of the

interaction effect.

 

 

 

2. A marketing research trainee in the national office of a chain of shoe stores used

the following response function to study seasonal (winter, spring, summer, fall) effects on sales of a certain line of shoes: E(Y) = Beta0 + Beta1*X1 + Beta2*X2 + Beta3*X3

The X’s are indicator variables defined as follows: X1 = 1 if winter and 0 otherwise, X2 = 1 if spring and 0 otherwise, X3 = 1 if fall and 0 otherwise. After fitting the model, she tested the regression coefficients Betaj ( j = 0,…,3 ) and came to the following set of conclusions at an 0.05 family level of significance:

Beta0 is not 0, Beta1 is 0, Beta2 is not 0, Beta3 is not 0.

In her report she then wrote: “Results of regression analysis show that climatic and other seasonal factors have no influence in determining sales of this shoe line in the winter. Seasonal influences do exist in the other seasons.” Do you agree with this interpretation of the test results?

 

 

 

3. Refer to Muscle Mass problem (problem 2 Exercise 1). The results are in file

MUSCLE and a brief description of the variables is in the file MUSCLEd .

The nutritionist conjectures that the regression of muscle mass on age follows a

two-piece linear relation, with the slope changing at age 60 without discontinuity.

  1. State the regression model that applies if the nutritionist’s conjecture is correct. What are the respective response functions when age is 60 or less and when age is over 60?

  2. Fit the regression model specified in part (a) and state the estimated regression function.

  3. Test whether a two-piece linear regression function is needed; use Alfa = 0.10. State the alternatives, decision rule, and conclusion. What is the P-value of the test?