Exercise 5

 

  1. Set up the X matrix and Beta vector for each of the following models (assume

    2. i = 1,…,4):

    a. Yi = Beta0 + Beta1 * Xi1 + Beta2 * Xi1 * Xi2 + Epciloni

    b. logYi = Beta0 + Beta1 * Xi1 + Beta2 * Xi2 + Epciloni

     

     

  2. In a small-scale study of the relation between degree of brand liking (Y) and

moisture content (X1) and sweetness (X2) of the product, the following results

were obtained (data are coded):

 

i

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

Xi1

4

4

4

4

6

6

6

6

8

8

8

8

10

10

10

10

Xi2

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

Yi

64

73

61

76

72

80

71

83

83

89

86

93

88

95

94

100

 

 

Assume that the first order linear regression model with independent normal error terms is appropriate (i.e. the model with two independent X's and no interaction).

    1. Find the estimated regression coefficients. State the estimated regression

      2. function. How is b1 interpreted here?

    2. Test whether there is a regression relation using a level of significance of 0.01. State the alternatives, decision rule, and conclusion. What does your test imply about Beta1 and Beta2 ?

    3. What is the P-value of the test in part (b)?

    4. Estimate Beta1 and Beta2 jointly by the Bonferroni procedure using a 99 percent family confidence coefficient. Interpret your results.

 

 

  1. Refer to the brand problem (problem 2 Exercise 5)

    1. Calculate the coefficient of the multiple determination R2 . How is it

      2. intepretated here?

    2. Calculate the coefficient of the simple determination r2 between Yi

    and Yhati . Does it equal R2 ?

     

     

     

  2. Refer to the brand problem (problem 2 Exercise 5)

    1. Obtain an interval estimate of E(Yh) when Xh1 = 5 and Xh2 = 4. Use a 99 percent confidence coefficient. Interpret your interval estimate.

    2. Obtain a prediction interval for a new observation Yh(new) when

    Xh1 = 5 and Xh2 = 4. Use a 99 percent confidence coefficient.

     

     

  3. Refer to the brand problem (problem 2 Exercise 5)

    1. Obtain the residuals.

    2. Plot the residuals against Yhat, X1, and X2 on the separate graphs. Also prepare a normal probability plot . Analyze the plots and summarize your findings.

    3. Conduct a formal test for lack of fit of the first-order regression function; use Alfa = 0.01. State the alternatives, decision rule, and conclusion.