1. State whether each of the following statements is true or false. For the statements that are false, explain why they are false.
(a) A clear linear trend in a scatterplot with residuals on the vertical axis and fitted values on the horizontal axis suggests that the multiple linear regression linearity assumption is likely satisfied.
(b) A small p-value associated with the Anderson-Darling test for error normality indicates that the regression errors are normally distributed.

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 and fall) effects on sales of a certain line of shoes: E(Y) = Î²0 + Î²1X1 + Î²2X2+ Î²3X3. The Xs 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, the trainee tested the regression coefficients Î²k (k=0,1,2,3) and come to the following set of conclusions at a 0.05 level of significance: Î²0 â‰ 0, Î²1 = 0, Î²2 â‰ 0 and Î²3 â‰ 0. In the report the trainee then wrote: â€œResults 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.â€ Do you agree with this interpretation of the test results? Discuss.

3. Suppose we have a data set with five predictors, X1= GPA, X2= IQ, X3 = Gender (1 for Female and 0 for Male), X4= Interaction between GPA and IQ, and X5 = Interaction between GPA and Gender. The response variable is starting salary after graduation (in thousands of dollars). Suppose we use least squares to fit the model, and get b0 = 50, b1= 20, b2= 0.07, b3 = 35, b4 = 0.01 and b5= âŽ¼10.
(a) Predict the salary (in thousands of dollars) of a female with a GPA of 4.0 and an IQ of 110.
(b) Only one of the following statements is true. Say which and justify your answer briefly.
i. For fixed values of GPA and IQ, males earn more on average than females.
ii. For fixed values of GPA and IQ, females earn more on average than males.
iii. For fixed values of GPA and IQ, males earn more on average than females provided that their GPA is higher than 3.5.
iv. For fixed values of GPA and IQ, females earn more on average than males provided that their GPA is higher than 3.5.
(c) Do you agree with the following statement? Briefly justify your answer.
â€œSince the estimated coefficient for the GPA/IQ interaction term is very small, there is very little evidence of an interaction effect.â€

4. Three different drug treatments are compared in a medical experiment. Each treatment is used on 30 patients (90 patients in total). The response variable is Y = blood pressure. A useful predictor is X1 = body mass index. Define the indicator variables:
X2 = 1 if Treatment 2, 0 otherwise
X3 = 1 if Treatment 3, 0 otherwise.
Treatment 1 is a placebo. Consider the following population regression equation for the mean of Y: E(Y) = Î²0 + Î²1X1 + Î²2X2 + Î²3X3.
a) Substitute 0s and 1s into the regression equation above to determine equations for E(Y) for Treatment 1, Treatment 2, and Treatment 3 separately.
b) Use the answers to part (a) to explain what each of these regression coefficients measures:
i. Î²1
ii. Î²2
iii. Î²3
c) Suppose that we wish to test H0: Î²2 = Î²3 = 0. In the context of this situation. Explain what it would mean if we fail to reject this null hypothesis at a selected significance level. Explain what it would mean if we reject this null hypothesis.
d) Briefly explain how you would carry out the test of the null hypothesis given in the previous part. Describe the full and reduced models you would use for the test, the formula of the F test statistic, and the actual values of the numerator and denominator degrees of freedom for the F-distribution.
e) Suppose that we think there might be an interaction between treatments and the variable X1 (body mass index). Write the appropriate interaction model equation for E(Y).
f) For the interaction model that you wrote in the previous part, consider testing whether there is an interaction between treatments and body mass index. State the null hypothesis for this test.

1. State whether each of the following statements is true or false....