4.14. Refer to Grade point average Problem 1.19. Assume that linear regression through the origin model (4.10) is appropriate.

1. Fit regression model (4.10) and state the estimated regression function.
2. Estimate \(\beta_{1}\) with a 95 percent confidence interval. Interpret your interval estimate.
3. Estimate the mean frestman GPA for students whose ACT test score is 30 . Use a 95 percent confidence interval.

4.22. Derive an extension of the Bonferroni inequality (4.2a) for the case of three statements, each with statement confidence coefficient \(1-\alpha\).

6.5. Brand preference. In a small-scale experimental study of the relation between degree of brand liking \((Y)\) and moisture content \(\left(X_{1}\right)\) and sweetness \(\left(X_{2}\right)\) of the product, the following results were obtained from the experiment based on a completely randomized design (data are coded):

1. Obtain the scatter plot matrix and the correlation matrix. What information do these diagnostic aids provide here?
2. Fit regression model (6.1) to the data. State the estimated regression function. How is \(b_{1}\) interpreted here?
3. Obtain the residuals and prepare a box plot of the residuals. What information does this plot provide?
4. Plot the residuals against \(\hat{Y}, X_{1}, X_{2}\), and \(X_{1} X_{2}\) on separate graphs. Also prepare a normal probability plot. Interpret the plots and summarize your findings.

6.6. Refer to Brand preference Problem 6.5. Assume that regression mode1 (6.1) with independent normal error terms is appropriate.

1. Test whether there is a regression relation, using \(\alpha=.01\). State the alternatives, decision rale and conclusion. What does your test imply about \(\beta_{1}\) and \(\beta_{2}\) ?
2. What is the \(P\) -value of the test in part (a)?
3. Estimate \(\beta_{1}\) and \(\beta_{2}\) jointly by the Bonferroni procedure, using a 99 percent family confidence coefficient. Interpret your results.

6.7. Refer to Brand preference Problem 6.5

1. Calculate the coefficient of multiple determination \(R^{2}\). How is it interpreted here?
2. Calculate the coefficient of simple determination \(R^{2}\) between \(Y_{i}\) and \(\hat{Y}_{i}\). Does it equal the coefficient of multiple determination in part (a)?

6.8. Refer to Brand preference Problem 6.5. Assume that regression model (6.1) with independent normal error terms is appropriate.

1. Obtain an interval estimate of \(E\left\{Y_{k}\right\}\) when \(X_{h 1}=5\) and \(X_{h 2}=4\). Use a 99 percent confidence coefficient. Interpret your interval estimate.
2. Obtain a prediction interval for a new observation \(Y_{h(n c w)}\) when \(X_{h 1}=5\) and \(X_{h 2}=4\). Use a 99 percent confidence coefficient.

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