Use your answers to Exercise 5.1 to

a. Compute a \(95 \%\) interval estimate for \(\beta_{2}\).

b. Test the hypothesis \(H_{0}: \beta_{2}=1.25\) against the alternative that \(H_{1}: \beta_{2} eq 1.25\).

Data From Exercise 5.1:-

Consider the multiple regression model

with the seven observations on \(y_{i}, x_{i 1}, x_{i 2}\), and \(x_{i 3}\) given in Table 5.5.

Use a hand calculator or spreadsheet to answer the following questions:

a. Calculate the observations in terms of deviations from their means. That is, find \(x_{i 2}^{*}=x_{i 2}-\bar{x}_{2}\), \(x_{i 3}^{*}=x_{i 3}-\bar{x}_{3}\), and \(y_{i}^{*}=y_{i}-\bar{y}\).

b. Calculate \(\sum y_{i}^{*} x_{i 2}^{*}, \sum x_{i 2}^{* 2}, \sum y_{i}^{*} x_{i 3}^{*}, \sum x_{i 2}^{*} x_{i 3}^{*}\), and \(\sum x_{i 3}^{* 2}\).

c. Use the expressions in Appendix \(5 \mathrm{~A}\) to find least squares estimates \(b_{1}, b_{2}\), and \(b_{3}\).

d. Find the least squares residuals \(\hat{e}_{1}, \hat{e}_{2}, \ldots, \hat{e}_{7}\).

e. Find the variance estimate \(\hat{\sigma}^{2}\).

f. Find the sample correlation between \(x_{2}\) and \(x_{3}\).

g. Find the standard error for \(b_{2}\).

h. Find \(S S E, S S T, S S R\), and \(R^{2}\).

Transcribed Image Text:

y = x+x2 + x 13 3 + ei