Given below are a set of paired data points for \( X \) and \( Y \). \begin{tabular}{|l|l|l|l|l|l|l|l|l|} \hline \( \mathrm{X} \) & 12 & 15 & 15 & 18 & 21 & 23 & 24 & 25 \\ \hline \( \mathrm{Y} \) & 33 & 41 & 32 & 45 & 60 & 58 & 54 & 60 \\ \hline \end{tabular} (a) Find the regression line for the data. (b) Test at the \( 5 \% \) level if the slope is zero. (c) Find a 90\% confidence interval for the mean of \( Y \) when \( X=30 \).
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The data from exercise 2 follow. $$ \begin{array}{lllllll}{x_{i}} & {3} & {12} & {6} & {20} & {14}\end{array} $$ $$ \begin{array}{l|lllll}{55} & {40} & {55} & {10} & {15}\end{array} $$ $$ \begin{array}{l}{\text { a. Compute an estimate of the standard deviation of } \hat{y}^{* \prime} \text { when } x=8 \text { . }} \\ {\text { b. Develop a } 95 \% \text { confidence interval for the expected value of } y \text { when } x=8 \text { . }}\end{array} $$ $$ \begin{array}{l}{\text { c. Estimate the standard deviation of an individual value of } y \text { when } x=8 \text { . }} \\ {\text { d. Develop a } 95 \% \text { prediction interval for } y \text { when } x=8 \text { . }}\end{array} $$
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