The following data resulted from an experiment to assess the potential of unburnt colliery spoil as a medium for plant growth. The variables are x=acid extractable cations and y=exchangeable acidity/total cation exchange capacity.

\begin{array}{l}\begin{array} { c c c c c c c c } \hline x & - 23 & - 5 & 16 & 26 & 30 & 38 & 52 \\\hline x & 1.50 & 1.46 & 1.32 & 1.17 & .96 & .78 & .77 \\\hline\end{array}\\\\\begin{array} { l l l l l l l } \hline x & 58 & 67 & 81 & 96 & 100 & 113 \\\hline x & .91 & .78 & .69 & .52 & .48 & .55 \\\hline\end{array}\end{array}

Standardizing the independent variable x to obtain

x ^ { t } = ( x - \bar { x } ) / s _ { x }

and fitting the regression function

y = \vec { \beta } _ { 0 } + \vec\beta _ { 1 } x ^ { t } + \vec { \beta } _ { 2 } ( x ^{t}) ^ { 2 }

yielded the accompanying computer output. $The following data resulted from an experiment to assess the potential of unburnt colliery spoil as a medium for plant growth. The variables are x=acid extractable cations and y=exchangeable acidity/total cation exchange capacity. \begin{array}{l} \begin{array} { c c c c c c c c } \hline x & - 23 & - 5 & 16 & 26 & 30 & 38 & 52 \\ \hline x & 1.50 & 1.46 & 1.32 & 1.17 & .96 & .78 & .77 \\ \hline \end{array}\\\\ \begin{array} { l l l l l l l } \hline x & 58 & 67 & 81 & 96 & 100 & 113 \\ \hline x & .91 & .78 & .69 & .52 & .48 & .55 \\ \hline \end{array} \end{array} Standardizing the independent variable x to obtain x ^ { t } = ( x - \bar { x } ) / s _ { x } and fitting the regression function y = \vec { \beta } _ { 0 } + \vec\beta _ { 1 } x ^ { t } + \vec { \beta } _ { 2 } ( x ^{t}) ^ { 2 } yielded the accompanying computer output. a. Estimate \mu _ { y ^ { 50 } } . b. Compute the value of the coefficient of multiple determination. c. What is the estimated regression function \hat { \beta } _ { 10 } + \hat { \beta } _ { 1 } x + \hat { \beta } _ { 2 } x ^ { 2 } using the unstandardized variable x? d. What is the estimated standard deviation of \hat { \beta } _ { 2 } computed in part ( c )? e. Carry out a test using the standardized estimates to decide whether the quadratic term should be retained in the model. Repeat using the unstandardized estimates. Do your conclusions differ?$
a. Estimate

\mu _ { y ^ { 50 } }

.
b. Compute the value of the coefficient of multiple determination.
c. What is the estimated regression function

\hat { \beta } _ { 10 } + \hat { \beta } _ { 1 } x + \hat { \beta } _ { 2 } x ^ { 2 }

using the unstandardized variable x?
d. What is the estimated standard deviation of

\hat { \beta } _ { 2 }

computed in part ( c )?
e. Carry out a test using the standardized estimates to decide whether the quadratic term should be retained in the model. Repeat using the unstandardized estimates. Do your conclusions differ?

When the Numbers of Predictors Is Too Large to Allow

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