Chapter 8, Problem 34E

(a)

To determine

To explain is a regression model appropriate for predicting mortgage amount from interest rates.

Answer to Problem 34E

Yes, it is appropriate.

Explanation of Solution

In the question the plot of mortgage amount versus the interest rates at various times is given. Also, it is mentioned that:

  $\begin{array}{l}r=-0.84\\ {\mu }_1=151.9\\ {\mu }_2=8.88\\ {\sigma }_1=23.86\\ {\sigma }_2=2.58\end{array}$

Thus, we can say that the regression model is appropriate for predicting mortgage amount from interest rates because both interest rate and total mortgage are quantitative and the scatterplot looks fairly straight. The spread is fairly constant and there are no outliers.

(b)

To determine

To find out what is the equation that predicts mortgage amount from interest rates.

Answer to Problem 34E

  $\text{<mover accent="true"><mo>M</mo><mo>^</mo></mover> ortgage}=220.88-7.768\text{Interest}$ .

Explanation of Solution

In the question the plot of mortgage amount versus the interest rates at various times is given. Also, it is mentioned that:

  $\begin{array}{l}r=-0.84\\ {\mu }_1=151.9\\ {\mu }_2=8.88\\ {\sigma }_1=23.86\\ {\sigma }_2=2.58\end{array}$

Thus, the equation that predicts mortgage amount from interest rates can be calculated as:

  $\begin{array}{l}\beta =r\times \frac{{\sigma }_1}{{\sigma }_2}\\ =-0.84\times \frac{23.86}{2.58}\\ =-7.786\\ \alpha ={\mu }_1-\beta \times {\mu }_2\\ =151.9+7.768\times 8.88\\ =220.88\end{array}$

And therefore the regression equation will be:

  $\begin{array}{l}\text{<mover accent="true"><mo>M</mo><mo>^</mo></mover> ortgage}=\alpha +\beta (\text{Interest)}\\ =220.88-7.768\text{Interest}\end{array}$

(c)

To determine

To find out what would you predict the mortgage amount would be if the interest rates climbed to $20\%$ .

Answer to Problem 34E

The mortgage amount will be of $ $65.52$ million.

Explanation of Solution

In the question the plot of mortgage amount versus the interest rates at various times is given. Also, it is mentioned that:

  $\begin{array}{l}r=-0.84\\ {\mu }_1=151.9\\ {\mu }_2=8.88\\ {\sigma }_1=23.86\\ {\sigma }_2=2.58\end{array}$

And therefore the regression equation will be:

  $\begin{array}{l}\text{<mover accent="true"><mo>M</mo><mo>^</mo></mover> ortgage}=\alpha +\beta (\text{Interest)}\\ =220.88-7.768\text{Interest}\end{array}$

Thus, the mortgage amount would be if the interest rates climbed to $20\%$ can be calculated as:

  $\begin{array}{l}\text{<mover accent="true"><mo>M</mo><mo>^</mo></mover> ortgage}=\alpha +\beta (\text{Interest)}\\ =220.88-7.768\text{Interest}\\ =220.88-7.768\times 20\\ =65.52\end{array}$

Thus, with the interest rate of $20\%$ , we would predict a mortgage amount of $ $65.52$ million.

(d)

To determine

To explain do you have any reservations about your prediction in part (c).

Answer to Problem 34E

No, we do not have reservations.

Explanation of Solution

In the question the plot of mortgage amount versus the interest rates at various times is given. Also, it is mentioned that:

  $\begin{array}{l}r=-0.84\\ {\mu }_1=151.9\\ {\mu }_2=8.88\\ {\sigma }_1=23.86\\ {\sigma }_2=2.58\end{array}$

And therefore the regression equation will be:

  $\begin{array}{l}\text{<mover accent="true"><mo>M</mo><mo>^</mo></mover> ortgage}=\alpha +\beta (\text{Interest)}\\ =220.88-7.768\text{Interest}\end{array}$

Thus, we do not have any reservations about prediction in part (c) because we are extrapolating beyond the range of our data and cannot be sure that the linear relationship still remains.

(e)

To determine

To find out what would be the regression equation that predicts standardized mortgage amount from standardized interest rates.

Answer to Problem 34E

  ${\widehat{z}}_{\text{Mortgage}}=-0.84z_{\text{interest}}$ .

Explanation of Solution

In the question the plot of mortgage amount versus the interest rates at various times is given. Also, it is mentioned that:

  $\begin{array}{l}r=-0.84\\ {\mu }_1=151.9\\ {\mu }_2=8.88\\ {\sigma }_1=23.86\\ {\sigma }_2=2.58\end{array}$

Thus, if we standardized both the variables that is, mortgage amount and the interest rates, then the regression equation that predicts standardized mortgage amount from standardized interest rates will be as:

  ${\widehat{z}}_{\text{Mortgage}}=-0.84z_{\text{interest}}$

(f)

To determine

To find out what would be the regression equation that predicts standardized interest rates from standardized mortgage amount.

Answer to Problem 34E

  ${\widehat{z}}_{\text{Interest}}=-0.84z_{\text{Mortgage}}$ .

Explanation of Solution

In the question the plot of mortgage amount versus the interest rates at various times is given. Also, it is mentioned that:

  $\begin{array}{l}r=-0.84\\ {\mu }_1=151.9\\ {\mu }_2=8.88\\ {\sigma }_1=23.86\\ {\sigma }_2=2.58\end{array}$

Thus, if we standardized both the variables that is, mortgage amount and the interest rates, then the regression equation that predicts standardized interest rates from standardized mortgage amount will be as:

  ${\widehat{z}}_{\text{Interest}}=-0.84z_{\text{Mortgage}}$