Chapter 8, Problem 33E

(a)

To determine

To explain is a regression analysis appropriate.

Answer to Problem 33E

Yes, it is appropriate.

Explanation of Solution

The scatterplot of the Housing cost index (HCI) and Median family income (MFI) is given in the question. Yes, regression analysis is appropriate because the relationship is straight enough with a few outliers. And the spread of the data increases a bit for states with large median incomes but we can still fit a regression line.

(b)

To determine

To find out what is the equation that predicts housing cost index from median family income.

Answer to Problem 33E

  $\text{<mover accent="true"><mo>H</mo><mo>^</mo></mover> CI}=-156.50+0.0107\text{MFI}$ .

Explanation of Solution

The scatterplot of the Housing cost index (HCI) and Median family income (MFI) is given in the question. It is also given that:

  $\begin{array}{l}r=0.65\\ {\mu }_1=338.2\\ {\mu }_2=46234\\ {\sigma }_1=116.55\\ {\sigma }_2=7072.47\end{array}$

Thus, now we will calculate the slope and the constant for the regression equation as:

  $\begin{array}{l}\beta =r\times \frac{{\sigma }_1}{{\sigma }_2}\\ =0.65\times \frac{116.55}{7072.47}\\ =0.0107\\ \alpha ={\mu }_1-\beta \times {\mu }_2\\ =338.2-0.0107\times 46234\\ =-156.50\end{array}$

Thus, the regression equation will be as:

  $\begin{array}{l}\text{<mover accent="true"><mo>H</mo><mo>^</mo></mover> CI}=\alpha +\beta \text{MFI}\\ =-156.50+0.0107\text{MFI}\end{array}$

(c)

To determine

To find out what would be the predicted HCI.

Answer to Problem 33E

HCI= $324.93$ .

Explanation of Solution

The scatterplot of the Housing cost index (HCI) and Median family income (MFI) is given in the question. It is also given that:

  $\begin{array}{l}r=0.65\\ {\mu }_1=338.2\\ {\mu }_2=46234\\ {\sigma }_1=116.55\\ {\sigma }_2=7072.47\end{array}$

Now we have to predict the value of HCI when $\text{MFI}=44993$ , then it is calculated as:

  $\begin{array}{l}\text{<mover accent="true"><mo>H</mo><mo>^</mo></mover> CI}=\alpha +\beta \text{MFI}\\ =-156.50+0.0107\text{MFI}\\ =-156.50+0.0107\times 44993\\ =324.93\end{array}$

Thus, the value of HCI is $324.93$ .

(d)

To determine

To explain how far off is the prediction in part (b) from the actual HCI.

Answer to Problem 33E

The difference is $223.09$ .

Explanation of Solution

The scatterplot of the Housing cost index (HCI) and Median family income (MFI) is given in the question. It is also given that:

  $\begin{array}{l}r=0.65\\ {\mu }_1=338.2\\ {\mu }_2=46234\\ {\sigma }_1=116.55\\ {\sigma }_2=7072.47\end{array}$

Now we have to predict the value of HCI when $\text{MFI}=44993$ , then it is calculated as:

  $\begin{array}{l}\text{<mover accent="true"><mo>H</mo><mo>^</mo></mover> CI}=\alpha +\beta \text{MFI}\\ =-156.50+0.0107\text{MFI}\\ =-156.50+0.0107\times 44993\\ =324.93\end{array}$

The actual value of HCI is given as $548.02$ .

Thus, the difference of the predicted value and the actual value is:

  $\begin{array}{l}=\text{ Actual}-\text{ Predicted}\\ =548.02-324.93\\ =223.09\end{array}$

(e)

To determine

To explain what would be the regression equation that predicts standardized HCI from standardized MFI.

Answer to Problem 33E

  ${\widehat{z}}_{\text{HCI}}=0.65z_{\text{MFI}}$ .

Explanation of Solution

The scatterplot of the Housing cost index (HCI) and Median family income (MFI) is given in the question. It is also given that:

  $\begin{array}{l}r=0.65\\ {\mu }_1=338.2\\ {\mu }_2=46234\\ {\sigma }_1=116.55\\ {\sigma }_2=7072.47\end{array}$

Thus, if we standardized both the variables that is HCI and MFI, then the regression equation that predicts standardized HCI from standardized MFI will be as:

  ${\widehat{z}}_{\text{HCI}}=0.65z_{\text{MFI}}$

(f)

To determine

To explain what would be the regression equation that predicts standardized MFI from standardized HCI.

Answer to Problem 33E

  ${\widehat{z}}_{\text{MFI}}=0.65z_{\text{HCI}}$ .

Explanation of Solution

The scatterplot of the Housing cost index (HCI) and Median family income (MFI) is given in the question. It is also given that:

  $\begin{array}{l}r=0.65\\ {\mu }_1=338.2\\ {\mu }_2=46234\\ {\sigma }_1=116.55\\ {\sigma }_2=7072.47\end{array}$

Thus, if we standardized both the variables that is HCI and MFI, then the regression equation that predicts standardized MFI from standardized HCI will be as:

  ${\widehat{z}}_{\text{MFI}}=0.65z_{\text{HCI}}$