Chapter 8, Problem 35E

(a)

To determine

To find out what is the linear regression equation for predicting total yearly purchase from age.

Answer to Problem 35E

  $\text{<mover accent="true"><mo>T</mo><mo>^</mo></mover> otal purchase}=539.803+1.103\text{Age}$ .

Explanation of Solution

In the question, an online clothing company retailer keeps track of its customers’ purchases and the scatter plot is given for the total yearly purchases versus the age. Also, it is mentioned in the question that:

  $\begin{array}{l}r=0.037\\ {\mu }_1=572.52\\ {\mu }_2=29.67\\ {\sigma }_1=253.62\\ {\sigma }_2=8.51\end{array}$

Thus, the linear regression equation for predicting total yearly purchase from age can be calculated as:

  $\begin{array}{l}\beta =r\times \frac{{\sigma }_1}{{\sigma }_2}\\ =0.037\times \frac{253.62}{8.51}\\ =1.103\\ \alpha ={\mu }_1-\beta \times {\mu }_2\\ =572.52-1.103\times 29.67\\ =539.803\end{array}$

Thus, the regression equation is as:

  $\begin{array}{l}\text{<mover accent="true"><mo>T</mo><mo>^</mo></mover> otal purchase}=\alpha +\beta (\text{Age)}\\ =539.803+1.103\text{Age}\end{array}$

(b)

To determine

To explain do the assumptions and conditions for regression appear to be met.

Answer to Problem 35E

Yes, they are met.

Explanation of Solution

In the question, an online clothing company retailer keeps track of its customers’ purchases and the scatter plot is given for the total yearly purchases versus the age. Also, it is mentioned in the question that:

  $\begin{array}{l}r=0.037\\ {\mu }_1=572.52\\ {\mu }_2=29.67\\ {\sigma }_1=253.62\\ {\sigma }_2=8.51\end{array}$

Thus, the regression equation is as:

  $\begin{array}{l}\text{<mover accent="true"><mo>T</mo><mo>^</mo></mover> otal purchase}=\alpha +\beta (\text{Age)}\\ =539.803+1.103\text{Age}\end{array}$

The assumptions and conditions for regression appear to be met as both the variables are quantitative and the plot appears to be straight. Moreover, there are no apparent outliers and the scatterplot does not appear to change spread throughout the range of age.

(c)

To determine

To find out what is the predicted average total yearly purchase for an $18$ year old and for a $50$ year old.

Answer to Problem 35E

The predicted average total yearly purchase for an $18$ year old is $ $559.65$ and for a $50$ year old is $ $594.94$ .

Explanation of Solution

In the question, an online clothing company retailer keeps track of its customers’ purchases and the scatter plot is given for the total yearly purchases versus the age. Also, it is mentioned in the question that:

  $\begin{array}{l}r=0.037\\ {\mu }_1=572.52\\ {\mu }_2=29.67\\ {\sigma }_1=253.62\\ {\sigma }_2=8.51\end{array}$

Thus, the regression equation is as:

  $\begin{array}{l}\text{<mover accent="true"><mo>T</mo><mo>^</mo></mover> otal purchase}=\alpha +\beta (\text{Age)}\\ =539.803+1.103\text{Age}\end{array}$

Now, the predicted average total yearly purchase for an $18$ year old can be calculated as:

  $\begin{array}{l}\text{<mover accent="true"><mo>T</mo><mo>^</mo></mover> otal purchase}=539.803+1.103\text{Age}\\ =539.803+1.103\times 18\\ =559.65\end{array}$

And the predicted average total yearly purchase for a $50$ year old can be calculated as:

  $\begin{array}{l}\text{<mover accent="true"><mo>T</mo><mo>^</mo></mover> otal purchase}=539.803+1.103\text{Age}\\ =539.803+1.103\times 50\\ =594.94\end{array}$

(d)

To determine

To explain what percent of the variability in total early purchases accounted for by this model.

Answer to Problem 35E

  $0.14\%$ variability in total early purchases accounted for by this model.

Explanation of Solution

In the question, an online clothing company retailer keeps track of its customers’ purchases and the scatter plot is given for the total yearly purchases versus the age. Also, it is mentioned in the question that:

  $\begin{array}{l}r=0.037\\ {\mu }_1=572.52\\ {\mu }_2=29.67\\ {\sigma }_1=253.62\\ {\sigma }_2=8.51\end{array}$

Thus, the regression equation is as:

  $\begin{array}{l}\text{<mover accent="true"><mo>T</mo><mo>^</mo></mover> otal purchase}=\alpha +\beta (\text{Age)}\\ =539.803+1.103\text{Age}\end{array}$

Thus, the percent of the variability in total early purchases accounted for by this model can be calculated by:

  $\begin{array}{l}{R}^2=r^2\\ ={(}^{0.037}\\ =0.0014\\ =0.14\%\end{array}$

Thus, we can say that $0.14\%$ variability in total early purchases accounted for by this model.

(e)

To determine

To explain do you think the regression might be useful one for the company.

Answer to Problem 35E

No, the regression might not be useful one for the company.

Explanation of Solution

In the question, an online clothing company retailer keeps track of its customers’ purchases and the scatter plot is given for the total yearly purchases versus the age. Also, it is mentioned in the question that:

  $\begin{array}{l}r=0.037\\ {\mu }_1=572.52\\ {\mu }_2=29.67\\ {\sigma }_1=253.62\\ {\sigma }_2=8.51\end{array}$

Thus, the regression equation is as:

  $\begin{array}{l}\text{<mover accent="true"><mo>T</mo><mo>^</mo></mover> otal purchase}=\alpha +\beta (\text{Age)}\\ =539.803+1.103\text{Age}\end{array}$

Thus, we think that the regression might not be useful one for the company because the plot is nearly flat and the model explains almost none of the variations in total yearly purchases.