Chapter 8, Problem 36E

(a)

To determine

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

Answer to Problem 36E

  $\text{<mover accent="true"><mo>T</mo><mo>^</mo></mover> otal purchase}=28.81+28.81(\text{Income)}$ .

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 income. Also, it is mentioned in the question that:

  $\begin{array}{l}r=0.722\\ {\mu }_1=572.52\\ {\mu }_2=50343.40\\ {\sigma }_1=253.62\\ {\sigma }_2=16952.50\end{array}$

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

  $\begin{array}{l}\beta =r\times \frac{{\sigma }_1}{{\sigma }_2}\\ =0.722\times \frac{253.62}{16952.50}\\ =0.0108\\ \alpha ={\mu }_1-\beta \times {\mu }_2\\ =572.52-0.0108\times 50343.40\\ =28.81\end{array}$

Thus, the regression equation is as:

  $\begin{array}{l}\text{<mover accent="true"><mo>T</mo><mo>^</mo></mover> otal purchase}=\alpha +\beta (\text{Income)}\\ =28.81+28.81(\text{Income)}\end{array}$

(b)

To determine

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

Answer to Problem 36E

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 income. Also, it is mentioned in the question that:

  $\begin{array}{l}r=0.722\\ {\mu }_1=572.52\\ {\mu }_2=50343.40\\ {\sigma }_1=253.62\\ {\sigma }_2=16952.50\end{array}$

Thus, the regression equation is as:

  $\begin{array}{l}\text{<mover accent="true"><mo>T</mo><mo>^</mo></mover> otal purchase}=\alpha +\beta (\text{Income)}\\ =28.81+28.81(\text{Income)}\end{array}$

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 someone with a yearly income of $\$20000$ and for someone with an annual income of $\$80000$ .

Answer to Problem 36E

The predicted average total yearly purchase for someone with a yearly income of $\$20000$ is $244.81$ and for someone with an annual income of $\$80000$ is $892.81$ .

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 income. Also, it is mentioned in the question that:

  $\begin{array}{l}r=0.722\\ {\mu }_1=572.52\\ {\mu }_2=50343.40\\ {\sigma }_1=253.62\\ {\sigma }_2=16952.50\end{array}$

Thus, the regression equation is as:

  $\begin{array}{l}\text{<mover accent="true"><mo>T</mo><mo>^</mo></mover> otal purchase}=\alpha +\beta (\text{Income)}\\ =28.81+28.81(\text{Income)}\end{array}$

Now, the predicted average total yearly purchase for someone with a yearly income of $\$20000$ can be calculated as:

  $\begin{array}{l}\text{<mover accent="true"><mo>T</mo><mo>^</mo></mover> otal purchase}=28.81+28.81(\text{Income)}\\ =28.81+28.81\times 20000\\ =244.81\end{array}$

And the predicted average total yearly purchase for someone with an annual income of $\$80000$ can be calculated as:

  $\begin{array}{l}\text{<mover accent="true"><mo>T</mo><mo>^</mo></mover> otal purchase}=28.81+28.81(\text{Income)}\\ =28.81+28.81\times 80000\\ =892.81\end{array}$

(d)

To determine

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

Answer to Problem 36E

  $52.13\%$ 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 income. Also, it is mentioned in the question that:

  $\begin{array}{l}r=0.722\\ {\mu }_1=572.52\\ {\mu }_2=50343.40\\ {\sigma }_1=253.62\\ {\sigma }_2=16952.50\end{array}$

Thus, the regression equation is as:

  $\begin{array}{l}\text{<mover accent="true"><mo>T</mo><mo>^</mo></mover> otal purchase}=\alpha +\beta (\text{Income)}\\ =28.81+28.81(\text{Income)}\end{array}$

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

  $\begin{array}{l}{R}^2=r^2\\ ={(}^{0.722}\\ =0.5213\\ =52.13\%\end{array}$

Thus, we can say that $52.13\%$ 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 36E

Yes, the regression might possibly 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 income. Also, it is mentioned in the question that:

  $\begin{array}{l}r=0.722\\ {\mu }_1=572.52\\ {\mu }_2=50343.40\\ {\sigma }_1=253.62\\ {\sigma }_2=16952.50\end{array}$

Thus, the regression equation is as:

  $\begin{array}{l}\text{<mover accent="true"><mo>T</mo><mo>^</mo></mover> otal purchase}=\alpha +\beta (\text{Income)}\\ =28.81+28.81(\text{Income)}\end{array}$

Thus, we think that the regression might possibly be useful one for the company because the value of $R^2$ is not extremely close to one and the line is still able to provide a general idea about the relationship.