Chapter 8, Problem 43E

(a)

To determine

To find the equation of the regression line.

Answer to Problem 43E

  $\text{<mover accent="true"><mo>P</mo><mo>^</mo></mover> rice}=14286-959(\text{Age)}$ .

Explanation of Solution

In the question it shows the relationship between the age and the price of the cars of used Corolla. In the above scatterplot we can see that the points are in the downward direction. Thus, we can say that the relationship between the age and the price is negative. Thus, it is given that $R^2=94.4\%$ . So, we have,

  $\begin{array}{l}{r}^2=R^2\\ \Rightarrow r=-\sqrt{0.944}\\ =-0.972\end{array}$

Also, we have given the information:

Age	Price
1	13990
1	13495
3	12999
4	9500
4	10495
5	8995
5	9495
6	6999
7	6950
7	7850
8	6999
8	5995
10	4950
10	4495
13	2850

Thus, we will calculate the mean and standard deviation of the age and price by using excel as,

Formula used:

The formula for mean is as:

  $\text{AVERAGE}\left(\text{number1,}\left[\text{number2}\right]\text{, }\mathrm{...}\right)$

And the formula for the standard deviation is as:

  $\text{STDEV}\left(\text{number1,}\left[\text{number2}\right]\text{,}\mathrm{...}\right]\text{)}$

Calculation:

The mean and standard deviation is calculated as:

For the mean,

	Age	Price
Mean=	=AVERAGE(I8:I22)	=AVERAGE(J8:J22)

For the standard deviation,

	Age	Price
Standard deviation =	=STDEV(I8:I22)	=STDEV(J8:J22)

And the result of the following is as:

	Age	Price
Mean=	6.13	8,403.80

	Age	Price
Standard deviation =	3.38	3,333.58

Thus, we have,

  $\begin{array}{l}r=-0.972\\ {\mu }_1=8403.80\\ {\mu }_2=6.13\\ {\sigma }_1=3333.58\\ {\sigma }_2=3.38\end{array}$

Thus, the regression line will be calculated as:

  $\begin{array}{l}\beta =r\times \frac{{\sigma }_1}{{\sigma }_2}\\ =-0.972\times \frac{3333.58}{3.38}\\ =-959\\ \alpha ={\mu }_1-\beta \times {\mu }_2\\ =8403.80+959\times 6.13\\ =14286\end{array}$

And the regression line is as:

  $\begin{array}{l}\text{<mover accent="true"><mo>P</mo><mo>^</mo></mover> rice}=\alpha +\beta (\text{Age)}\\ =14286-959(\text{Age)}\end{array}$

(b)

To determine

To explain the meaning of the slope of the line.

Explanation of Solution

In the question it shows the relationship between the age and the price of the cars of used Corolla. In the above scatterplot we can see that the points are in the downward direction. Thus, we can say that the relationship between the age and the price is negative. Thus, it is given that,

  $\begin{array}{l}r=-0.972\\ {\mu }_1=8403.80\\ {\mu }_2=6.13\\ {\sigma }_1=3333.58\\ {\sigma }_2=3.38\end{array}$

And the regression line is as:

  $\begin{array}{l}\text{<mover accent="true"><mo>P</mo><mo>^</mo></mover> rice}=\alpha +\beta (\text{Age)}\\ =14286-959(\text{Age)}\end{array}$

Thus, the slope of the line means that every extra year of age decreases on average value by $\$959$ . As the graph in the scatterplot the points have a downward slope.

(c)

To determine

To explain the meaning of the y -intercept of the line.

Explanation of Solution

In the question it shows the relationship between the age and the price of the cars of used Corolla. In the above scatterplot we can see that the points are in the downward direction. Thus, we can say that the relationship between the age and the price is negative. Thus, it is given that,

  $\begin{array}{l}r=-0.972\\ {\mu }_1=8403.80\\ {\mu }_2=6.13\\ {\sigma }_1=3333.58\\ {\sigma }_2=3.38\end{array}$

And the regression line is as:

  $\begin{array}{l}\text{<mover accent="true"><mo>P</mo><mo>^</mo></mover> rice}=\alpha +\beta (\text{Age)}\\ =14286-959(\text{Age)}\end{array}$

Thus, the y -intercept means that the average new Corolla costs a predicted $\$14286$ when the age in years of the car used is zero, which is practically not possible, as it represents the $\alpha$ value of the regression line.

(d)

To determine

To find out what price seems appropriate if you want to sell a $7$ -year old Corolla.

Answer to Problem 43E

The price that seems appropriate if you want to sell a $7$ -year old Corolla is $\$7573$ .

Explanation of Solution

In the question it shows the relationship between the age and the price of the cars of used Corolla. In the above scatterplot we can see that the points are in the downward direction. Thus, we can say that the relationship between the age and the price is negative. Thus, it is given that,

  $\begin{array}{l}r=-0.972\\ {\mu }_1=8403.80\\ {\mu }_2=6.13\\ {\sigma }_1=3333.58\\ {\sigma }_2=3.38\end{array}$

And the regression line is as:

  $\begin{array}{l}\text{<mover accent="true"><mo>P</mo><mo>^</mo></mover> rice}=\alpha +\beta (\text{Age)}\\ =14286-959(\text{Age)}\end{array}$

Thus, the price that seems appropriate if you want to sell a $7$ -year old Corolla can be calculated as:

  $\begin{array}{l}\text{<mover accent="true"><mo>P</mo><mo>^</mo></mover> rice}=14286-959(\text{Age)}\\ =14286-959\times 7\\ =7573\end{array}$

The price that seems appropriate if you want to sell a $7$ -year old Corolla is $\$7573$ .

(e)

To determine

To explain would you rather buy the one with a positive residual or one with the negative residual.

Answer to Problem 43E

We would rather buy one with the negative residual.

Explanation of Solution

In the question it shows the relationship between the age and the price of the cars of used Corolla. In the above scatterplot we can see that the points are in the downward direction. Thus, we can say that the relationship between the age and the price is negative. Thus, it is given that,

  $\begin{array}{l}r=-0.972\\ {\mu }_1=8403.80\\ {\mu }_2=6.13\\ {\sigma }_1=3333.58\\ {\sigma }_2=3.38\end{array}$

And the regression line is as:

  $\begin{array}{l}\text{<mover accent="true"><mo>P</mo><mo>^</mo></mover> rice}=\alpha +\beta (\text{Age)}\\ =14286-959(\text{Age)}\end{array}$

Thus, we can say that we would rather buy the one with the negative residual because the price of Corolla is below the predicted value for its age in years.

(f)

To determine

To calculate what is the residual.

Answer to Problem 43E

The residual is $\$-1196$ .

Explanation of Solution

In the question it shows the relationship between the age and the price of the cars of used Corolla. In the above scatterplot we can see that the points are in the downward direction. Thus, we can say that the relationship between the age and the price is negative. Thus, it is given that,

  $\begin{array}{l}r=-0.972\\ {\mu }_1=8403.80\\ {\mu }_2=6.13\\ {\sigma }_1=3333.58\\ {\sigma }_2=3.38\end{array}$

And the regression line is as:

  $\begin{array}{l}\text{<mover accent="true"><mo>P</mo><mo>^</mo></mover> rice}=\alpha +\beta (\text{Age)}\\ =14286-959(\text{Age)}\end{array}$

It is given that a $10$ year old Corolla stating the asking price as $\$3500$ . Thus, the predicted price for a $10$ year old Corolla is calculated as:

  $\begin{array}{l}\text{<mover accent="true"><mo>P</mo><mo>^</mo></mover> rice}=14286-959(\text{Age)}\\ =14286-959\times 10\\ =4696\end{array}$

Thus, the residual is calculated as:

  $\begin{array}{l}=\text{Actual}-\text{ Predicted}\\ =3500-4696\\ =-1196\end{array}$

(g)

To determine

To explain would this regression model be useful in establishing a fair price for a $20$ year old model.

Answer to Problem 43E

No, we would not use this regression model.

Explanation of Solution

In the question it shows the relationship between the age and the price of the cars of used Corolla. In the above scatterplot we can see that the points are in the downward direction. Thus, we can say that the relationship between the age and the price is negative. Thus, it is given that,

  $\begin{array}{l}r=-0.972\\ {\mu }_1=8403.80\\ {\mu }_2=6.13\\ {\sigma }_1=3333.58\\ {\sigma }_2=3.38\end{array}$

And the regression line is as:

  $\begin{array}{l}\text{<mover accent="true"><mo>P</mo><mo>^</mo></mover> rice}=\alpha +\beta (\text{Age)}\\ =14286-959(\text{Age)}\end{array}$

Thus, we can say that we would not use this regression model in establishing a fair price for a $20$ year old model because after the age of $14$ years of the car, the model predicts negative prices and the relationship between the age and the price of the car is no longer linear.