Chapter 3, Problem 4CRE

(a)

To determine

To find: The equation of least square regression line.

Answer to Problem 4CRE

The equation is $\hat{y}=7288.84+11630.6x$

Explanation of Solution

Given information:

A scatter plot with least square regression and residual plot are given.

The regression equation is to determine as follows,

From the given information, the variables used in the context of the study are as follows:

Let  Y  denotes the mileage of cars, the dependent variable

Let  X  denotes the car age, the independent variable

The regression equation is as follows:

  $\hat{y}=7288.84+11630.6x$

Conclusion:

The equation is $\hat{y}=7288.84+11630.6x$

(b)

To determine

To find: The residual value.

Answer to Problem 4CRE

The residual value is −12,072.44.

Explanation of Solution

Given information:

The 6 year old car contains 65,000 miles on it.

Formula used:

Substitution method is used.

Calculation:

The residual value is obtained as below,

Given the values of the data point  x =6, y =65,000.

From the data point given, it is clear that the value of  x  is 6.

  $\begin{array}{l}\hat{y}=7288.84+11630.6x\\ =7288.84+11630.6\left(6\right)\\ =7288.84+69783.6\\ \hat{y}=77072.44\end{array}$

Let  e  denotes the residual value.

Therefore, the residual is obtained below:

  $\begin{array}{l}e=y-\hat{y}\\ =65,000-77072.44\\ e=-12,072.44\end{array}$

Thus, the residual value is −12,072.44.

Conclusion:

The residual value is −12,072.44.

(c)

To determine

To analyse: The slope of the line in the context.

Answer to Problem 4CRE

The one year increase in the age of the car gives the independent variable x and the dependent variable y is get to increase by 11630.6 miles.

Explanation of Solution

Given information:

A scatter plot with least square regression and residual plot are given.

Formula used:

It follows x and y variables.

Calculation:

Interpret the slope of the regression line.

The slope is interpreted as follows:

From the given information, the value of slope is 11630.6

Interpret:

That is, it is clear that for every one year increase in the age of car the independent variable (x), the dependent variable mileage of car (y) is expected to increase by 11630.6 miles.

Conclusion:

The one year increase in the age of the car gives the independent variable x and the dependent variable y is get to increase by 11630.6 miles.

(d)

To determine

To find: The correlation between the car age and mile age.

Answer to Problem 4CRE

The correlation is close to 1

Explanation of Solution

Given information:

The correlation between car age and mile age is to be interpreted.

Formula used:

  $r=\sqrt{r^2}$

Calculation:

From the output, the correlation is,

  $\begin{array}{l}r=\sqrt{r^2}\\ =\sqrt{0.82}\\ r=0.91\end{array}$

Here, the correlation is close to 1. Hence, there is a perfect positive relationship between car age and mileage.

Conclusion:

There is a perfect positive relationship between car age and mileage.

(e)

To determine

To find: The regression line that fit the data properly.

Answer to Problem 4CRE

The residuals are not approximately equally distributed above and below the horizontal axis.

Explanation of Solution

Given information:

The regression line that fit the data properly.

The properties of the residual plot are given below:

• A residual plot drawn against the predictor variables must be somewhat centred and symmetric about the horizontal axis.
• A residual plot drawn against the predicted values of the response variable must be somewhat cantered and symmetric about the horizontal axis.
• The normal probability plot of the residuals must be linear or close to linear.

From the graph of the residual versus the fitted values, it can be observed that the one horizontal axis about the symmetry is represented. It indicates that the variation among the residuals is increased. There is a slight difference between the values about the symmetry.

The residuals are not approximately equally distributed above and below the horizontal axis. Thus, the assumption of constancy of the conditional standard deviation is likely to be violated.

Conclusion:

The assumption of constancy of the conditional standard deviation is likely to be violated.