Chapter 15.4, Problem 10WE

To determine

To calculate: The estimate of the value of the automobile if its age was 2 years.

Answer to Problem 10WE

The estimate of the value of the automobile if its age was 2 yearswas $\$8850$ .

Explanation of Solution

Given information:

For a total of 50 automobilesthe correlation coefficient between their values and agesis $-0.85$ . The average values of automobiles is $\$8000$ and average age is 4 years. Standard deviation of values is $\$1000$ and for age is 2 years.

Formula used:

The correlation coefficient r is denoted as $r=\frac{M_{xy}-M_x\cdot M_y}{{\sigma }_x\cdot {\sigma }_y}$ , where $M_{xy}$ is the mean of the product of values of x and y , $M_x$ is the mean of x -values, $M_y$ is the mean of y -values, ${\sigma }_x$ is standard deviation of x -values, ${\sigma }_y$ is the standard deviation of y -values.

The regression line that relates x and y , contains the point $\left(M_x,M_y\right)$ has slope $r\left(\frac{{\sigma }_y}{{\sigma }_x}\right)$ and the equation is $y-M_y=r\left(\frac{{\sigma }_y}{{\sigma }_x}\right)\left(x-M_x\right)$ .

Calculation:

Consider the provided information that for a total of 50 automobiles the correlation coefficient between their values and ages is $-0.85$ . The average values of automobiles is $\$8000$ and average age is 4 years. Standard deviation of values is $\$1000$ and for age is 2 years.

Let y denote the value and x denote the age.

Here, $r=-0.85,M_x=4,M_y=8000,{\sigma }_x=2\text{ and }{\sigma }_y=1000$ .

Next, recall that the regression line that relates x and y , contains the point $\left(M_x,M_y\right)$ has slope $r\left(\frac{{\sigma }_y}{{\sigma }_x}\right)$ and the equation is $y-M_y=r\left(\frac{{\sigma }_y}{{\sigma }_x}\right)\left(x-M_x\right)$ .

Since, age of automobile is 2 yearsso $x=2$ .

Substitute the provided values in the expression provided above,

  $\begin{array}{c}y-8000=-0.85\left(\frac{1000}{2}\right)\left(2-4\right)\\ y-8000=850\\ y=8850\end{array}$

Therefore, approximately the value of $y\approx 8850$ .

Thus, the estimate of the value of the automobile if its age was 2 years was $\$8850$ .