Chapter 1.6, Problem 8E

a.

To determine

To graph: the data on the scatter plot.

Answer to Problem 8E

  

Explanation of Solution

Given information:

The table of values given is as follows:

Weight (100 pounds) (x)	Fuel economy (mpg) (y)
17.5	65.4
20.0	49.0
22.5	59.2
22.5	41.1
22.5	38.9
25.0	40.7
27.5	46.9
35.0	27.7
45.0	?

Calculation:

The graph for the given data is shown below.

  

b.

To determine

To find: best fit line using two ordered pairs.

Answer to Problem 8E

  $y=-3.29x+123.03$

Explanation of Solution

Given information:

The table of values given is as follows:

Weight (100 pounds) (x)	Fuel economy (mpg) (y)
17.5	65.4
20.0	49.0
22.5	59.2
22.5	41.1
22.5	38.9
25.0	40.7
27.5	46.9
35.0	27.7
45.0	?

Calculation:

Let’s select any two ordered pair like $\left(17.5,65.4\right)$ and $\left(25.0,40.7\right)$ . Plug in the values in the slope equation and solve m as follows:

  $\begin{array}{c}m=\frac{y_2-y_1}{x_2-x_1}\\ =\frac{40.7-65.4}{25.0-17.5}\\ =\frac{-24.7}{7.5}\\ =-3.29\end{array}$

Now, plug the value of m and any one point coordinate into the slope intercept equation and evaluate b as follows:

  $\begin{array}{c}y=mx+b\\ 40.7=\left(-3.29\right)\left(25\right)+b\\ 40.7=-82.25+b\\ b=40.7+82.25\\ =123.03\end{array}$

Now, plug the values of m and b in the slope intercept formula as follows:

  $\begin{array}{l}y=mx+b\\ y=-3.29x+123.03\end{array}$

Thus, the line equation is $y=-3.29x+123.03$ .

c.

To determine

To find: the equation for regression line and correlation value using the graphing calculator.

Answer to Problem 8E

  $\begin{array}{c} y=87.590588235294-1.7237647058824x\\ r=-0.77026683516305\end{array}$

Explanation of Solution

Given information:

The table of values given is as follows:

Weight (100 pounds) (x)	Fuel economy (mpg) (y)
17.5	65.4
20.0	49.0
22.5	59.2
22.5	41.1
22.5	38.9
25.0	40.7
27.5	46.9
35.0	27.7
45.0	?

Calculation:

The graph for the above equation is as follows:

  

Using the graphing calculator, the regression line equation is as follows:

  $y=87.590588235294-1.7237647058824x$

Using the graphing calculator, the correlation coefficient comes out to be, $r=-0.77026683516305$

d.

To determine

To find: the missing value if regression line equation has strong or moderate relationship, and explains whether the prediction is reliable.

Answer to Problem 8E

  $\approx 10$

Prediction is somewhat reliable.

Explanation of Solution

Given information:

The table of values given is as follows:

Weight (100 pounds) (x)	Fuel economy (mpg) (y)
17.5	65.4
20.0	49.0
22.5	59.2
22.5	41.1
22.5	38.9
25.0	40.7
27.5	46.9
35.0	27.7
45.0	?

  $\begin{array}{c} y=87.590588235294-1.7237647058824x\\ r=-0.77026683516305\end{array}$

Calculation:

The equation of regression line shows strong relationship because $0.75\le r\le 1$ .

Plugging $x=45.0$ into the regression formula and evaluate y as follows:

  $\begin{array}{c} y=87.590588235294-1.7237647058824x\\ =87.590588235294-1.7237647058824\left(45.0\right)\\ =87.590588235294-77.569411764708\\ =10.021176470586013\\ \approx 10\end{array}$

The fuel efficiency is affected by the weight of a car. The data somewhat reflects that the heaviest car has the least mpg and the lightest car has the most mpg. Since the correlation coefficient is on the low side of strong, the given data shows more of a moderately strong relationship. So, the predicted value is somewhat reliable.