Chapter 11.3, Problem 9E

a.

To determine

Draw the data points on a graph paper.

Answer to Problem 9E

The plot of data points on graph paper is represented as follows:

Explanation of Solution

Step-by-step procedure to obtain the graph:

• Mark the values of x along the horizontal axis from 1.5 to 2.5 at intervals of 0.25 units.
• Mark the values of y along the vertical axis from 50 to 150 at intervals of 25 units.
• Plot the point for each value of x with the corresponding value of y.

Thus, the graph is obtained.

b.

To determine

Obtain the least-square line for the data.

Answer to Problem 9E

The least-square line for the data is $\widehat{y}=-15.452+65.172x$.

Explanation of Solution

Calculation:

The formula for the mean of y is $\overline{y}=\frac{1}{n}{\displaystyle \sum _{i=1}^n{y}_i}$. The formula for the mean of x is $\overline{x}=\frac{1}{n}{\displaystyle \sum _{i=1}^n{x}_i}$.

The formula for $S_{xy}$ is $S_{xy}={\displaystyle \sum _{i=1}^n\left(x_i-\overline{x}\right)\left(y_i-\overline{y}\right)}$. The formula for $S_{xx}$ is $S_{xx}={\displaystyle \sum _{i=1}^n{\left(x_i-\overline{x}\right)}^2}$. The formula for ${\widehat{\beta }}_1$ is ${\widehat{\beta }}_1=\frac{S_{xy}}{S_{xx}}$. The formula for ${\widehat{\beta }}_0$ is ${\widehat{\beta }}_0=\overline{y}-{\widehat{\beta }}_1\overline{x}$.

 The calculation is shown below:

Horsepower (y)	Cylinder Volume (x)
51	1.8
51	1.5
115	2
150	2.5
126	1.8
150	2.5
118	1.6
106	1.5
${\displaystyle \sum _{i=1}^8{y}_i}=867$	${\displaystyle \sum _{i=1}^{8}x}=15.2$

Thus, the mean of y and mean of x are obtained as follows:

$\begin{array}{c}\overline{y}=\frac{1}{n}{\displaystyle \sum _{i=1}^n{y}_i}\\ =\frac{867}{8}\\ =108.375\end{array}$

$\begin{array}{c}\overline{x}=\frac{1}{n}{\displaystyle \sum _{i=1}^n{x}_i}\\ =\frac{15.2}{8}\\ =1.9\end{array}$

The value of $\overline{y}$ is 108.375; the value of $\overline{x}$ is 1.9.

The values of $S_{xy}$ and $S_{xx}$ are calculated below:

$\begin{array}{c}{S}_{xy}={\displaystyle \sum _{i=1}^n\left(x_i-\overline{x}\right)\left(y_i-\overline{y}\right)}\\ =\left[\begin{array}{l}\left(1.8-1.9\right)\left(51-108.375\right)+\left(1.5-1.9\right)\left(51-108.375\right)\\ +\mathrm{...}+\left(1.5-1.9\right)\left(106-108.375\right)\end{array}\right]\\ =75.6\end{array}$

$\begin{array}{c}{S}_{xx}={\displaystyle \sum _{i=1}^n{\left(x_i-\overline{x}\right)}^2}\\ =\left[\begin{array}{l}{\left(1.8-1.9\right)}^2+{\left(1.5-1.9\right)}^2+\mathrm{..}\\ +{\left(1.5-1.9\right)}^2\end{array}\right]\\ =1.16\end{array}$

The value of $S_{xy}$ is 75.6; the value of $S_{xx}$ is 1.16.

Thus, the values of ${\widehat{\beta }}_1$ and ${\widehat{\beta }}_0$ are obtained as follows:

$\begin{array}{c}{\widehat{\beta }}_1=\frac{S_{xy}}{S_{xx}}\\ =\frac{75.6}{1.16}\\ =65.172\end{array}$

$\begin{array}{c}{\widehat{\beta }}_0=\overline{y}-{\widehat{\beta }}_1\overline{x}\\ =108.375-\left(65.172\right)1.9\\ =108.375-123.827\\ =-15.452\end{array}$

The value of ${\widehat{\beta }}_0$ is –15.452; the value of ${\widehat{\beta }}_1$ is 65.172.

The straight line is obtained as follows:

$\begin{array}{c}\widehat{y}={\widehat{\beta }}_0+{\widehat{\beta }}_{1}x\\ =-15.452+65.172x\end{array}$

Thus, the least-square line for the data is $\widehat{y}=-15.452+65.172x$.

c.

To determine

Draw the least-square line on a graph paper.

Answer to Problem 9E

The graph of a least-square line is represented as follows:

Explanation of Solution

Step-by-step procedure to obtain the straight line:

• Mark the values of x along the horizontal axis from 1.5 to 2.5 at intervals of 0.25 units.
• Mark the values of y along the vertical axis from 50 to 150 at intervals of 25 units.
• Plot the point for each value of x with the corresponding value of y.
• Draw the line based on the estimated values of ${\beta }_0$ and ${\beta }_1$.

Thus, the graph is obtained.

d.

To determine

Obtain the estimated value of the mean horsepower rating for a fuel-efficient automobile with cylinder volume 1.9 L.

Answer to Problem 9E

The estimated value of the mean horsepower rating for a fuel-efficient automobile with cylinder volume 1.9 L is 108.375 Horsepower.

Explanation of Solution

Calculation:

From the part (b), the straight line is $\widehat{y}=-15.452+65.172x$.

The estimated value of the mean horsepower rating for a fuel-efficient automobile with cylinder volume 1.9 L is calculated as follows:

$\begin{array}{c}\widehat{y}=-15.452+65.172x\\ =-15.452+65.172\left(1.9\right)\\ =-15.452+123.827\\ =108.375\end{array}$

Therefore, the estimated value of the mean horsepower rating for a fuel-efficient automobile with cylinder volume 1.9 L is 108.375 Horsepower.