Chapter 9.2, Problem 12P

(a)

To determine

Construct a scatter diagram for the data.

Answer to Problem 12P

The scatter diagram for data is,

Explanation of Solution

Calculation:

The variable x denotes the age of a licensed driver in years and y denotes the percentage of all fatal accidents due to failure to yield the right-of-way.

Step by step procedure to obtain scatter plot using MINITAB software is given below:

• Choose Graph > Scatterplot.
• Choose Simple. Click OK.
• In Y variables, enter the column of x.
• In X variables, enter the column of y.
• Click OK.

(b)

To determine

Verify the values of ${\displaystyle \sum x}$, ${\displaystyle \sum x^2}$, ${\displaystyle \sum y}$, ${\displaystyle \sum y^2}$, ${\displaystyle \sum xy}$ and r.

Explanation of Solution

Calculation:

The formula for correlation coefficient is,

$r=\frac{n{\displaystyle \sum xy}-\left({\displaystyle \sum x}\right)\left({\displaystyle \sum y}\right)}{\sqrt{n{\displaystyle \sum x^2}-{\left({\displaystyle \sum x}\right)}^2}\sqrt{n{\displaystyle \sum y^2}-{\left({\displaystyle \sum y}\right)}^2}}$

In the formula, n is the sample size.

The values are verified in the table below,

x	y	$x^2$	$y^2$	xy
37	5	1369	25	185
47	8	2209	64	376
57	10	3249	100	570
67	16	4489	256	1072
77	30	5929	900	2310
87	43	7569	1849	3741
${\displaystyle \sum x}=372$	${\displaystyle \sum y}=112$	${\displaystyle \sum x^2}=24,814$	${\displaystyle \sum y^2}=3,194$	${\displaystyle \sum xy}=8,254$

Hence, the values are verified.

The number of data pairs are $n=6$. The value of correlation r is,

$\begin{array}{c}r=\frac{n{\displaystyle \sum xy}-\left({\displaystyle \sum x}\right)\left({\displaystyle \sum y}\right)}{\sqrt{n{\displaystyle \sum x^2}-{\left({\displaystyle \sum x}\right)}^2}\sqrt{n{\displaystyle \sum y^2}-{\left({\displaystyle \sum y}\right)}^2}}\\ =\frac{6\left(8,254\right)-\left(372\times 112\right)}{\sqrt{6\left(24,814\right)-{\left(372\right)}^2}\sqrt{6\left(3,194\right)-{\left(112\right)}^2}}\\ =\frac{49,524-41,664}{\sqrt{10,500}\sqrt{6,620}}\\ =\frac{7,860}{102.4695\times 81.3634}\\ =\frac{7,860}{8,337.2669}\\ =0.943\end{array}$

The value of r is 0.943 this shows that r is not –0.943.

Hence, the value of r is verified as approximately 0.943.

(c)

To determine

Find the value of $\overline{x}$.

Find the value of $\overline{y}$.

Find the value of a.

Find the value of b.

Find the equation of the least-squares line.

Answer to Problem 12P

The value of $\overline{x}$ is 62 years.

The value of $\overline{y}$ is 18.67%.

The value of a is –27.768.

The value of b is 0.749.

The equation of the least-squares line is $\widehat{y}=-27.768+0.749x$.

Explanation of Solution

Calculation:

From part (b), the values are ${\displaystyle \sum x}=372$, ${\displaystyle \sum y}=112$, ${\displaystyle \sum x^2}=24,814$, ${\displaystyle \sum y^2}=3,194$, ${\displaystyle \sum xy}=8,254$ and $n=6$

The value of $\overline{x}$ is,

$\begin{array}{c}\overline{x}=\frac{{\displaystyle \sum x}}{n}\\ =\frac{372}{6}\\ =62\end{array}$

Hence, the value of $\overline{x}$ is 62 years.

The value of $\overline{y}$ is,

$\begin{array}{c}\overline{y}=\frac{{\displaystyle \sum y}}{n}\\ =\frac{112}{6}\\ =18.67\end{array}$

Hence, the value of $\overline{y}$ is 18.67%.

The value of b is,

$\begin{array}{c}b=\frac{n{\displaystyle \sum xy}-\left({\displaystyle \sum x}\right)\left({\displaystyle \sum y}\right)}{n{\displaystyle \sum x^2}-{\left({\displaystyle \sum x}\right)}^2}\\ =\frac{6\left(8,254\right)-\left(372\times 112\right)}{6\left(24,814\right)-{\left(372\right)}^2}\\ =\frac{49,524-41,664}{148,884-138,384}\\ =\frac{7,860}{10,500}\\ =0.749\end{array}$

Hence, the value of b is 0.749.

The value of a is,

$\begin{array}{c}a=\overline{y}-b\overline{x}\\ =18.67-\left(0.749\right)62\\ =18.67-46.438\\ =-27.768\end{array}$

Hence, the value of a is –27.768.

The equation of the least-squares line is,

$\begin{array}{c}\widehat{y}=a+bx\\ =-27.768+0.749x\end{array}$

Hence, the equation of the least-squares line is $\widehat{y}=-27.768+0.749x$.

(d)

To determine

Construct a scatter diagram with least squares line.

Locate the point $\left(\overline{x},\overline{y}\right)$ on the line.

Answer to Problem 12P

The scatter diagram with least squares line with point $\left(\overline{x},\overline{y}\right)$ is,

Explanation of Solution

Calculation:

In the dataset of failure of yield, also include the point $\left(\overline{x},\overline{y}\right)$. That is, $\left(62,18.67\right)$.

Step by step procedure to obtain scatter plot using MINITAB software is given below:

• Choose Graph > Scatterplot.
• Choose With regression. Click OK.
• In Y variables, enter the column of x.
• In X variables, enter the column of y.
• Click OK.

(e)

To determine

Calculate the value of the coefficient of determination $r^2$.

Mention percentage of the variation in y that can be explained by variation in x.

Mention percentage of the variation in y that cannot be explained by variation in x.

Answer to Problem 12P

The value of the coefficient of determination $r^2$ is 0.889.

The percentage of the variation in y that can be explained by variation in x is 88.9%.

The percentage of the variation in y that cannot be explained by variation in x is 11.1%.

Explanation of Solution

Calculation:

Coefficient of determination$\left(r^2\right)$:

The coefficient of determination $\left(r^2\right)$ determines the percent of the variation in response variable that is explained by the predictor variables. A larger value of $r^2$ indicates that the model is a good fit.

From part (b), the value of correlation coefficient r is 0.943. The value of $r^2$ is,

$\begin{array}{c}r=0.943\\ r^2={\left(0.943\right)}^2\\ =0.889\end{array}$

Hence, the value of the coefficient of determination $r^2$ is 0.889.

About 88.9% of the variation in y (percentage of all fatal accidents due to failure to yield the right-of-way) is explained by x (age of a licensed driver in years). Since the value of $r^2$ is large the model is a good fit for the data.

Hence, the percentage of the variation in y that can be explained by variation in x is 92%.

About 11.1% $\left(=100\%-88.9\%\right)$ of the variation y (percentage of all fatal accidents due to failure to yield the right-of-way) cannot be explained by x (age of a licensed driver in years).

Hence, the percentage of the variation in y that cannot be explained by variation in x is 11.1%.

(f)

To determine

Find the percentage of all fatal accidents due to failing to yield the right-of-way for 70-year-olds.

Answer to Problem 12P

The percentage of all fatal accidents due to failing to yield the right-of-way for 70-year-olds is 24.662%.

Explanation of Solution

Calculation:

From part (c), the equation of the least-squares line is $\widehat{y}=-27.768+0.749x$.

Substitute $x=70$ in equation of the least-squares line.

$\begin{array}{c}\widehat{y}=-27.768+0.749\left(70\right)\\ =-27.768+52.43\\ =24.662\end{array}$

Hence, the percentage of all fatal accidents due to failing to yield the right-of-way for 70-year-olds is 24.662%.