Chapter 4.2, Problem 11P

Auto Accidents: Age Data for this problem are based on information taken from The Wall Street Journal. Let x be the age in years of a licensed automobile driver. Let y be the percentage of all fatal accidents (for a given age) due to speeding. For example, the first data pair indicates that 36% of all fatal accidents involving 17-year-olds are due to speeding.

x	17	27	37	47	57	67	77
y	36	25	20	12	10	7	5

Complete parts (a) through (e), given ${\displaystyle \sum x}=329,{\displaystyle \sum y=115,{\displaystyle \sum x^2=18,263,{\displaystyle \sum y^2=2639,{\displaystyle \sum xy=4015,\text{and}r\approx -\mathrm{0.959.}}}}}$

(f) Predict the percentage of all fatal accidents due to speeding for 25-year-olds.

To determine

To graph: The scatter diagram for age in years of a licensed automobile driver (x) and percentage of all fatal accidents (y).

Explanation of Solution

Given: The data which consists of variables, ‘age of a licensed automobile driver (in years)’ and ‘the percentage of all fatal accidents (for a given age)’ due to speeding, represented by x and y respectively, is provided.

Graph:

Follow the steps given below in Excel to obtain the scatter diagram of the data.

Step 1: Enter the data into Excel sheet. The screenshot is given below.

Step 2: Select the data and click on ‘Insert’. Go to charts and select the chart type ‘Scatter’.

Step 3: Select the first plot and then click ‘add chart element’ provided in the left corner of the menu bar. Insert the ‘Axis titles’ and ‘Chart title’. The scatter plot for the provided data is shown below:

Interpretation: The scatterplot shows that, there is a negative correlation between the age of a licensed automobile drivers and the speeding- related fatal percentage. So, as the age increases, the fatal percentage related to speeding decreases and vice versa.

To determine

To test: Whether the provided values of ${\displaystyle \sum x}$, ${\displaystyle \sum y}$, ${\displaystyle \sum x^2}$ $,{\displaystyle \sum y^2}$, ${\displaystyle \sum xy}$ and $r$ are correct or not.

Answer to Problem 11P

Solution: The provided values, that is, ${\displaystyle \sum x}=329$, ${\displaystyle \sum y=115}$, ${\displaystyle \sum x^2=18263}$, ${\displaystyle \sum y^2}=2639$ ${\displaystyle \sum xy}=4015$ and $r\approx -0.959$ are correct.

Explanation of Solution

Given: The provided values are: ${\displaystyle \sum x}=329$, ${\displaystyle \sum y=115}$, ${\displaystyle \sum x^2=18263}$, ${\displaystyle \sum y^2}=2639$ ${\displaystyle \sum xy}=4015$ and $r\approx -0.959$.

Calculation:

To compute ${\displaystyle \sum x}$, ${\displaystyle \sum y}$, ${\displaystyle \sum x^2}$ $,{\displaystyle \sum y^2}$ and ${\displaystyle \sum xy}$, it is easy to organize the data in a table of five columns and then add the values in each column. The table is given below:

$x$	$y$	$x^2$	$y^2$	$xy$
17	36	289	1296	612
27	25	729	625	675
37	20	1369	400	740
47	12	2209	144	564
57	10	3249	100	570
67	7	4489	49	469
77	5	5929	25	385
${\displaystyle \sum x}=329$	${\displaystyle \sum y=115}$	${\displaystyle \sum x^2=18263}$	${\displaystyle \sum y^2}=2639$	${\displaystyle \sum xy}=4015$

Now, the value of $r$ can be calculated by using the formula below:

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

Substituting the values in the above formula. Thus:

$\begin{array}{c}r=\frac{7\left(4015\right)-\left(329\right)\left(115\right)}{\sqrt{\left(7\right)\left(18263\right)-{\left(329\right)}^2}\sqrt{\left(7\right)\left(2639\right)-{\left(115\right)}^2}}\\ =-\frac{139}{144.89}\\ =-0.959\end{array}$

Thus, the value of $r\approx -0.959$.

Conclusion: The provided values, ${\displaystyle \sum x}=329$, ${\displaystyle \sum y=115}$, ${\displaystyle \sum x^2=18263}$, ${\displaystyle \sum y^2}=2639$ ${\displaystyle \sum xy}=4015$ and $r\approx -0.959$ have been verified.

To determine

To find: The values of $\overline{x},\overline{y}$, a, b and the equation for the least-squares line.

Answer to Problem 11P

Solution: The calculated values are, $\overline{x}=47\text{ years},\overline{y}=16.43\%,a\approx 39.761$ and $b\approx -0.496$. The least-squares line is,

$\widehat{y}=39.761-\mathrm{0..496}x$.

Explanation of Solution

Given: The provided values are, ${\displaystyle \sum x}=329$, ${\displaystyle \sum y=115}$, ${\displaystyle \sum x^2=18263}$, ${\displaystyle \sum y^2}=2639$ ${\displaystyle \sum xy}=4015$ and $r\approx -0.959$. The number of pairs of (x,y) in the data set (n) is 7.

Calculation:

The value of $\overline{x}$ can be calculated as:

$\begin{array}{c}\overline{x}=\frac{{\displaystyle \sum x}}{n}\\ =\frac{329}{7}\\ =47\end{array}$

The value of $\overline{y}$ can be calculated as:

$\begin{array}{c}\overline{y}=\frac{{\displaystyle \sum y}}{n}\\ =\frac{115}{7}\\ \approx 16.43\end{array}$

The value of $b$ can be calculated as:

$\begin{array}{c}b=\frac{n{\displaystyle \sum xy}-\left({\displaystyle \sum x}\right)\left({\displaystyle \sum y}\right)}{n{\displaystyle \sum x^2-{\left({\displaystyle \sum x}\right)}^2}}\\ =\frac{7\left(4015\right)-\left(329\right)\left(115\right)}{7\left(18263\right)-{\left(329\right)}^2}\\ =\frac{-9730}{19600}\\ \approx -0.4964\end{array}$

The value of $a$ can be calculated as:

$\begin{array}{c}a=\overline{y}-b\overline{x}\\ =16.43-\left(-0.4964\times 47\right)\\ =16.43+23.3308\\ \approx 39.761\end{array}$

Therefore, the values are, $\overline{x}=47\text{ years},\overline{y}=16.43\%,a\approx 39.761$ and $b\approx -0.496$.

The general formula for the least-squares line is,

$\widehat{y}=a+bx$

Here, a is the y-intercept and b is the slope.

Substitute the values of a and b in the general equation, to get the least-squares line of the data. That is,

$\begin{array}{c}\widehat{y}=a+bx\\ =39.761+\left(-0.496\right)x\\ =39.761-0.496x\end{array}$

Therefore, the least-squares line is $\widehat{y}=39.761-0.496x$.

To determine

To graph: The least-squares line on the scatter diagram which passes through the point $\left(\overline{x},\overline{y}\right)$.

Explanation of Solution

Given: The data which consists of variables, ‘age of a licensed automobile driver (in years)’ and ‘the fatal accidents percentage (for a given age) due to speeding’, represented by x and y respectively, is provided.

Graph:

Follow the steps given below in Excel to obtain the scatter diagram of the data.

Step 1: Enter the data into Excel sheet. The screenshot is given below.

Step 2: Select the data and click on ‘Insert’. Go to charts and select the chart type ‘Scatter’.

Step 3: Select the first plot and then click ‘add chart element’ provided in the left corner of the menu bar. Insert the ‘Axis titles’ and ‘Chart title’. The scatter plot for the provided data is shown below:

Step 3: Right click on any data point and select ‘Add Trendline’. In the dialogue box, select ‘linear’ and check ‘Display Equation on Chart’

Interpretation: The least-squares line passes through the point $\left(\overline{x},\overline{y}\right)=\left(47,16.43\right)$ and the scatterplot displays the equation of the least-squares line as, $\widehat{y}=39.761-0.4964x$.

To determine

The value of $r^2$, the explained percentage of variation and the percentage of variation that is unexplained.

Answer to Problem 11P

Solution: The value of $r^2$ is 0.920. The percentage of variation in fatal accident percentage due to speeding that can be explained by the age of the licensed automobile driver using the least-squares line is 92% and the rest 8% variation remains unexplained.

Explanation of Solution

Given: The value of the correlation coefficient (r) is $-0.959$.

Calculation: The coefficient of determination $\left(r^2\right)$ can be calculated as:

$\begin{array}{c}{r}^2={\left(r\right)}^2\\ ={\left(-0.959\right)}^2\\ \approx 0.920\end{array}$

Therefore, the value of $r^2$ is 0.920.

The value of $r^2$ indicates the proportion of total variation in y that can be explained by using the least-squares line. So, the explained percentage of variation in y is 92%.

Further, the proportion of variation in y that is unexplained can be calculated as:

$\begin{array}{c}1-r^2=1-0.920\\ =0.08\\ =8\%\end{array}$

Hence, the percentage of variation in y that is unexplained is 8%.

Interpretation: About 92% of the variation in y (fatal % due to speeding) can be explained by the corresponding variation in x (age) and the least-squares line. The rest 8% of variation remains unexplained.

To determine

To find: The predicted percentage of speeding related fatal accidents $\left(\widehat{y}\right)$ for 25-year-olds.

Answer to Problem 11P

Solution: The predicted value is approximately 27.36%.

Explanation of Solution

Given: The least-squares line from part (c) is, $\widehat{y}=39.761-0.496x$.

Calculation:

The predicted value $\left(\widehat{y}\right)$ for $x=25$ can be calculated as follows:

$\begin{array}{c}\widehat{y}=39.761-0.496x\\ =39.761-0.496\left(25\right)\\ =39.761-12.4\\ \approx 27.361\end{array}$

Thus, the value of $\widehat{y}\approx 27.36$.

Interpretation: The percentage of all fatal accidents due to speeding for 25-years-olds is predicted to be 27.36%.