Chapter 4, Problem 13CR

In Problems 9-14,

(a) Draw a scatter diagram for the data.

(b) Find $\bar{x},\bar{y},b$, and the equations of the least-squares line. Plot the line on the scatter diagram of part (a).

(c) Find the sample correlation coefficient r and the coefficient of determination $r^2$. What percentage of variation in y is explained by the least-squares model?

Marketing: Coupons Each box of Healthy Crunch breakfast cereal contains a coupon entitling you to a free package of garden seeds. At the Healthy Crunch home office, they use the weight of incoming mail to determine how many of their employees are to be assigned to collecting coupons and mailing out seed packages on a given day. (Healthy Crunch has a policy of answering all its mail on the day it is received.)

Let x = weight of incoming mail and y = number of employees required to process the mail in one working day. A random sample of 8 days gave the following data:

x(lb)	11	20	16	6	12 18 23	25
y (Number of employees)	6	10	9	5	8 14 13	16

Complete parts (a) through (c), given

${\displaystyle \sum x}=131;{\displaystyle \sum y}=81:x{\displaystyle \sum {}^2}=2435; {\displaystyle \sum y^2}-927;{\displaystyle \sum xy}=1487$

(d) If Healthy Crunch receives 15 pounds of mail, how many employees should be assigned mail duty that day?

To determine

To graph: The scatter diagram.

Explanation of Solution

Given: The data which consists of variables, ‘weight of incoming mail in pounds’ and ‘number of employess required to process the mail in 1 working day’, 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 on 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 scatter plot shows that there is a high positive correlation between the weight of mail and the number of employees required to process the mail.

To determine

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

Answer to Problem 13CR

Solution: The calculated values are $\overline{x}=16.375$, $\overline{y}=10.125$, and $b=0.554$, and the least-squares line is,

$\widehat{y}=1.051+0.554x$

Explanation of Solution

Given: The provided values are ${\displaystyle \sum x}=131$, ${\displaystyle \sum y=81}$, ${\displaystyle \sum x^2=2435}$, ${\displaystyle \sum y^2}=927$, and ${\displaystyle \sum xy}=1487$, and the number of pairs $\left(x,y\right)$ in a data set (n) is 8.

Calculation:

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

$\begin{array}{c}\overline{x}=\frac{{\displaystyle \sum x}}{n}\\ =\frac{131}{8}\\ =16.375\end{array}$

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

$\begin{array}{c}\overline{y}=\frac{{\displaystyle \sum y}}{n}\\ =\frac{81}{8}\\ =10.125\end{array}$

The value of $b$ can be calculated as follows:

$\begin{array}{c}b=\frac{n{\displaystyle \sum xy}-\left({\displaystyle \sum x}\right)\left({\displaystyle \sum y}\right)}{n{\displaystyle \sum x^2-{\left({\displaystyle \sum x}\right)}^2}}\\ =\frac{8\left(1487\right)-\left(131\right)\left(81\right)}{8\left(2435\right)-{\left(131\right)}^2}\\ \approx 0.554118\\ \approx 0.554\end{array}$

Therefore, the values are $\overline{x}=16.375$, $\overline{y}=10.125$, and $b=0.554$.

The general formula for the least-squares line is,

$\widehat{y}=a+bx$

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

The value of $a$ can be calculated as follows:

$\begin{array}{c}a=\overline{y}-b\overline{x}\\ =10.125-\left(0.554118\times 16.375\right)\\ \approx 1.051\end{array}$

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

$\begin{array}{c}\widehat{y}=a+bx\\ =1.051+\left(0.554\right)x\\ =1.051+0.554x\end{array}$

Therefore, the least-squares line is $\widehat{y}=1.051+0.554x$.

Follow the steps given below in Excel to obtain the least-squares line on the scatter diagram.

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 4: Right click on any data point and select ‘Add Trend line’. A dialogue box will appear then select ‘linear’ and check ‘Display Equation on Chart’.

Interpretation: The scatter plot displays the equation of the least-squares line, $\widehat{y}=0.5541x+1.0513$.

To determine

The values of r and $r^2$, and the percentage of variation in y that can be explained by the least-squares model.

Answer to Problem 13CR

Solution: The values of r and $r^2$ are 0.913 and 0.833, respectively. The percentage of variation in the number of employees required, explained by the least-squares line, is 83.3%.

Explanation of Solution

Given: The provided values are ${\displaystyle \sum x}=131$, ${\displaystyle \sum y=81}$, ${\displaystyle \sum x^2=2435}$, ${\displaystyle \sum y^2}=927$, and ${\displaystyle \sum xy}=1487$, and the number of pairs $\left(x,y\right)$ in a data set (n) is 8.

Calculation: The value of $r$ can be calculated by using the formula below:

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

Substitute the values in the above formula. Thus,

$\begin{array}{c}r=\frac{8\left(1487\right)-\left(131\right)\left(81\right)}{\sqrt{\left(8\right)\left(2435\right)-{\left(131\right)}^2}\sqrt{\left(8\right)\left(927\right)-{\left(81\right)}^2}}\\ =0.9126\\ \approx 0.913\end{array}$

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

$\begin{array}{c}{r}^2={\left(r\right)}^2\\ ={\left(0.913\right)}^2\\ \approx 0.833\end{array}$

Therefore, the value of $r^2$ is 0.833.

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 83.3%.

Interpretation: The correlation coefficient value $r=0.913$ indicates that there is a high positive linear relationship between the weight of mail and the number of employees required to process the mail. About 83.3% variation in the number of employees required can be explained by the corresponding variation in the weight of mail.

To determine

To find: The least-squares line forecast for y at $x=15$.

Answer to Problem 13CR

Solution: The forecasted value is about 9.

Explanation of Solution

Given: The least-squares line from part (b) is $\widehat{y}=1.051+0.554x$.

Calculation:

The predicted value $\left(\widehat{y}\right)$ for $x=15$ can be calculated as follows:

$\begin{array}{c}\widehat{y}=1.051+0.554x\\ =1.051+0.554\left(15\right)\\ =9.3\\ \approx 9\end{array}$

Thus, the value of $\widehat{y}=9$.

Interpretation: If Healthy Crunch receives 15 pounds of mail, it is predicted using the least-squares line that about 9 employees should be assigned mail duty for a day.