Chapter 4, Problem 6CR

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?

Sociology: Job Changes A sociologist is interested in the relation between x = number of job changes and y = annual salary (in thousands of dollars) for people living in the Nashville area. A random sample of 10 people employed in Nashville provided the following information:

x (Number of job changes)	4	7	5	6	1	5	9	10	10	3
y (Salary in $1000)	33	37	34	32	32	38	43	37	40	33

Complete parts (a) through (c), given

${\displaystyle \sum x}=60;{\displaystyle \sum y}=359;{\displaystyle \sum x^2}=442;{\displaystyle \sum y^2}=13.013;{\displaystyle \sum xy}=2231.$

(d) If someone had x= 2 job changes, what does the least-squares line predict for y, the annual salary?

To determine

To graph: The scatter diagram.

Explanation of Solution

Given: The data which consists of variables, ‘number of job changes’ and ‘annual salary (in thousads of dollars)’, 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 positive correlation between number of job changes and annual salary.

To determine

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

Answer to Problem 6CR

Solution: The calculated values are $\overline{x}=6$, $\overline{y}=35.9$, and $b=0.939$, and the least-squares line is,

$\widehat{y}=30.266+0.939x$

Explanation of Solution

Given: The provided values are ${\displaystyle \sum x}=60$, ${\displaystyle \sum y=359}$, ${\displaystyle \sum x^2=442}$, ${\displaystyle \sum y^2}=13013$, and ${\displaystyle \sum xy}=2231$, and the number of pairs $\left(x,y\right)$ in a data set (n) is 10.

Calculation:

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

$\begin{array}{c}\overline{x}=\frac{{\displaystyle \sum x}}{n}\\ =\frac{60}{10}\\ =6\end{array}$

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

$\begin{array}{c}\overline{y}=\frac{{\displaystyle \sum y}}{n}\\ =\frac{359}{10}\\ =35.9\end{array}$

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

$\begin{array}{c}b=\frac{n{\displaystyle \sum xy}-\left({\displaystyle \sum x}\right)\left({\displaystyle \sum y}\right)}{n{\displaystyle \sum x^2-{\left({\displaystyle \sum x}\right)}^2}}\\ =\frac{10\left(2231\right)-\left(60\right)\left(359\right)}{10\left(442\right)-{\left(60\right)}^2}\\ \approx 0.939\end{array}$

Therefore, the values are $\overline{x}=6$, $\overline{y}=35.9$, and $b=0.939$.

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:

$\begin{array}{c}a=\overline{y}-b\overline{x}\\ =35.9-\left(0.939\times 6\right)\\ =30.266\end{array}$

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

$\begin{array}{c}\widehat{y}=a+bx\\ =30.266+\left(0.939\right)x\\ =30.266+0.939x\end{array}$

Therefore, the least-squares line is $\widehat{y}=30.266+0.939x$.

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 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:

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

Interpretation: The scatter plot displays the equation of the least-squares line, $\widehat{y}=30.266+0.939x$.

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 6CR

Solution: The values of r and $r^2$ are 0.761 and 0.579, respectively. The percentage of variation in the annual salary explained by the least-squares model is 57.9%.

Explanation of Solution

Given: The provided values are ${\displaystyle \sum x}=60$, ${\displaystyle \sum y=359}$, ${\displaystyle \sum x^2=442}$, ${\displaystyle \sum y^2}=13013$, and ${\displaystyle \sum xy}=2231$, and the number of pairs $\left(x,y\right)$ in a data set (n) is 10.

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,

$\begin{array}{c}r=\frac{10\left(2231\right)-\left(60\right)\left(359\right)}{\sqrt{\left(10\right)\left(442\right)-{\left(60\right)}^2}\sqrt{\left(10\right)\left(13013\right)-{\left(359\right)}^2}}\\ =0.7608\\ \approx 0.761\end{array}$

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

$\begin{array}{c}{r}^2={\left(r\right)}^2\\ ={\left(0.761\right)}^2\\ \approx 0.579\end{array}$

Therefore, the value of $r^2$ is 0.579.

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 57.9%.

Interpretation: The correlation coefficient value $r=0.761$ indicates that there is a fairly high positive linear relationship between the number of job changes and the annual salary for people living in Nashville area, and about 57.9% variation in salary can be explained by the corresponding variation in the number of job changes.

To determine

To find: The least-squares line forecast for y at $x=2$.

Answer to Problem 6CR

Solution: The forecasted value is $32,144.

Explanation of Solution

Given: The least-squares line from part (b) is $\widehat{y}=30.266+0.939x$.

Calculation:

The predicted value $\left(\widehat{y}\right)$ for $x=2$ can be calculated as follows:

$\begin{array}{c}\widehat{y}=30.266+0.939x\\ =30.266+0.939\left(2\right)\\ =32.144\end{array}$

Thus, the value of $\widehat{y}=32.144$.

As the values of y are in 1000 dollars,

$\begin{array}{c}\widehat{y}=\$32.144\times 1000\\ =\$32144\end{array}$

Interpretation: For a person who had 2 job changes, the predicted value of the annual salary using the least-squares line is $32144.