Chapter 2, Problem 144E

(a)

To determine

To graph: The scatterplot.

Explanation of Solution

Graph: To draw the scatterplot for the provided data set, the below steps are followed in Minitab software.

Step 1: Open the “MEIS” file in Minitab software.

Step 2: Go to Graph $\to$ Scatterplot.

Step 3: Choose the option “Simple” and click “OK”.

Step 4: Select “Sales” as Y variables and “Dwellpermit” as X variables.

Step 5: Click “OK” twice.

The scatterplot is obtained as:

To determine

To explain: The relationship between the variables and the presence of the outliers.

Answer to Problem 144E

Solution: Though the relationship between the variables is very weak, but there is no outlier in the data set.

Explanation of Solution

The scatterplot obtained in part (a) shows that the data are very much scattered. That is there is very weak relationship between the response variable sales and the explanatory variable, issued permits. But there is no outlier present in the data set.

(b)

To determine

To find: The least-squares regression line for the data.

Answer to Problem 144E

Solution: The least-squares regression line is obtained as $\text{Sales}=109.8+0.1263\text{DwellPermit}$, which is shown in the below diagram.

Explanation of Solution

Calculation: To draw the regression line and obtain the sketch of the regression line, the below steps are followed in the Minitab software.

Step 1: Right click on the obtained graph in the part (a).

Step 2: Go to Add $\to$ Regression Fit.

Step 3: Click on “Linear” under the menu “Model Order” and tick the “Fit intercept”.

Step 4: Click on “OK” to obtain the regression line.

The obtained regression equation is $\text{Sales}=109.8+0.1263\text{DwellPermit}$.

Graph: The below graph shows the regression line.

(c)

To determine

To explain: The slope of the obtained line.

Answer to Problem 144E

Solution: The slope can be interpreted as if there is an increase of 1unit of the variable, issued permit for dwelling then there is an increase of 0.1263 units in the value of the variable sales.

Explanation of Solution

The slope of the least-square regression line indicates the amount of increase (decrease) in the response variable due to the 1-unit increase (decrease) in the value of the explanatory variable. In the provided equation, the amount of increase in the value of the response variable, sales is 0.1263 due to the 1-unit increase in the value of the variable issued permit for dwelling.

(d)

To determine

To explain: The intercept of the line.

Answer to Problem 144E

Solution: The intercept can be interpreted as the value of the response variable when the value of the explanatory variable is zero. As the value of the response variable is 109.8 for the zero issued permits, it is appropriate to use the intercept for explaining the relationship between the variables.

Explanation of Solution

The intercept of the least-squares linear regression provided the value of the response variable when the value of the explanatory variable is zero. The explanatory variable can be zero because the index of issued permits for new dwelling can be zero and the value of the response variable for zero issued permit is 109.8. Therefore, it is appropriate to use the intercept for explaining the relationship between the variables.

(e)

To determine

The sales for an index of 224 dwelling permits.

Answer to Problem 144E

Solution: The predicted value of the sales is 138.0912.

Explanation of Solution

The predicted value of the sales can be obtained by substituting the value of the index of dwelling permit in the least-squares linear regression equation.

The predicted value can be calculated as:

$\begin{array}{c}\text{Sales}=109.8+0.1263\text{DwellPermit}\\ =109.8+0.1263\times 224\\ =138.0912\end{array}$

(f)

To determine

To find: The residual value for Canada.

Answer to Problem 144E

Solution: The obtained value of the residual is $\underset{\_}{-8.0912}$.

Explanation of Solution

Calculation: The sale for the Canada is provided as 122. The residual value for the sales for Canada whose index of dwelling permits is 224 is calculated as:

$\begin{array}{c}\text{Residual}=\text{Observed Sales}-\text{Predicted Sales}\\ =122-138.0912\\ =-16.0912\end{array}$

(g)

To determine

The percentage of variability in sales is explained by dwelling permit.

Answer to Problem 144E

Solution: The explained percentage of variability in sales is 10.30%.

Explanation of Solution

The variability is explained by $r^2$ which indicates that how better the model fits the data. The value of $r^2$ is obtained along with the regression line when the line is added to the obtained scatterplot in the Minitab software in part (b). The value is obtained as 10.30%.

Therefore, the proportion of variation that is explained by the explanatory variables of the model is 10.30%.