Chapter 5, Problem 5.42E

(a)

To determine

To find: The correlation for all four data sets.

To find: The least-squares regression line for all four data sets.

To find: The predicted value for $x=10$ using least-squares regression line for all four data sets.

Answer to Problem 5.42E

The correlation for the data set A is 0.816.

The correlation for the data set B is 0.816.

The correlation for the data set C is 0.816.

The correlation for the data set D is 0.8176.

The least-squares regression line for the data set A is $\widehat{y}=3.00+0.500x$.

The least-squares regression line for the data set B is $\widehat{y}=3.00+0.500x$.

The least-squares regression line for the data set C is $\widehat{y}=3.00+0.500x$.

The least-squares regression line for the data set D is $\widehat{y}=3.00+0.500x$.

The predicted value for $x=10$ using least-squares regression line for the data set A is 8.001.

The predicted value for $x=10$ using least-squares regression line for the data set B is 8.001.

The predicted value for $x=10$ using least-squares regression line for the data set C is 8.000.

The predicted value for $x=10$ using least-squares regression line for the data set D is 8.001.

Explanation of Solution

Given info:

The four data sets are used to exploring the correlation and regression.

Calculation:

Correlation for Data set A:

Software procedure:

Step-by-step procedure to find the correlation between the x and y for data set A by using the MINITAB software:

• Select Stat >Basic Statistics > Correlation.
• In Variables, select x and y.
• Click OK.

Output using the MINITAB software is given below:

From the MINITAB output, the correlation between the x and y for data set A is 0.816.

Correlation for Data set B:

Software procedure:

Step-by-step procedure to find the correlation between the x and y for data set B by using the MINITAB software:

• Select Stat >Basic Statistics > Correlation.
• In Variables, select x and y.
• Click OK.

Output using the MINITAB software is given below:

From the MINITAB output, the correlation between the x and y for data set B is 0.816.

Correlation for Data set C:

Software procedure:

Step-by-step procedure to find the correlation between the x and y for data set C by using the MINITAB software:

• Select Stat >Basic Statistics > Correlation.
• In Variables, select x and y.
• Click OK.

Output using the MINITAB software is given below:

From the MINITAB output, the correlation between the x and y for data set C is 0.816.

Correlation for Data set D:

Software procedure:

Step-by-step procedure to find the correlation between the x and y for data set D by using the MINITAB software:

• Select Stat >Basic Statistics > Correlation.
• In Variables, select x and y.
• Click OK.

Output using the MINITAB software is given below:

From the MINITAB output, the correlation between the x and y for data set D is 0.817.

Equation of the least-squares line for Data set A:

Software procedure:

Step-by-step procedure to find the equation of the least-squares line by using the MINITAB software:

• Choose Stat > Regression > Regression.
• In Responses, enter the column of y.
• In Predictors, enter the column of x.
• Click OK.

Output using the MINITAB software is given below:

From the MINITAB output, the least-squares line for predicting y from x for data set A is $\widehat{y}=3.00+0.500x$.

Equation of the least-squares line for Data set B:

Software procedure:

Step-by-step procedure to find the equation of the least-squares line by using the MINITAB software:

• Choose Stat > Regression > Regression.
• In Responses, enter the column of y.
• In Predictors, enter the column of x.
• Click OK.

Output using the MINITAB software is given below:

From the MINITAB output, the least-squares line for predicting y from x for data set B is $\widehat{y}=3.00+0.500x$.

Equation of the least-squares line for Data set C:

Software procedure:

Step-by-step procedure to find the equation of the least-squares line by using the MINITAB software:

• Choose Stat > Regression > Regression.
• In Responses, enter the column of y.
• In Predictors, enter the column of x.
• Click OK.

Output using the MINITAB software is given below:

From the MINITAB output, the least-squares line for predicting y from x for data set C is $\widehat{y}=3.00+0.500x$.

Equation of the least-squares line for Data set D:

Software procedure:

Step-by-step procedure to find the equation of the least-squares line by using the MINITAB software:

• Choose Stat > Regression > Regression.
• In Responses, enter the column of y.
• In Predictors, enter the column of x.
• Click OK.

Output using the MINITAB software is given below:

From the MINITAB output, the least-squares line for predicting y from x for data set D is $\widehat{y}=3.00+0.500x$.

Predicted value for $x=10$ for Data set A:

Software procedure:

Step-by-step procedure to find the predicted value for $x=10$ using least-squares regression line for the data set A by using the MINITAB software:

• Choose Stat > Regression > Regression.
• In Responses, enter the column of y.
• In Predictors, enter the column of x.
• In option, enter 10 under prediction.
• Click OK.

Output using the MINITAB software is given below:

From the MINITAB output, the predicted value for $x=10$ using least-squares regression line for the data set A is 8.001.

Predicted value for $x=10$ for Data set B:

Software procedure:

Step-by-step procedure to find the predicted value for $x=10$ using least-squares regression line for the data set B by using the MINITAB software:

• Choose Stat > Regression > Regression.
• In Responses, enter the column of y.
• In Predictors, enter the column of x.
• In option, enter 10 under prediction.
• Click OK.

Output using the MINITAB software is given below:

From the MINITAB output, the predicted value for $x=10$ using least-squares regression line for the data set B is 8.001.

Predicted value for $x=10$ for Data set C:

Software procedure:

Step-by-step procedure to find the predicted value for $x=10$ using least-squares regression line for the data set C by using the MINITAB software:

• Choose Stat > Regression > Regression.
• In Responses, enter the column of y.
• In Predictors, enter the column of x.
• In option, enter 10 under prediction.
• Click OK.

Output using the MINITAB software is given below:

From the MINITAB output, the predicted value for $x=10$ using least-squares regression line for the data set C is 8.000.

Predicted value for $x=10$ for Data set D:

Software procedure:

Step-by-step procedure to find the predicted value for $x=10$ using least-squares regression line for the data set D by using the MINITAB software:

• Choose Stat > Regression > Regression.
• In Responses, enter the column of y.
• In Predictors, enter the column of x.
• In option, enter 10 under prediction.
• Click OK.

Output using the MINITAB software is given below:

From the MINITAB output, the predicted value for $x=10$ using least-squares regression line for the data set D is 8.001.

From the results, it can be observed that the correlation for all four data sets, the least-squares regression line and the predicted value for $x=10$ all four data sets are similar.

(b)

To determine

To construct: The scatterplot for each of the data sets with regression line.

Answer to Problem 5.42E

Scatterplot for Data set A:

Output using the MINITAB software is given below:

Scatterplot for Data set B:

Output using the MINITAB software is given below:

Scatterplot for Data set C:

Output using the MINITAB software is given below:

Scatterplot for Data set D:

Output using the MINITAB software is given below:

Explanation of Solution

Calculation:

Scatterplot:

Software procedure:

Step-by-step procedure to construct scatterplot for x and y for all four data sets by using the MINITAB software:

• Choose Graph > Scatter plot.
• Choose With Regression, and then click OK.
• Under Y variables, enter a column of y.
• Under X variables, enter a column of x.
• Click OK.

Observation:

The scatterplot shows that the predicted values are passed through the regression line of the model. Moreover, there is outlier that appears in the x and y directions for the data set A, B, and C. Also, the scatterplot for the data set D shows that the most of the points are plotted around 8.

(c)

To determine

To identify: Which of the four cases would you be willing to use the regression line to describe the dependence of y on x.

Answer to Problem 5.42E

The data set A would use the regression line to describe the dependence of y on x.

Explanation of Solution

From the scatterplots for all data sets, it can be observed that the points for data set A are scattered around the straight line when compared to the other data sets. Hence, the data set A would use the regression line to describe the dependence of y on x.