4. Explained and Unexplained Variation What is the difference between explained variation and unexplained variation?

Interpreting the Coefficient of Determination. In Exercises 5–8, use the value of the linear correlation coefficient r to find the coefficient of determination and the percentage of the total variation that can be explained by the linear relationship between the two variables.

5. r
0.3 6. r
0.2 7. r
0.901 8. r
0.747 Using Technology STATDISK STATDISK can be used to find the linear correlation coefficient r, the equation of the regression line, the standard error of estimate se, the total variation, the explained variation, the unexplained variation, and the coefficient of determination.

Enter the paired data in columns of the STATDISK Data Window, select Analysis from the main menu bar, then select Correlation and Regression. Enter a value for the significance level and select the columns of data. Click on the Evaluate button.

The STATDISK display will include the linear correlation coefficient, the coefficient of determination, the regression equation, and the value of the standard error of estimate se.

MINITAB Minitab can be used to find the regression equation, the standard error of estimate se (labeled S), the value of the coefficient of determination (labeled R-sq), and the limits of a prediction interval. Enter the x-data in column C1 and the y-data in column C2, then select the options Stat, Regression, and Regression. Enter C2 in the box labeled “Response” and enter C1 in the box labeled “Predictors.” If you want a prediction interval for some given value of x, click on the Options box and enter the desired value of x0 in the box labeled “Prediction intervals for new observations.”

EXCEL Excel can be used to find the regression equation, the standard error of estimate se, and the coefficient of determination

(labeled as R square). Enter the paired data in columns A and B.

To use Excel’s Data Analysis add-in, select Tools from the main menu, then select Data Analysis, followed by Regression, and then click OK. Enter the range for the y values, such as B1:B8. Enter the range for the x values, such as A1:A8. Click OK.

To use the Data Desk XL add-in, click DDXL and select Regression, then click on the Function Type box and select Simple Regression.

Click on the pencil icon for the response variable and enter the range of values for the y variable. Click on the pencil icon for the explanatory variable and enter the range of values for the x variable. Click on OK.

TI-83/84 PLUS The TI-83 84 Plus calculator can be used to find the linear correlation coefficient r, the equation of the regression line, the standard error of estimate se, and the coefficient of determination (labeled as r2). Enter the paired data in lists L1 and L2, then press STAT and select TESTS, and then choose the option LinRegTTest.

For Xlist enter L1, for Ylist enter L2, use a Freq (frequency) value of 1, and select  0.

Scroll down to Calculate, then press the ENTER key.

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10-4 Variation and Prediction Intervals 563 564 Chapter 10 Correlation and Regression