Chapter 12b Testing for significance—the t-test Developing confidence intervals for estimates of β 1. Testing for significance—the f-test Using Excel’s regression tool.

Testing for Significance To test for a significant regression relationship, we To test for a significant regression relationship, we must conduct a hypothesis test to determine whether must conduct a hypothesis test to determine whether the value of  1 is zero. the value of  1 is zero. To test for a significant regression relationship, we To test for a significant regression relationship, we must conduct a hypothesis test to determine whether must conduct a hypothesis test to determine whether the value of  1 is zero. the value of  1 is zero. Two tests are commonly used: Two tests are commonly used: t Test and F Test Both the t test and F test require an estimate of  2, Both the t test and F test require an estimate of  2, the variance of  in the regression model. the variance of  in the regression model. Both the t test and F test require an estimate of  2, Both the t test and F test require an estimate of  2, the variance of  in the regression model. the variance of  in the regression model.

An Estimate of  Testing for Significancewhere: s 2 = MSE = SSE/( n  2) The mean square error (MSE) provides the estimate of  2, and the notation s 2 is also used.

Testing for Significance An Estimate of  To estimate  we take the square root of  2. To estimate  we take the square root of  2. The resulting s is called the standard error of The resulting s is called the standard error of the estimate. the estimate.

Hypotheses Test Statistic Testing for Significance: t Test

n Rejection Rule Testing for Significance: t Test where: t  is based on a t distribution with n - 2 degrees of freedom Reject H 0 if t t 

1. Determine the hypotheses. 2. Specify the level of significance. 3. Select the test statistic.  = State the rejection rule. Reject H 0 if | t| > Using the Test Statistic Using the Test Statistic Testing for Significance: t Test (3 degrees of freedom)

Testing for Significance: t Test 5. Compute the value of the test statistic. 6. Determine whether to reject H 0. At the.05 level of significance, the sample At the.05 level of significance, the sample evidence indicates that there is a significant relationship between the number of TV ads aired and the number of cars sold. Because t = 4.63 > 3.182, we reject H 0. Using the Test Statistic Using the Test Statistic

Confidence Interval for  1 H 0 is rejected if the hypothesized value of  1 is not H 0 is rejected if the hypothesized value of  1 is not included in the confidence interval for  1. included in the confidence interval for  1. We can use a 95% confidence interval for  1 to test We can use a 95% confidence interval for  1 to test the hypotheses just used in the t test. the hypotheses just used in the t test.

The form of a confidence interval for  1 is: Confidence Interval for  1 where is the t value providing an area of  /2 in the upper tail of a t distribution with n - 2 degrees of freedom b 1 is the pointestimator is the margin of error

Rejection Rule 95% Confidence Interval for  1 Conclusion Confidence Interval for  1 Reject H 0 if 0 is not included in the confidence interval for  1. 0 is not included in the confidence interval. Reject H 0 = 5 +/ (1.08) = 5 +/ or 1.56 to 8.44

n Hypotheses n Test Statistic Testing for Significance: F Test F = MSR/MSE

n Rejection Rule Testing for Significance: F Test where: F  is based on an F distribution with 1 degree of freedom in the numerator and n - 2 degrees of freedom in the denominator Reject H 0 if F > F 

1. Determine the hypotheses. 2. Specify the level of significance. 3. Select the test statistic.  = State the rejection rule. Reject H 0 if F > Using the Test Statistic Using the Test Statistic Testing for Significance: F Test F = MSR/MSE (1 d.f. in numerator, 3 d.f. in denominator) 3 d.f. in denominator)

Testing for Significance: F Test 5. Compute the value of the test statistic. 6. Determine whether to reject H 0. Because F = > 10.13, we reject H 0. Using the Test Statistic Using the Test Statistic F = MSR/MSE = 100/4.667 = At the.05 level of significance, the statistical At the.05 level of significance, the statistical evidence is sufficient to conclude that we have a significant relationship between the number of TV ads aired and the number of cars sold.

Some Cautions about the Interpretation of Significance Tests Just because we are able to reject H 0 :  1 = 0 and Just because we are able to reject H 0 :  1 = 0 and demonstrate statistical significance does not enable demonstrate statistical significance does not enable us to conclude that there is a linear relationship between x and y. Rejecting H 0 :  1 = 0 and concluding that the Rejecting H 0 :  1 = 0 and concluding that the relationship between x and y is significant does not enable us to conclude that a cause-and-effect relationship is present between x and y.

Using Excel’s Regression Tool The Regression tool can be used to perform a The Regression tool can be used to perform a complete regression analysis. complete regression analysis. Excel also has a comprehensive tool in its Data Excel also has a comprehensive tool in its Data Analysis package called Regression. Analysis package called Regression. Up to this point, you have seen how Excel can be Up to this point, you have seen how Excel can be used for various parts of a regression analysis. used for various parts of a regression analysis.

Using Excel’s Regression Tool n Formula Worksheet (showing data)

Using Excel’s Regression Tool n Performing the Regression Analysis Step 3 Choose Regression from the list of Analysis Tools Analysis Tools Step 2 Choose the Data Analysis option Step 1 Select the Tools pull-down menu

Using Excel’s Regression Tool n Performing the Regression Analysis Step 4 When the Regression dialog box appears: Enter C1:C6 in the Input Y Range box Enter C1:C6 in the Input Y Range box Enter B1:B6 in the Input X Range box Enter B1:B6 in the Input X Range box Select Labels Select Labels Select Confidence Level Select Confidence Level Enter 95 in the Confidence Level box Enter 95 in the Confidence Level box Select Output Range Select Output Range Enter A9 (any cell) in the Ouput Range box Enter A9 (any cell) in the Ouput Range box Click OK to begin the regression analysis Click OK to begin the regression analysis

Using Excel’s Regression Tool n Regression Dialog Box

Using Excel’s Regression Tool n Value Worksheet ANOVA Output Regression Statistics Output Data Estimated Regression Equation Output

Using Excel’s Regression Tool Note: Columns F-I are not shown. n Estimated Regression Equation Output (left portion)

Using Excel’s Regression Tool Note: Columns C-E are hidden. n Estimated Regression Equation Output (right portion)

Using Excel’s Regression Tool n ANOVA Output

Using Excel’s Regression Tool n Regression Statistics Output

n Confidence Interval Estimate of E ( y p ) n Prediction Interval Estimate of y p Using the Estimated Regression Equation for Estimation and Prediction where: confidence coefficient is 1 -  and t  /2 is based on a t distribution with n - 2 degrees of freedom

If 3 TV ads are run prior to a sale, we expect the mean number of cars sold to be: Point Estimation^ y = (3) = 25 cars

Using Excel to Develop Confidence and Prediction Interval Estimates n Formula Worksheet (confidence interval portion)

Using Excel to Develop Confidence and Prediction Interval Estimates n Value Worksheet (confidence interval portion)

The 95% confidence interval estimate of the mean number of cars sold when 3 TV ads are run is: Confidence Interval for E ( y p ) = to cars

Using Excel to Develop Confidence and Prediction Interval Estimates n Formula Worksheet (prediction interval portion)

Using Excel to Develop Confidence and Prediction Interval Estimates n Value Worksheet (prediction interval portion)

The 95% prediction interval estimate of the number of cars sold in one particular week when 3 TV ads are run is: Prediction Interval for y p = to cars