1 1 Slide © 2011 Cengage Learning Assumptions About the Error Term  1. The error  is a random variable with mean of zero. 2. The variance of , denoted by  2, is the same for all values of the independent variable. all values of the independent variable. 2. The variance of , denoted by  2, is the same for all values of the independent variable. all values of the independent variable. 3. The values of  are independent. 4. The error  is a normally distributed random variable. variable. 4. The error  is a normally distributed random variable. variable. Chapter 12 – Simple Linear Regression

2 2 Slide © 2011 Cengage Learning 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.

3 3 Slide © 2011 Cengage Learning n An Estimate of  2 Testing for Significance where: s 2 = MSE = SSE/( n  2) The mean square error (MSE) provides the estimate of  2, and the notation s 2 is also used.

4 4 Slide © 2011 Cengage Learning Testing for Significance An Estimate of  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.

5 5 Slide © 2011 Cengage Learning Testing for Significance: t Test n Hypotheses n Test Statistic

6 6 Slide © 2011 Cengage Learning n Rejection Rules where: t  is based on a t distribution with n - 2 degrees of freedom p -Value Approach: Reject H 0 if p -value <  Critical Value Approach: Reject H 0 if t t  /2 Testing for Significance: t Test

7 7 Slide © 2011 Cengage Learning 1. Determine the hypotheses. 2. Specify the level of significance. 3. Select the test statistic.  = State the rejection rule. Reject H 0 if p -value <.05 or | t| > (with 3 degrees of freedom) Testing for Significance: t Test

8 8 Slide © 2011 Cengage Learning 5. Compute the value of the test statistic. 6. Determine whether to reject H 0. t = provides an area of.01 in the upper tail. Hence, the p -value is less than.02. (Also, t = 4.63 > ) We can reject H 0. Testing for Significance: t Test

9 9 Slide © 2011 Cengage Learning 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.

10 Slide © 2011 Cengage Learning The form of a confidence interval for  1 is: 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

11 Slide © 2011 Cengage Learning 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  3.182(1.08) = 5  3.44 or 1.56 to 8.44 n Rejection Rule 95% Confidence Interval for  1 95% Confidence Interval for  1 n Conclusion

12 Slide © 2011 Cengage Learning n Hypotheses n Test Statistic F = MSR/MSE MSR = SSR/1 Testing for Significance: F Test

13 Slide © 2011 Cengage Learning n Rejection Rule where F  is the value leaving an area if  in the upper tail of the F distribution with 1 degree of freedom in the numerator and n - 2 degrees of freedom in the denominator. Testing for Significance: F Test p -Value Approach: Reject H 0 if p -value <  Critical Value Approach: Reject H 0 if F > F 

14 Slide © 2011 Cengage Learning 1. Determine the hypotheses. 2. Specify the level of significance. 3. Select the test statistic.  = State the rejection rule. p -Value Method: Reject H 0 if p -value <.05 Critical Value Method: Reject H 0 if F > (1 d.f. in numerator and 3 d.f. in denominator) (1 d.f. in numerator and 3 d.f. in denominator) F = MSR/MSE Example: Reed Auto Sales

15 Slide © 2011 Cengage Learning 5. Compute the value of the test statistic. 6. Determine whether to reject H 0. p -value Method: F = lies between the values of and in the F table, so the p -value is between.025 and.01. Thus, the p -value is below.05 and we reject H 0. Critical Value Method: > so we reject H 0. F = MSR/MSE = 100/4.667 = We have a significant relationship between the number of TV ads aired and the number of cars sold. Example: Reed Auto Sales

16 Slide © 2011 Cengage Learning Caution about the Interpretation of Significance Tests 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.

17 Slide © 2011 Cengage Learning End