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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
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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.
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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.
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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.
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5 5 Slide © 2011 Cengage Learning Testing for Significance: t Test n Hypotheses n Test Statistic
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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
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7 7 Slide © 2011 Cengage Learning 1. Determine the hypotheses. 2. Specify the level of significance. 3. Select the test statistic. =.05 4. State the rejection rule. Reject H 0 if p -value <.05 or | t| > 3.182 (with 3 degrees of freedom) Testing for Significance: t Test
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8 8 Slide © 2011 Cengage Learning 5. Compute the value of the test statistic. 6. Determine whether to reject H 0. t = 4.541 provides an area of.01 in the upper tail. Hence, the p -value is less than.02. (Also, t = 4.63 > 3.182.) We can reject H 0. Testing for Significance: t Test
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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.
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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
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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
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12 Slide © 2011 Cengage Learning n Hypotheses n Test Statistic F = MSR/MSE MSR = SSR/1 Testing for Significance: F Test
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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
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14 Slide © 2011 Cengage Learning 1. Determine the hypotheses. 2. Specify the level of significance. 3. Select the test statistic. =.05 4. State the rejection rule. p -Value Method: Reject H 0 if p -value <.05 Critical Value Method: Reject H 0 if F > 10.13 (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
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15 Slide © 2011 Cengage Learning 5. Compute the value of the test statistic. 6. Determine whether to reject H 0. p -value Method: F = 21.43 lies between the values of 17.44 and 34.12 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: 21.43 > 10.13 so we reject H 0. F = MSR/MSE = 100/4.667 = 21.43 We have a significant relationship between the number of TV ads aired and the number of cars sold. Example: Reed Auto Sales
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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.
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17 Slide © 2011 Cengage Learning End
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