ch7-1 - Chapter 7 Hypothesis Tests and Condence Intervals...

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Chapter 7 Hypothesis Tests and Confidence Intervals in Multiple Regression (Part 1)
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Hypothesis Tests and Confidence Intervals for a Single Coefficient Recall the model we examined in Ch.6. We had a multiple regression model: TestScore = β 0 + β 1 ° STR + β 2 ° PctEL + u . Applying OLS estimation, the slope β 0 , β 1 and β 2 could be estimated: \ TestScore = 686.0 ± 1.10 ° STR ± 0.650 ° PctEL . You may want to test whether the estimated b β 1 is statistically significantly different from zero.
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Hypothesis Tests and Confidence Intervals for a Single Coefficient Testing the hypothesis H 0 : β j = β j ,0 vs. H 1 : β j 6 = β j ,0 1. Compute the standard error of b β j , SE ( b β j ) . 2. Compute the t -statistic, t = b β j ± β j ,0 SE ( b β j ) . 3. Compute the p -value = 2 Φ ° ± ± ± t act ± ± ² ,where t act is the value of the t -statistic actually computed. Reject the hypothesis at the 5% significance level if the p -value is less than 0.05 or, equivalently, if ± ± t act ± ± > 1.96. 95% confidence interval for β j : β j = [ b β j ± 1.96 SE ( b β j ) , b β j + 1.96 SE ( b β j )]
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Confidence Intervals for a Single Coefficient For our example, suppose SE ( b β 0 ) = 8.7, SE ( b β 1 ) = 0.43 and SE ( b β 2 ) = 0.031, \ TestScore = 686.0 ± 1.10 ° STR ± 0.650 ° PctEL . ( 8.7 ) ( 0.43 ) ( 0.031 ) To test β 1 = 0, the t -statistic is t act = ( ± 1.10 ± 0 ) /0.43 = ± 2.54.
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