Testing Hypothesis about Coefficients n Use t tests for individual coefficients

Testing hypothesis about coefficients n use t tests

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Testing Hypothesis about Coefficients n Use t-tests for individual coefficients n Shows if a specific independent variable is relevant n Hypotheses: n H0: βj = 0 (no linear relationship) n H1: βj ≠ 0 (linear relationship does exist between xj and y) Ch. 12-20 12.4
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Evaluating Individual Regression Coefficients H0: βj = 0 (no linear relationship) H1: βj ≠ 0 (linear relationship does exist between xi and y) Test Statistic: ( df = n – k – 1) j b j S 0 b t - = (continued) Ch. 12-21
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Evaluating Individual Regression Coefficients Regression Statistics Multiple R 0.72213 R Square 0.52148 Adjusted R Square 0.44172 Standard Error 47.46341 Observations 15 ANOVA df SS MS F Significance F Regression 2 29460.027 14730.013 6.53861 0.01201 Residual 12 27033.306 2252.776 Total 14 56493.333 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% t-value for Price is t = -2.306, with p-value .0398 t-value for Advertising is t = 2.855, with p-value .0145 (continued) Ch. 12-22
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H0: βj = 0 H1: βj 0 d.f. = 15-2-1 = 12 α = .05 t12, .025 = 2.1788 The test statistic for each variable falls in the rejection region (p-values < .05) There is evidence that both Price and Advertising affect pie sales at = .05 From Excel output: Reject H0 for each variable Coefficients Standard Error t Stat P-value Price -24.97509 10.83213 -2.30565 0.03979 Advertising 74.13096 25.96732 2.85478 0.01449 Decision: Conclusion: Reject H0 Reject H0 α /2=.025 - tα/2 Do not reject H0 0 tα/2 α /2=.025 -2.1788 2.1788 Example: Evaluating Individual Regression Coefficients Ch. 12-23
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Confidence Interval Estimate for the Slope Confidence interval limits for the population slope βj Example: Form a 95% confidence interval for the effect of changes in price (x1) on pie sales: -24.975 ± (2.1788)(10.832) So the interval is -48.576 < β1 < -1.374 j b α/2 1, K n j S t b - - ± Coefficients Standard Error Intercept 306.52619 114.25389 Price -24.97509 10.83213 Advertising 74.13096 25.96732 where t has (n – K – 1) d.f. Here, t has (15 – 2 – 1) = 12 d.f. Ch. 12-24
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Confidence Interval Estimate for the Slope Confidence interval for the population slope βi Example: Excel output also reports these interval endpoints: Weekly sales are estimated to be reduced by between 1.37 to 48.58 pies for each increase of $1 in the selling price Coefficients Standard Error Lower 95% Upper 95% Intercept 306.52619 114.25389 57.58835 555.46404 Price -24.97509 10.83213 -48.57626 -1.37392 Advertising 74.13096 25.96732 17.55303 130.70888 (continued) Ch. 12-25
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Test on All Coefficients n F-Test for Overall Significance of the Model n Shows if there is a linear relationship between all of the X variables considered together and Y n Use F test statistic n Hypotheses: H0: β1 = β2 = … = βk = 0 (no linear relationship) H1: at least one βi ≠ 0 (at least one independent variable affects Y) Ch. 12-26 12.5
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F-Test for Overall Significance n Test statistic: where F has k (numerator) and (n – K – 1) (denominator) degrees of freedom n The decision rule is 1) K SSE/(n SSR/K s MSR F 2 e - - = = α 1, K n k, 0 F F if H Reject - - Ch. 12-27
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F-Test for Overall Significance 6.5386 2252.8 14730.0 MSE MSR F = = = Regression Statistics Multiple R 0.72213 R Square 0.52148 Adjusted R Square 0.44172 Standard Error 47.46341 Observations 15 ANOVA df SS MS F Significance F Regression 2 29460.027 14730.013 6.53861 0.01201 Residual 12 27033.306 2252.776 Total 14 56493.333 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% (continued) With 2 and 12 degrees of freedom P-value for the F-Test Ch. 12-28
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