# That is a and the null hypothesis is h y x p p for

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An interesting application is testing for equal population means. That is, a = 0 and the null hypothesis is: 0 : H Y X 0 P 0 P For this test, the test statistic is: n s d t d Econ 325 – Chapter 10 3 Choose a significance level D (the probability of a Type I error). When testing against a two-sided alternative, a decision rule can be set by one of three equivalent methods. (1) Use the Appendix Table of the t-distribution to find a critical value that satisfies: c t 2 ) t t ( P c ) 1 n ( D " 0 Reject the null hypothesis if: c t t " For = 0.05 (a 5 % significance level), the graph below shows the critical value and the rejection region. D PDF of ) 1 n ( t 0 tc 0 -tc Upper Tail Area = 0.05 / 2 = 0.025 Lower Tail Area = 0.025 Reject 0 H m o do not reject m o Reject 0 H Econ 325 – Chapter 10 4

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(2) Calculate a p-value for the test as: p-value ) t t ( P ) 1 n ( " ¡ 0 2 The decision rule is reject the null hypothesis if: p-value < D The p-value must be calculated with computer software. (3) Calculate a ) 1 ( D 0 100 % confidence interval estimate for the difference in population means Y X P 0 P . The method was presented in Chapter 8.1. The decision rule is to reject the null hypothesis of equal populations means if the value zero is not contained between the lower and upper limits of the confidence interval estimate. Econ 325 – Chapter 10 5 Example: The stock market data set introduced in Chapter 8.1 contained observations for 20 successive business days described by: , , . . . , daily percentage returns for a company, and 1 x 2 x n x , , . . . , daily percentage returns for a market portfolio. 1 y 2 y n y The differences are generated as: i i i y x d 0 for i = 1, 2, . . . , 20 The sample mean and standard deviation of the differences were calculated as: 0.173 0 d and 1.391 d s To test the null hypothesis of equal population means for the two samples against a two-sided alternative the t-test statistic is: 0.556 20 1.391 0.173 0 0 n s d t d Econ 325 – Chapter 10 6
With Microsoft Excel, the p-value calculation ) t t ( P ) 1 n ( " ¡ 0 2 is obtained by selecting Insert Function: TDIST(0.556, 19, 2) This returns a p-value of 0.58. That is, for the t-distribution with 19 degrees of freedom, the area under the probability density function to the right of the value 0.556

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