hypothesis_testing2

hypothesis_testing2 - Chapter 10.1 Tests of the Difference...

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Chapter 10.1 Tests of the Difference Between Two Means Interval estimation for the difference between two population means was presented in the lecture notes for Chapters 8.1 and 8.2. An extension to hypothesis testing is of interest for applied work. For two random samples from populations with means and consider testing the null hypothesis: X μ Y μ population means are equal Y X 0 : H μ = μ against the two-sided alternative hypothesis: population means are not equal Y X 1 : H μ μ An equivalent way of expressing the problem is test: against 0 : H Y X 0 = μ μ 0 : H Y X 1 μ μ A one-sided alternative hypothesis can also be entertained. Econ 325 – Chapter 10 1

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The test method depends on the particular data set. Separate cases are: ‘matched pairs’ . The numeric observations are: ) y , x ( i i for i = 1, 2, . . . , n. independent samples with sample sizes and . That is, the two samples can have different sample sizes. x n y n First, develop results for the matched pairs data set. Descriptive statistics are denoted by: x , y sample means, , sample variances, and 2 x s 2 y s sample covariance. xy s From the observations calculate the differences: for i = 1, 2, . . . , n i i i y x d = Following the discussion in the lecture notes for Chapter 8.1, the sample mean and variance of the differences can be calculated as: y x d = and xy 2 y 2 x 2 d s 2 s s s + = Note the role of the covariance term in the variance calculation. With positive covariance, the variance of the differences will be reduced compared to using independent samples. Econ 325 – Chapter 10 2
For a comparison of the two population means, for some value a , test the null hypothesis: a : H Y X 0 = μ μ against a two-sided alternative. The test statistic is: n s a d t d = With the assumption of normal population distributions for the two sample means, the test statistic can be compared with a t-distribution with (n–1) degrees of freedom. An interesting application is testing for equal population means.

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This note was uploaded on 12/16/2010 for the course ECON 257 taught by Professor Galbreath during the Spring '10 term at McGill.

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hypothesis_testing2 - Chapter 10.1 Tests of the Difference...

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