Unformatted text preview: mple mean and variance of the differences can be calculated as: d = x − y and 2
sd = s 2 + s 2 − 2 s xy x
y 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. 2 Chapter 11 For a comparison of the two population means, for some value a, test the null hypothesis: H0 : μ X − μ Y = a against a two‐sided alternative. The test statistic is: t= d−a sd
n 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. That is, a = 0 and the null hypothesis is: H0 : μ X − μ Y = 0 For this test, the test statistic is: 3 t= d
sd n Chapter 11 Choose a significance level α (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 t c that satisfies: P(t ( n − 1 ) > t c ) = α 2 Reject the null hypothesis if: t > tc For α = 0.05 (a 5% significance level), the graph below shows the critical value and the rejection region. PDF...
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- Spring '10
- Null hypothesis, Trigraph, XY sex-determination system