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Chapter19

The Basic Practice of Statistics (Paper) & Student CD

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Two-sample problems for population means BPS chapter 19 © 2006 W.H. Freeman and Company
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Objectives (BPS chapter 19) Comparing two population means Two-sample t procedures Examples of two-sample t procedures Using technology Robustness again Details of the t approximation Avoid the pooled two-sample t procedures Avoid inference about standard deviations The F test for comparing two standard deviations
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Conditions for inference comparing two means We have two independent SRSs (simple random samples) coming from two distinct populations (like men vs. women) with ( μ 1 , σ 1 ) and ( μ 2 , σ 2 ) unknown. Both populations should be Normally distributed. However, in practice, it is enough that the two distributions have similar shapes and that the sample data contain no strong outliers.
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SE = s 1 2 n 1 + s 2 2 n 2 s 1 2 n 1 + s 2 2 n 2 df μ 1 μ 2 x 1 " x 2 The two-sample t statistic follows approximately the t distribution with a standard error SE (spread) reflecting variation from both samples: Conservatively, the degrees of freedom is equal to the smallest of ( n 1 1, n 2 1).
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t = ( x 1 " x 2 ) " ( μ 1 " μ 2 ) SE Two-sample t -test The null hypothesis is that both population means μ 1 and μ 2 are equal, thus their difference is equal to zero. H 0 : μ 1 = μ 2 <=> μ 1 μ 2 = 0 with either a one-sided or a two-sided alternative hypothesis. We find how many standard errors (SE) away from ( μ 1 μ 2 ) is ( 1 2 ) by standardizing with t: Because in a two-sample test H 0 poses ( μ 1 - μ 2 ) = 0, we simply use with df = smallest( n 1 1, n 2 1) t = x 1 " x 2 s 1 2 n 1 + s 2 2 n 2 x x
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Does smoking damage the lungs of children exposed to parental smoking? Forced Vital Capacity (FVC) is the volume (in milliliters) of air that an individual can exhale in 6 seconds.
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