# IPS6eCh07_2-1 - Inference for Distributions Comparing Two...

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Inference for Distributions Comparing Two Means IPS Chapter 7.2 © 2009 W.H. Freeman and Company

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Objectives (IPS Chapter 7.2) Comparing two means Two-sample z statistic Two-samples t procedures Two-sample t significance test Two-sample t confidence interval Robustness Details of the two-sample t procedures
Comparing two samples Which is it? We often compare two treatments used on independent samples. Is the difference between both treatments due only to variations from the random sampling (B), or does it reflect a true difference in population means (A)? Independent samples: Subjects in one samples are completely unrelated to subjects in the other sample. Population 1 Sample 1 Population 2 Sample 2 (A) Population Sample 2 Sample 1 (B)

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Two-sample  statistic We have two independent SRSs (simple random samples) possibly coming from two distinct populations with ( μ 1 , σ 1 ) and ( μ 2 , σ 2 ). We use 1 and 2 to estimate the unknown μ 1 and μ 2 . When both populations are normal, the sampling distribution of ( 1 2 ) is also normal, with standard deviation : Then the two-sample z statistic has the standard normal N( 0 , 1 ) sampling distribution. 2 2 2 1 2 1 n n σ σ + 2 2 2 1 2 1 2 1 2 1 ) ( ) ( n n x x z σ σ μ μ + - - - = x x x x
Two independent samples  distribution We have two independent SRSs (simple random samples) possibly coming from two distinct populations with ( μ 1 , σ 1 ) and ( μ 2 , σ 2 ) unknown. We use ( 1 , s 1 ) and ( 2 , s 2 ) to estimate ( μ 1 , σ 1 ) and ( μ 2 , σ 2 ), respectively. To compare the means, 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. x x

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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
t = ( x 1 - x 2 ) - ( μ 1 - μ 2 ) SE Two-sample  t  significance test  The null hypothesis is that both population means μ 1 and μ 2 are equal, thus their difference is equal to zero. 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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