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# 7.2 - Comparingtwomeans(7.2 Comparing two means Two-sample...

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Inference for distributions: Comparing two means (7.2)

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Comparing two means Two-sample z- test Two sample t -test Two-sample t -confidence interval Robustness
Two Independent samples Population 1 Sample 1 Population 2 Sample 2

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Compare the means of two normal distributions based on two independent random samples Completely randomized experiments with two treatments
Two-sample  z  statistic We have two independent simple random samples of sizes n 1 and n 2 from two 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

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When σ 1 and σ 2 are known, the two-sample z-statistic can be used to test hypotheses about μ 1 - μ 2 For example, for testing H 0 : μ 1 = μ 2 vs H a : μ 1 > μ 2 , the null Hypothesis is rejected at 5% significance level if A 95% confidence interval for μ 1 - μ 2 is
When σ 1 and σ 2 are unknown, does not have a t-distribution. It is only approximately t

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Use t when the sample sizes are small. How many degrees of freedom?
A better approximation is df = s 1 2 n 1 + s 2 2 n 2 2 1 n 1 - 1 s 1 2 n 1 2 + 1 n 2 - 1 s 2 2 n 2 2 This is more lengthy to calculate. That’s why df = smaller of ( n 1 − 1, n 2 1) is often used, which errs on the conservative side, producing smaller df and therefore wider confidence interval.

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7.2 - Comparingtwomeans(7.2 Comparing two means Two-sample...

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