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Paul Cartledge

Ch. 24 - Comparing Two Population Means Assumptions: 1. The two samples are random and independent. 2. At least one of the following is also true: i. Both samples are large (n ≥130 and n ≥ 32) ii. If either one or both sample sizes are small, then both populations from which the samples are drawn are normally distributed. 3. The standard deviations σ a1d σ of 2he two populations are unknown and unequal to each other; that is, σ1≠ σ 2 Checking the Assumptions: The first assumption can be “checked” by analyzing the experimental design. The 2 nd assumption can be “checked” just like in Ch. 23. The third should require a formal test s that is highly sensitive, but for now, check if max ≥ 2 . s min Hypotheses: As in Ch. 22, there are two population means (a.k.a. parameters) in our data structure and we consider them together as ONE parameter: µ – µ . 1hus,2we have H 0 µ 1 µ =20 H A µ 1 µ ≠20 Note that we could use any value to compare to, but zero has a ‘special’ interpretation. Also, tests can be one-sided, too. Test statistic: If the assumptions hold, then we may use the t-distribution. Thus, the standard error of y − y is 1 2 s2 s2 SE(y −1y ) 2 1 + 2 n1 n 2 and the test statistic 0 is y − y − ( µ − µ ) t0= 1 2 1 2 SE(y −1y ) 2 and t follows a t-distribution with a complicated df (see footnote on p. 657). Thus, we will instead use a conservative lower bound: df ≥ min{n – 1,1n – 1}.2 P-value: No different than how we calculated it in Ch. 23. Conclusion: Reject/do not reject as in one-sample test; answer hypotheses/question posed. Confidence Interval The (1 – α)100% CI for µ – 1 is2 y1− y 2± tα/2, dfSE(y 1 y )2 Assumptions: as per hypothesis test. Notes: - CI tends to be more informative than a test. - check if zero falls within the interval; check sign and magnitude. The Pooled t-Test rd Recall the three assumptions from the previous test. The 3 assumption now changes to 3. The standard deviations σ 1nd σ o2 the two populations are unknown and equal to each other; that is, 1 = σ2. (Checking the assumption reverses as well.) The consequence of this change is that the standard error of y1− y 2 now uses the pooled sample standard deviation, or s p n −1) s2 + (n −1)s2 s = 1 1 2 2
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