Difference between means

Sunday, 18 August 2019

8:08 PM

9. Inferences concerning a difference of means 9.2 Independent samples Hypothesis test for We know (Central Limit Theorem) that —Exa a N (//2,— and X2 nt2 i=l 02 n2 i=l ,(a), ( means that these are exact results for any m , n2 if the populations are normal, approximate results for large m , if they are not) We also know that if Xl M (111 , 01) and M 02) are independent, then aX1 + + b/t2, a2a2i + + we deduce the sampling distribution of Xl — X2: X - & (2M -112, 2 2 21 + Now, as testing for Ho : = exactly amounts to testing for Ho — = 0, the one-sample procedure we introduced in Chapter 7 can be used up to some light adaptation, with Xl — as an estimator for — MATH2099/2859 (Statistics) Dr Jia Deng Term 2019 - Lecture 8 8/48

 

 

 

9. Inferences concerning a difference of means 9.2 Independent samples Hypothesis test for (with Assume for now that 01 = a, but a is unknown —+ estimate it ! Each squared deviation (Xli — Xl)2 is an estimator for 02 in population 1, and each squared deviation (X2i — X2)2 is an estimator for 02 in population 2 —+ we estimate 02 by pooling the sums of squared deviations from the respective sample means, thus we estimate 02 by the pooled variance estimator: s; ni + n2—2 I — X2)2 where S? 1 (Xli —Xl)2 and 1 Note: the pooled variance estimator has + — 2 degrees of freedom, because we have ni — 1 independent deviations from the mean in the first sample, and — 1 independent deviations from the mean in the second 2 sample —+ altogether, ni + — 2 independent deviations to estimate a MATH2099/2859 (Statistics) Dr Jia Deng Term 2019 - Lecture 8 15/48

 

 

 

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