lecture12note - Psych 100A Winter 2010 1 Inference on two...

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1 Psych 100A Winter 2010 Lecture 12: Statistical Inference 1. Inference on two population means 2. Estimates of independent groups’ 1 - 2 a. Known variances b. Unknown but equal variance 3. Hypothesis testing for 1 - 2 4. Inference on two population means: Dependent samples 1 1. Inference on two population means To find the effect of a new treatment, we wish to compare at least two groups: One with treatment One sans treatment (or treatment #2) Experiment can be designed in two ways: 1. Independent (unpaired) samples. Objects selected from population #1 have no bearing on objects selected from population #2 2. Dependent (paired) samples. Each object selected from population #1 is naturally or forced paired with an object selected from population #2 2 2. Point estimates of 1 - 2 , independent groups Population of units If 1 - 2 , treatment 1 “better” than treatment 2 If 1 - 2 =0, treatment 1 equals than treatment 2 Inference Independent selection If 1 - 2 , treatment 2 “better” than treatment 1 Point Estimate: 2 1 2 1 x Treatment groups Sample #2 2 2 , n SRS Sample #1 1 1 , 3 2. Interval estimates of 1 - 2 , independent groups Require a model for the chance variation of 1 - 2 Assume that each sample came from a normal population, or that n 1 and n 2 are both sufficiently large (eg, >25 each) such that the CLT holds, then 2 D Can show that 4 2 2 2 1 2 1 2 1 2 1 , ~ N  2 2 2 2 2 1 1 1 1 , ~ , ~ Theorem: confidence interval for the population mean difference, 2 ’s known If x 11 , x 12 , …, x 1n 1 and x 21 , x 22 , …, x 2n 2 are independent SRSs from normal populations (or large independent SRSs) with unknown means ( 1 and 2 ) and known variances ( 1 2 and 2 2 ), then is a (1- )100% confidence interval for 1 - 2 (CI for 1 - 2 ). 5 2 1 2 2 2 1 2 1 2 2 1 n n z x x Example. Suppose we wish to compare two diets to determine the difference between their average weight losses. Based on the following SRSs, obtain a point estimate of 1 - 2 and 95% CI for 1 - 2 . Data: diet loss in gm A. 448, 229, 316, 105, 516, 496, 130, 242, 470, 195, 389, 97, 458, 347, 340, 212 B. 232, 200, 184, 180, 265, 125, 193, 322, 211 Assumptions: 1. Independent SRS of dieters A and B. 2. Weight losses are normally distributed 3. 2 s are known (say, 1 2 = 2 2 = 14,000 gm 2 ) 6
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2 Solution. Difference in mean weight loss A.
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lecture12note - Psych 100A Winter 2010 1 Inference on two...

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