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handout_7_2 - 7.2 Comparing Two Means two-sample problems...

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7.2 Comparing Two Means two-sample problems are among the most commonly encountered in statistics compare control group and treatment group Two-sample problems •the goal of inference is to compare the responses in two groups •each group is considered to be a sample from a distinct population •the responses in each group are independent of those in the other group Notation used to describe the two populations: Population Variable Mean Standard deviation 1 x 1 μ 1 σ 1 2 x 2 μ 2 σ 2 Inference is based on two independent SRSs, one from each population. Population Sample size Sample mean Sample sd 1 n 1 x 1 s 1 2 n 2 x 2 s 2 We are interested in the difference μ 1 – μ 2 . The difference x 1 - x 2 is a natural estimator.

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The mean and variance of x select an SRS of size n from a population and measure variable X on each individual in sample the n measurements are values of n random variables- X 1 , X 2 X n •if population is large relative to the size of the sample- X 1 , X 2 X n considered independent •X i is measurement on one individual selected at random from pop→ has the pop distribution x = n 1 ( X 1 + X 2 +…+ X n ) 2
Two-sample z statistic Suppose that x 1 is the mean of an SRS of size n 1 drawn from an N(μ 1 , σ 1 ) population and that x 2 is the mean of an SRS of size n 2 drawn from an N(μ 2 , σ 2 ) population. Then the two-sample z statistic z = 2 2 2 1 2 1 2 1 2 1 ) ( ) ( n n x x σ σ μ μ + - - - has the standard normal N(0,1) distribution.

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handout_7_2 - 7.2 Comparing Two Means two-sample problems...

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