Comparing Two Population Means: Elementary Statistical Inference by Vincent LeMoine, Study notes of Statistics

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STAT 201 Section 501

Elementary Statistical Inference

Vincent LeMoine Texas A&M University Department of Statistics

April 17, 2004

Chapter 7

Section 7.2 : Comparing Two Means

Two–sample problems are among the most com- mon sitations encountered in statistical prac- tice.

Two Sample Problems:

• The goal of inferece is to compare the re- sponses 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.

The Two Sample z Statistic

The natural estimator of μ 1 − μ 2 is ¯x 1 − x¯ 2.

Now the addition rule for variance states that the variance of ¯x 1 − ¯x 2 is

σ 12 n 1

σ^22 n 2

Now if the two population distributions are both normal, then the distribution of ¯x 1 − ¯x 2 is also normal.

See Example 7.13 on pages 527-528.

Definition: The 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 independent SRS of size n 2 drawn from an N (μ 2 , σ 2 ) population. Then the two-sample z statistic

z = (¯x^1 −^ √¯x^2 )^ −^ (μ^1 −^ μ^2 ) σ 12 n 1 +^

σ 22 n 2

has the standard normal N (0, 1) sampling dis- tribution.