Chapter7.2 - 7.2 Comparing Two Means Overview One of the...

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7.2 Comparing Two Means Overview One of the most commonly used significance tests is the comparison of two population means 1 μ and 2 . 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. A two sample problem can arise from a randomized comparative experiment that randomly divides the subjects into two groups and exposes each group to a different treatment. Unlike the matched pairs designs studied earlier, there is no matching of the units in the two samples, and the two samples may be of different sizes. Inference procedures for two-sample data differ from those for matched pairs.
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The two sample z statistic Example 1 A fourth-grade class has 10 girls and 7 boys. The difference 2 1 x x - between the female and male mean heights varies in different random samples. The sampling distribution has mean 4 . 0 1 . 54 5 . 54 2 1 = - = - μ μ inch
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Figure Distributions for Example 7.13. (a) Distributions of heights of 10-year-old boys and girls. (b) Distribution of the difference between means of 10 girls and 7 boys. Two-Sample z Statistic Suppose that 1 x is the mean of an SRS of size 1 n drawn from N( 1 μ , 1 σ ) population and that
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2 x is the mean of an SRS of size 2 n drawn from N( 2 μ , 2 σ ) population. Then the two- sample z statistic ( 29 ( 29 2 2 2 1 2 1 2 1 2 1 n n x x z + - - - = has the standard normal N(0,1) sampling distribution. In the unlikely event that both population standard deviations are known, the two- sample z statistic is the basis for inference about 2 1 μ- . Exact z procedures are seldom used because 1 and 2 are rarely known. The two-sample t procedures Suppose that the population standard deviations 1 and 2 are not known. We estimate them by the sample standard deviations 1 s and 2 s from our two samples. The Two-Sample t Significance test Suppose that an SRS of size 1 n is drawn from a normal population with unknown mean 1
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and that an independent SRS of size 2 n is drawn from another normal population with unknown mean 2 μ . To test the hypothesis 2 1 0 : μ= H , compute the two-sample t statistic 2 2 2 1 2 1 2 1 n s n s x x t + - = and use P-values or critical values for the t(k) distribution, where the degrees of freedom k are the smaller 1 1 - n and 1 2 - n .
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Chapter7.2 - 7.2 Comparing Two Means Overview One of the...

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