Class_12 - Testing Hypotheses about the Difference between...

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Testing Hypotheses about the Difference between Two Means
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Random samples, partitioned into two independent subsamples (e.g., men and women). Question: Are the means of some variable (such as salary) significantly different between the two subsamples? Key: The sampling distribution of all theoretically possible differences between subsample means. For large samples (i.e., when the Central Limit Theorem holds), this sampling distribution of mean differences is normally shaped; for smaller samples, the sampling distribution takes the shape of one of the Student’s t distributions, identified by degrees of freedom .
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The key is: The difference between two means is a single value. In the case of these so-called “means difference tests,” the null hypothesis is that the means in general (i.e., in the universe) do NOT differ. Symbolically, H 0 : μ 2 - μ 1 = 0.00 There are two possible alternate hypotheses : nondirectional ; H 1 : μ 2 - μ 1 0.0 and directional , either H 1 : μ 2 - μ 1 > 0.0 or H 1 : μ 2 - μ 1 < 0.0
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In the 1997 General Social Survey, female respondents were asked whether or not their mothers had attended college. Then these female respondents were asked about their own education levels. A reasonable (alternate) hypothesis (H 1 ) would be: Women whose mothers attended college will themselves have more formal education than women whose mothers did not attend college. Thus, our two hypotheses are : H 1 : μ 2 - μ 1 > 0.0 H 0 : μ 2 - μ 1 = 0.00
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Because these two subsamples combined exceed 100, we know that the Central Limit Theorem applies. We can convert the difference between the value of the sample mean differences and the presumed value of the mean difference in the universe under the null hypothesis (i.e., 0.0) to z-values by using the estimated standard error of the difference (the standard deviation of the sampling distribution of sample mean differences). Recall that in general standard errors are estimated by dividing the standard deviation of the sample by the square root of the sample size, N s Y = σ ˆ
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However, in the case of this means difference test, we
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