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Unformatted text preview: I. Comparison of Two Independent Populations Confidence Intervals : A 100( 1- ) % confidence interval for ( 1 - 2 ) when the population standard deviations are unknown is: If 1 and 2 are unknown but 1 = 2 , then a pooled estimate of the common variance is: Using this estimate, a 100(1- )% c.i. for ( 1 - 2 ) is: Example: Exercise 7.19, page 233. Hypothesis Testing Null and alternative hypotheses, type I and type II errors, test statistics To test H o : 1 = 2 compute t and reject H o in favor of H A of H A : 1 < 2 if t < - t H A : 1 > 2 if t > t H A : 1 2 if t < - t /2 or t > t /2 Example: Exercise 7.33, page 246 If 1 and 2 are unknown but 1 = 2 , then use the pooled Procedure . Sample size Calculation and Power A popular formula for computing the sample size for comparing two means (one-sided alternative) is: n 1 = ( 1 2 + 2 2 / k) (z + z ) 2 / ( 1 - 2 ) 2 n 2 = k n 1 The Wilcoxon-Mann-Whitney Test Nonparametric methods are used when: 1 1. The populations are not normal. 2 2. Data are qualitative. 3 3. Data are ranked. To compare two populations, suppose we have a sample of size n 1 from the first population and a sample of size n 2 from the second population (n 1 n 2 ). For each observation in sample 1, count the number of observations in sample 2 that are smaller in value. Let K 1 be the sum of these counts. Similarly, for each observation in sample 2, count the number of observations in sample 1 that are smaller in value. Let K 2 be the sum of these counts. The Wilcoxon-Mann-Whitney test statistic U s is the larger of K 1 and K 2 . We use Table 6 to find the critical values II. Comparison of Paired Samples Paired t-test and Confidence Interval In the matched pair designs we apply the one-sample t-procedure. Paired designs are used to free the comparisons from the effects of extraneous variables. Example: Problem #4 The Sign Test In many applications we may be interested in comparing matched pairs , when we have ranked or quantitative data....
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