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1 CHAPTER 13: INFERENCE ABOUT TWO POPULATIONS Stats & Prob. for Bus. Mgmt (Stat1100) Jochem Chap 13: Comparing 2 Populations box2 Goals of Chapter 13 Be able to do inference* about… box5 1. the difference between 2 populations means square6 Case1: σ 1 and σ 2 are unknown; it is assumed that σ 1 = σ 2 2 square6 Case2: σ 1 and σ 2 are unknown; it is assumed that σ 1 σ 2 square6 Both for independent samples and matched pairs . box5 2. the ratio of 2 population variances box5 3. the difference between 2 population proportions * i.e. computing an interval estimate and testing a hypothesis Chap 13: Comparing 2 Populations box2 1A. Difference b/w 2 means (indep. samples) box5 Recall from Chap 9 the sampling distribution of the difference between 2 means (slides 49-53): 3 square6 The difference between two RVs is … square6 normal if the two populations are normal. square6 approximately normal if the populations are non-normal and the sample sizes are large. Standard Error of the dif- ference between two means Chap 13: Comparing 2 Populations box2 1A. Difference b/w 2 means (indep. samples) box5 And thus we have that... square6 the z-test statistic is: 4 Problem: Wee need population variances! square6 the interval estimator is: Solution: If population variances unknown, then (as in Chap 12) we replace population variances ( σ 1 and σ 2 ) with sample variances (s 1 and s 2 ) and use t-values instead of z- values. (forthcoming)

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Chap 13: Comparing 2 Populations box2 1A. Difference b/w 2 means (indep. samples) box5 To use these formula we have to assume that we know the population standard deviations σ 1 and σ 2 . box5 The remainder of this chapter assumes that the 2 5 populations are:* square6 either normally distributed, or square6 non-normal but we have a large sample size. * If this is not satisfied, we can use a nonparametric test – the “Wilcoxon rank sum test”. We won’t cover this method in this course, but you find it in Chapter 19 of your book. Chap 13: Comparing 2 Populations box2 1A. Difference b/w 2 means (indep. samples) box5 When we replace σ 1 2 with s 1 2 and σ 2 2 with s 2 2 we can have 2 cases: square6 Case 1: We can assume that σ 1 = σ 2 6 square6 Case 2: We can assume that σ 1 ≠ σ 2 box5 For each case, the book gives us a test-statistic and an estimator. square6 (Note: we get more accurate estimates when σ 1 = σ 2 .) Chap 13: Comparing 2 Populations box2 1A. Difference b/w 2 means (indep. samples) box5 Test Statistic for μ 1 - μ 2 when σ 1 2 = σ 2 2 : square6 The t-test statistic is: 7 square6 where is the “pooled variance estimator” , which is the weighted average of the two sample variances (with the degrees of freedom of both samples used as the weights): This formula is called the “equal-variances test statistic.” Chap 13: Comparing 2 Populations box2 1A. Difference b/w 2 means (indep. samples) box5 Confidence Interval Estimator for μ 1 - μ 2 when σ 1 2 = σ 2 2 : square6 And subsequently: 8 This formula is called the “equal-variances confidence interval estimator.” Example
Chap 13: Comparing 2 Populations box2 1A. Difference b/w 2 means (indep. samples)

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