2379-chapter12(withWelch)

2379-chapter12(withWelch) - MAT 2379 Introduction to...

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MAT 2379, Introduction to Biostatistics, Lecture Notes for Section 12.2 1 MAT 2379, Introduction to Biostatistics Chapter 12. Comparison of Two Independent Samples In this chapter, we will compare the means corresponding to two independent populations. For this, we will use the following methods, called methods of statistical inference : (a) interval estimation; (b) hypothesis testing. The case of dependent populations will be considered in Chapter 13. We skip Section 12.1. 12.2. Confidence Intervals and Tests for Means We denote by μ 1 , μ 2 the means of the two populations and by σ 2 1 , σ 2 2 their variances. From the first population, we draw a sample of size n 1 , whose mean is ¯ X 1 . From the second population, we draw a sample of size n 2 , whose mean is ¯ X 2 . A point estimator for μ 1 - μ 2 is ¯ X 1 - ¯ X 2 . A positive (respectively negative) observed value for this estimator is an indication that μ 1 might be larger (respectively smaller) than μ 2 . Example 1. (from p.219 of “Biostatistics. How it works” by Selvin) We want to examine if there is any difference in the final grade obtained in a statistics course between the male and female student populations. A sample of 37 male students has the mean ¯ x 1 = 85 . 738. A sample of 30 female students has the mean ¯ x 2 = 89 . 4. A point estimate for μ 1 - μ 2 is ¯ x 1 - ¯ x 2 = 85 . 738 - 89 . 4 = - 3 . 662. Since this a negative value, we might infer that μ 1 (the average grade for the male population) is smaller than μ 2 (the average grade for the student population). In what follows, we would like to refine this conclusion. From the Central Limit Theorem, we know that the distribution of ¯ X 1 is approximately normal with mean μ 1 and variance σ 2 1 /n 1 . Similarly, the distribution of ¯ X 2 is approximately normal with mean μ 2 and variance σ 2 2 /n 2 . Since the populations are independent , we can deduce the following important fact: The distribution of ¯ X 1 - ¯ X 2 is approximately normal with mean μ 1 - μ 2 and variance σ 2 1 /n 1 + σ 2 2 /n 2 (the word “approximately” can be removed if the two populations are normal). By the standard- ization procedure, we obtain that: the distribution of ( ¯ X 1 - ¯ X 2 ) - ( μ 1 - μ 2 ) q σ 2 1 /n 1 + σ 2 2 /n 2 is approximately standard normal In practice, the variances σ 2 1 and σ 2 2 are unknown. Therefore, one has to replace them by suitable estimators. In order to do this, we have to know if the populations are normal or not. (If the populations are normal, we do not need large sample sizes; if they are not normal, we do.) Moreover, we need to know if the variances σ 2 1 and σ 2 2 are equal or not. In summary, we have the following 4 cases: Case (1). The two populations are normal with known variances σ 2 1 and σ 2 2 . (We skip this case.) Case (2). The two populations are normal with unknown and equal variances σ 2 1 = σ 2 2 Case (3). The two populations are normal with unknown and unequal variances σ 2 1 6 = σ 2 2 Case (4). The two populations are arbitrary and have unknown variances σ 2 1 and σ 2 2 . (We need large sample sizes.)
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MAT 2379, Introduction to Biostatistics, Lecture Notes for Section 12.2
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