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# Chapter7print - STAT503 Fall 2009 Lecture Notes Chapter 7 1...

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STAT503 — Fall 2009 Lecture Notes: Chapter 7 1 Chapter 7: Comparison of Two Independent Samples October 13, 2009 7.1 Introduction We are often interested in comparing two populations (or groups) based on a continuous measurement. Postmortem serotonin levels in patients who died of heart disease vs. those who died from other causes (control group). To evaluate a new dietary supplement for beef cattle, one group gets a standard diet and a second group gets the standard diet plus a supplement. Observe weight gains. (What are similarities and differences between these two studies?) Two samples may be compared in a number of different ways: 1. Comparison of means. 2. Comparison of standard deviations. 3. Comparison of (shapes of) distributions. We will be primarily concerned with the comparison of means . Some new notation: To differentiate between two populations or samples we will use subscripts. Population 1: Normal( μ 1 , σ 1 ) . I Take sample of size n 1 and calculate ¯ y 1 , s 1 , SE 1 = s 1 n 1 , SS 1 = ( n 1 - 1) s 2 1 . Population 2: Normal( μ 2 , σ 2 ) . I Take sample of size n 2 and calculate ¯ y 2 , s 2 , SE 2 = s 2 n 2 , SS 2 = ( n 2 - 1) s 2 2 .

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STAT503 — Fall 2009 Lecture Notes: Chapter 7 2 Always let the reader know which subscript goes with which population or group! When comparing two means it seems reasonable to ask if μ 1 = μ 2 or, equivalently, if μ 1 - μ 2 = 0. Our estimate of this difference of population means will be the difference of sample means, ¯ y 1 - ¯ y 2 . 7.2 Standard Error of y 1 - ¯ y 2 ) Recall that the precision of the sample mean ¯ y as an estimate of the population mean μ was expressed by standard error SE ¯ y = s n . To estimate the standard deviation of ¯ Y 1 - ¯ Y 2 . we need to define standard error of the difference (¯ y 1 - ¯ y 2 ). There are two methods: “ unpooled ” and “ pooled ”. Both are based on some combination of s 1 and s 2 . We use the “pooled” method only when we think σ 1 = σ 2 or when specifically told to do so. When n 1 = n 2 or s 1 = s 2 they give the same result. We will use the “unpooled” (easier) method most of the time. The unpooled standard error The ( unpooled ) standard error of ¯ y 1 - ¯ y 2 is defined to be SE ¯ y 1 - ¯ y 2 = q SE 2 1 + SE 2 2 SE ¯ y 1 - ¯ y 2 = s s 2 1 n 1 + s 2 2 n 2 = s SS 1 / ( n 1 - 1) n 1 + SS 2 / ( n 2 - 1) n 2 All the formulas give the same result. Which one you use depends on what is given in the problem. It is called “unpooled” because we calculate the SE’s separately and only then combine them (like in Pythagorean theorem). Example: Suppose we observed the following data for our postmortem serotonin level study Serotonin (ng/g) Heart Disease Controls n 8 12 ¯ y 3840 5310 SE 850 640
STAT503 — Fall 2009 Lecture Notes: Chapter 7 3 The (unpooled) standard error of ¯ y 1 - ¯ y 2 is SE ¯ y 1 - ¯ y 2 = p 850 2 + 640 2 = 1064 . 002 .

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Chapter7print - STAT503 Fall 2009 Lecture Notes Chapter 7 1...

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