250 W17 Lecture Notes Ch6 Two Proportions - Stats 250...

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Lecture Notes Page 99 Stats 250 Lecture Notes 6: Learning about the Difference in Population Proportions Part 1: Distribution for a Difference in Sample Proportions The Independent Samples Scenario Two samples are said to be independent samples when the measurements in one sample are not related to the measurements in the other sample. Independent samples are generated in a variety of ways. Some common ways: x Random samples are taken separately from two populations and the same response variable is recorded for each individual. x One random sample is taken and a variable is recorded for each individual, but then units are categorized as belonging to one population or another , e.g. old/young. x Participants are randomly assigned to one of two treatment conditions , and the same response variable, such as weight loss, is recorded for each individual unit. If the response variable is categorical , a researcher might compare two independent groups by looking at the difference between the two proportions There are usually two questions of interest about a difference in two population proportions. First, we want to estimate the value of the difference. Second, often we want to test the hypothesis that the difference is 0, which would indicate that the two proportions are equal. In either case, we will need to know about the sampling distribution for the difference in two sample proportions (from independent samples). Sampling Distribution for the Difference in Two Sample Proportions Example: Do Older People Snore More than Younger? Question of interest: How much of a difference is there between older adults and younger adults with regard to the proportion who snore? Study: Researchers at the National Sleep Foundation were interested in comparing the proportion of people who snore for two age populations (1 = older adults defined as over 50 years old and 2 = younger adults defined as between 18 and 30 years old). Let p 1 be the population proportion of all older adults who snore. Let p 2 be the population proportion of all younger adults who snore. We want to learn about p 1 and p 2 and how they compare to each other. We could estimate the difference p 1 p 2 with the corresponding difference in the sample proportions Will it be a good estimate? How close can we expect the difference in sample proportions to be to the true difference in population proportions (on average)? 2 1 ˆ ˆ p p ± . .
Lecture Notes Page 100

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