9-2 Two Means: Independent Samples 455 PART 1 Independent Samples: S1 and S2 Unknown and Not Assumed Equal This section involves two independent samples, and the following section deals with samples that are dependent because they consist of matched pairs. It is important to know the difference between independent samples and dependent samples. DEFINITIONS Two samples are independent if the sample values from one population are not related to or somehow naturally paired or matched with the sample values from the other population. Two samples are dependent (or consist of matched pairs) if the sample values are somehow matched, where the matching is based on some inherent relationship. (That is, each pair of sample values consists of two measurements from the same subject—such as before>after data—or each pair of sample values consists of matched pairs—such as husband>wife data—where the matching is based on some meaningful relationship.) Caution: “Dependence” does not require a direct cause>effect relationship. HINT If the two samples have different sample sizes with no missing data, they must be independent. If the two samples have the same sample size, the samples may or may not be independent. Here is an example of independent samples and another example of dependent samples consisting of matched pairs: ■ Independent Samples: Heights of Men and Women Data Set 1 “Body Data” in Appendix B includes the following heights (cm) of samples of men and women, and the two samples are not matched according to some inherent relationship. They are actually two independent samples that just happen to be listed in a way that might cause us to incorrectly think that they are matched. Heights (cm) of Men 172 154 156 158 169 Heights (cm) of Women 186 161 179 167 179 ■ Dependent Samples: Heights of Husbands and Wives Students of the author collected data consisting of the heights (cm) of husbands and the heights (cm) of their wives. Five of those pairs of heights are listed below. These two samples are dependent, because the height of each husband is matched with the height of his wife. Height (cm) of Husband 175 180 173 176 178 Height (cm) of Wife 160 165 163 162 166 For inferences about means from two independent populations, the following box summarizes key elements of a hypothesis test and a confidence interval estimate of the difference between the population means. situations: (1) The two population standard deviations are unknown but are assumed to be equal; (2) the unrealistic case in which two population standard deviations are both known. Do Real Estate Agents Get You the Best Prices? When a real estate agent sells a home, does he or she get the best price for the seller? This question was addressed by Steven Levitt and Stephen Dubner in Freakonomics. They collected data from thousands of homes near Chicago, including homes owned by the agents themselves. Here is what they write: “There’s one way to find out: measure the difference between the sales data for houses that belong to real-estate agents themselves and the houses they sold on behalf of clients. Using the data from the sales of those 100,000 Chicago homes, and controlling for any number of variables—location, age and quality of the house, aesthetics, and so on—it turns out that a real-estate agent keeps their own home on the market an average of ten days longer and sells it for an extra 3-plus percent, or $10,000 on a $300,000 house.” A conclusion such as this can be obtained by using the methods of this section. S
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