Comparing Two Populations
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Main Concepts: Two Samples
• This unit represents a milestone of sorts. There are no major conceptual topics to introduce. For the most part, from here on in, we're talking about applications of ideas. The Big Ideas themselves have already been covered. In this section we'll apply the Big Ideas in inference (sampling distributions, confidence intervals, hypothesis tests) to cover the particular problem of comparing the means or proportions of two populations.
• Don't forget to look at graphs! Often, the true research question is not "Are the means of these two groups different?" but simply "Are these two groups different?" Looking at a graph will give you additional insight into whether or not the shapes of the distributions are different, or whether the standard deviations are different, etc.
• You must be able to recognize when data are paired or not paired because your analysis depends on this. If the data are paired, the analysis will focus on the differences between pairs. While we would not go so far as to call this a Big Idea, it is frightfully important.
• You can also compare the proportions in two populations. At the AP level, we make heavy use of approximations to do this, and so a large sample size is required. Once you've mastered the basic steps of the hypothesis test, though, it's just a matter of adjusting a few of the details to fit this new situation.
• The format of the test statistic has not changed from the one-sample version. It is still (observed value - the null hypothesis value)/standard error. But the observed value will now be the difference between the sample means or proportions, and so you can apply the rules for standard deviations of random variables to figure out the standard error.
• Nor has the format of the confidence interval changed. It is still (estimator) +/- constant times standard error.
• More importantly, the interpretations of confidence intervals and hypothesis tests have not changed.