Comparing Two Populations
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• If the data are paired, students should think about what the null hypothesis says about the mean of the differences. When the data are paired we can use our one-sample analysis tools.
• Do not use the pooled standard deviation for inference on the means from independent samples. This is an old-fashioned approach which is no longer well accepted. In practice, standard deviations are rarely equal even if the means are equal. If you assume they are equal, and it turns out that you're wrong, then the tests are invalid. On the other hand, if you assume they are unequal and do not pool, then there are no bad consequences whether or not the standard deviations are really equal.
• Pooled inference procedures are also not acceptable for a confidence interval on the difference between two proportions.
• However, pooled inference procedures for proportions are acceptable for a hypothesis test on the difference between two proportions, when the null hypothesis is that the two proportions are equal. When we hypothesize that the two proportions are the same, we proceed as if all of the data come from one population.
• Don't get confused -- or allow your students to get confused -- by the seemingly large number of different formulas. They may look different, but these formulas all have very similar structures. For hypothesis tests, the test statistics are (estimator-null value)/standard error, and for confidence intervals the structure is estimator +/- constant * standard error.
• There are two popular methods for approximating the degrees of freedom for a t-test comparing means from independent populations. (In fact, the degrees of freedom cannot be calculated exactly!) The calculator or computer produces the best approximation (usually a non-integer number of degrees of freedom using a fairly complex calculation), but a good runner-up is to take the smallest of n1-1 and n2-1.
• When doing hypothesis tests with paired data, there are two equivalent ways of writing the null hypothesis. The first is mudiff = 0, which emphasizes that the data are paired; or you can write mu1-mu2=0, which emphasizes the research question.
• For that matter, there are two equivalent ways of writing null hypotheses for unpaired comparisons, too. You can write mu1=mu2 or you can write mu1-mu2=0.