Comparing Two Populations |
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Main Concepts | Demonstration | Activity | Teaching Tips | Data Collection & Analysis | Practice Questions | Milestone | Fathom Tutorial | ||

Teaching Tips • 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 • However, pooled inference procedures for • 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 n • When doing hypothesis tests with paired data, there are two
equivalent ways of writing the null hypothesis. The first is mu • For that matter, there are two equivalent ways of writing
null hypotheses for unpaired comparisons, too. You can write mu |