Unit 12: Comparing Two Populations
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Teaching Tips• There are three questions to that students might want to ask themselves before deciding what hypothesis test or confidence interval to use:
• 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.
• 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.
• 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.
• 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.
• It is important to keep stressing the same points from the previous two units over and over. These concepts can be difficult for students, so repetition is helpful.
Student Misconceptions and Confusions• Students will have a lot of trouble determining whether they need to do a one or two sample test; remind them that in order to do a two-sample test, they need two samples, which requires two sample sizes and two sets of sample statistics to work with.
• As in the previous unit, students find the null hypothesis difficult to identify in context. They need to practice writing null and alternative hypotheses. For a two-sample test, students should be comparing one population statistic to another, which is equivalent to the difference between two population statistics in comparison to zero. However, for a one-sample test, they would compare the population statistic to a specific value.
• 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.
• Re-stress that neither the p-value nor the significance level is the probability that the null hypothesis is true. The null hypothesis is either true or false and probability statements don't apply. Remember that the p-value is a conditional probability: we assume the null hypothesis is true when computing it.
• Re-stress that if the evidence is insufficient to reject the null hypothesis, don't write or say that you "accept" the null hypothesis. The best you can say is that you "fail to reject" the null hypothesis. The null hypothesis might still be false.
• As in the previous unit, students have trouble putting the conclusion of a hypothesis test in context because they get caught up with vocabulary and methodologies. Remind them that they are just answering a simple question. Also remind them that they are determining whether or not they have enough evidence to reject their null hypothesis, so being able to state that hypothesis in context is important for their conclusions.