Abstract:
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Student's t-test is appropriate for testing the difference of means for normally distributed data with equal variances, as is an approximate t-test for unequal variances. The conventional strategy chooses between them using Levene's test for equality of sample variances. Which test should be used if both samples have the same skewed distribution; what is the effect of unbalanced sample sizes? Examples have the dichotomous Bernoulli distribution with success probability .10. The sample means have binomial distributions, facilitating evaluation of significance comparing a sample of 20 with other samples, under the null hypothesis of the same distribution. Testing significance for an outcome from each sample is simplified by using a weighted data set. For this example, Levene's test is more sensitive than either t-test, so the conventional test is the same as unequal variances. For both sample sizes 20, each two-sided test is symmetric. As the larger sample size increases, the two t-tests favor opposite sides. This bias makes each t-test inappropriate for either two- or one-sided testing. Kurtosis associated with skewedness also makes the unequal variances test liberal.
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