Six Sigma and Beyond: Statistics and Probability, Volume III

To perform a statistical test of a hypothesis, you must make certain assumptions about the data. The particular assumptions you must make depend on the statistical test you are using. Some procedures require stricter assumptions than others. The assumptions are needed so that you (or your computer) can figure out what the distribution of the statistic is. Unless you know the distribution, you cannot determine the correct significance levels. For the pooled-variance t test, you need to assume that you have two random samples with the same population variance. You also need to assume that the distribution of the means is approximately normal, which can happen one of two ways:

• The variable is normally distributed, so the means will automatically be normally distributed.

• The sample size is large enough to allow you to rely on the Central Limit Theorem to make sure that the means are distributed normally.

Of course, some assumptions are more important than others. Moderate violation of some of them may not have very serious consequences. Therefore it is important to know, for each statistical procedure, not only what assumptions are needed but also how severely their violation may influence the results. We will talk about these things when we discuss the different statistical procedures. For example, as mentioned earlier, the F test for equality of variances is quite sensitive to departures from normality. The t test for equality of means is less so.

Based on the means observed in two independent samples, how can you test the hypothesis that two population means are equal? Here is the procedure:

• To test the null hypothesis that two population means are equal, you must calculate the probability of seeing a difference at least as large as the one you have observed if no difference exists in the population.

• The hypothesis that no difference exists between the two population means is called the null hypothesis.

• The probability of seeing a difference at least as large as the one you have observed, when the null hypothesis is true, is called the observed significance level.

• If the observed significance level is small, usually less than .05, you reject the null hypothesis.

• If you reject the null hypothesis when it is true, you make a Type 1 error. If you do not reject the null hypothesis when it is false, you make a Type 2 error.

• The t test is used to test the hypothesis that two population means are equal.