It is a good idea to test for equal variances. If the test fails, check the ratio of the largest sample standard deviation to the smallest sample standard deviation. If the sample sizes are equal, or nearly equal, there should not be a problem unless this ratio is larger than 4. Even when the sample sizes are not nearly equal, there should not be a problem unless this ratio is greater than 2. And, more importantly, remember that you are using these tools to detect changes in the mean of Y, and the “problem” you are having is that you have detected changes in the variation of Y at the same time. Of course, that may help you satisfy the ultimate goal of reducing the variation of Y.
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