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I'm investigating if two of our processes are producing products with the same or different average measurements.

The first process has been used for a very long time and thus I have a sample size of 430 measurements for it.

The second process is very new and only has a sample size of 15 measurements for it.

I've done an F-test to compare the sample variances for what I have now, and I have no evidence to suggest the variances of the samples are different.

When I perform a t-test for the differences of averages, I get results that indicate a very significant difference between the two processes' measurements.

Is there any test out there that will achieve the goal as the t-test, but will also control for or take into account the very large sample size differences?

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The Welch approximation t-test is designed to do the same thing as an independent samples t-test, but without relying on the assumption that the variances are equal. It is readily available in most standard statistical software. In R, it's actually the default for the t.test function (the argument var.equal controls whether a IST or Welch approximation is performed). Here's an interesting blog post advocating for the use of a Welch approximation regardless of whether or not you think you might be violating the assumption of equal variances.

Especially if you have very unequal sample sizes, you may wish to use bootstrapping instead since it doesn't make any assumptions at all about the distribution of the sample statistic. Here are instructions for a bootstrap t-test from another SE post.

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The t-test is not dependent on equal, similar, or even close sample sizes. A t-test can be done with any sample sizes. Go ahead and use the t-test you have. I wish I knew where people got the idea that a t-test requires equal sample sizes.

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    $\begingroup$ Since I am using sample variances the sample size effects the sample variance. $\endgroup$ – Omar123456789 Jun 2 '17 at 19:58
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    $\begingroup$ The reason sample size matters is that unequal variances don't pose a problem for a t-test with equal sample sizes. So as long as your sample sizes are equal, you don't have to worry about homogeneity of variances. Here's a detailed explanation with simulations: http://daniellakens.blogspot.ca/2015/01/always-use-welchs-t-test-instead-of.html $\endgroup$ – Rose Hartman Jun 2 '17 at 21:21

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