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I'm trying to analyse an experiment I performed. I executed 9 tests measurements, at two different settings (A and B); I expect it to be normal distributed. I had the following hypotheses:

  • $H_0:\; \mu_A = \mu_B$
  • $H_1:\; \mu_A \neq \mu_B$

I got the following results for my measurements

  • A: n=9, mean = 1.66, sd = 1.12
  • B: n=9, mean = 2.48, sd = 1.16

I would like to know whether this difference is significant (say at 95%), but I get a bit stuck - I only learned how to do this when I know $\mu_A$ or $\sigma_A$ (or have a very good estimate of it due to a large sample), but I only have a sample standard deviation for A.

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Your terminology is somewhat confusing here: it appears to suggest that you have done 9 different statistical tests, but after consideration, I assume you mean you have simply measured the value of interest 9 times for each setting.

Since there is no obvious reason for this (time dependence of measurements or similar) mentioned in your question, I'm also going to assume the measurements aren't paired in any way.

Your sample size is rather small, but if it is safe to say that you have normality (i.e. the measurements of the value of interest are distributed normally), then a two-sample T-test will do. I cannot tell you whether this is the case, because I don't have more information on your setting, but I can tell you not to rely upon tests that check for normality. You also have to consider whether the variance in both settings is the same (whether or not this is credibly the case changes the way the variances are used in the test).

An alternative, that doesn't require assumptions, but that tends to be less powerful, is a nonparametric test, like the Wilcoxon Rank Sum test (mentioned in the Wikipedia link above). Note however, that this doesn't really test for a difference in mean unless you have the assumption of location shift. Since you're probably not really interested in a difference in means, but in whether there is any difference between the two settings, this should not be an issue.

Most if not all statistical software includes these two tests. I believe even Excel has them, but I wouldn't trust that for the world.

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  • $\begingroup$ Thanks for your detailed comment, especially about terminology - that's useful for beginners to this field. I don't have a reason to assume the data is not normal distributed, I should have added that. $\endgroup$ – Frank Meulenaar Jul 14 '11 at 10:32

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