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I've recently started to brush up my statistics, and have begun doing so with two of the most often used statistical tools in my field: the t-test and ANOVA.

Reading up on popular information sources such as wikipedia (none of which provided citable references for the following statements - sadly) I have concluded that the null hypotheses tested by these tools are:

  • t-test: The t-test is a method used to test the null hypothesis that means of two compared groups are equal.

  • (one-way) ANOVA: Formally, ANOVA tests the null hypothesis that the means of multiple defined groups are equal.

My adviser told me both these statements are very wrong, and that “inference-statistical methods do not test the equality of means”; but offered no further clarification. I'm wondering whether I'm misrepresenting the information I have read.

So, could you tell me what the null hypotheses of these tests are then? also, do you know what sources I could cite for this (since it's obvious I'm not stating my own independent conclusions)?

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    $\begingroup$ Sounds like you've got the right idea to me. You should ask your advisor to clarify what he felt was wrong about your intuition and to provide an alternative interpretation of these tests instead of just telling you "You're wrong." $\endgroup$ – David Marx Dec 1 '13 at 1:49
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    $\begingroup$ You could replace 'mean' by 'true mean' or 'population mean' to make sure that we are not talking about the sample means. Otherwise you got it right. $\endgroup$ – Michael M Dec 1 '13 at 9:13
  • $\begingroup$ One- way ANOVA and t-test should give you same result. These tests are fixed-effects tests. Please specify the data as well as purpose of your study. Before you apply these tests, you should make sure that the data meets appropriate conditions - large or small sample, data is standard-normal or random etc. There is nothing wrong with these tests. $\endgroup$ – Subhash C. Davar Dec 1 '13 at 13:50
  • $\begingroup$ could any of you point me to some citable sources which explicitly name the null hypotheses? $\endgroup$ – TheChymera Dec 5 '13 at 9:50
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There is the distinction between the population mean(s) that you want to make a pronouncement about by investigating the sample mean(s) and sample variance(s). You description isn't sufficiently explicit on that front.

Another subtlety that could be seen as a mistake if one applies an unfavorable reading is that the t-test compares the means of two populations. While that is by far the most common scenario, t-tests can also be used to compare the mean of one population to an arbitrary constant number.

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