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I have some data on frog weights for a small sample ($n \approx 30$). I need to compare the mean frog weight from this sample $\mu_s$ to the mean of another, much larger sample of frog weights $\mu_r$ (since it comes from a $r$eference distribution).

I know everything about my sample: I have all the data points and I can compute any statistics. However, I do not have any specific data for the reference distribution other than the mean, the sample size, and the relative standard error.

What is the proper test or general method to to determine if $\mu_s$ is statistically different from $\mu_r$?

I should also note there may be a better way to approach this altogether -- I am open to suggestions!

kjetil b halvorsen
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MH765
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  • If you can assume that both samples are drawn from a normal population, the appropriate statistical test is the two-sample $t$-test for comparing means. If you apply it, you will have to decide first whether or not the variances of both populations can be assumed equal. If you have no clue about this, you can infer it from the data by using e.g. a Chi Square test, or Levene's test. – StijnDeVuyst May 18 '15 at 23:01
  • @StijnDeVuyst numerous papers that have studied the performance of such a test-the-assumption-to-choose-the-test approach recommend *not* using a test of variances to decide between a equal-variance vs pooled (Welch-type) test, since it adversely affects the properties of the following t-tests. Their recommendation is that if you aren't fairly sure the populations are close to equally variable, use Welch (though if the sample sizes are the same it won't really matter). – Glen_b May 19 '15 at 02:14
  • mah765 -- What's RSE? Is that sample standard deviation, or something else? (please edit your response into your question) – Glen_b May 19 '15 at 02:16
  • Glen_b, thank you -- I've clarified that RSE is relative standard error. – MH765 May 19 '15 at 13:49
  • OK -- it sounds like the Welch t-test is the way to go. Thanks! – MH765 May 22 '15 at 22:49
  • @Glen_b: I was not aware of such studies, thank you. I usually entertain both assumptions in parallel and most of the time, the results are very close. Your remark: "if the sample sizes are the same it won't really matter", does that mean the Satterthwaite-Welch approximation becomes exact if the sample sizes are equal? I do not think this is so. – StijnDeVuyst May 23 '15 at 20:13
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    @StijnDeVuyst I mean "won't really matter" in that there's not much difference in performance (significance level and power) between the two tests when sample sizes are equal, in spite of unequal variances. In that case, both tests have approximately correct significance levels and their power properties are similar. On equality of variance and choosing between the tests see for example [here](http://stats.stackexchange.com/questions/97098/practically-speaking-how-do-people-handle-anova-when-the-data-doesnt-quite-mee/97120#97120) (1 ref, plus a link to an answer with some others). – Glen_b May 24 '15 at 00:23

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