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I am doing a two tailed exact binomial goodness of fit test, see, e.g., here. I am looking for something similar as for Chi-square tests, which, as a rule of thumb, require a frequency of 5 for each cell in the table. So, what is the minimum sample size for an exact binomial test of goodness of fit (two-tailed)?

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  1. You have the "rule of thumb" wrong. It doesn't say you need a frequency of 5 in each cell. It relates to expected frequency under the null hypothesis. You could have 10 in one cell and 0 in the other, but if $p_0=\frac12$ then the expected frequencies are both 5.

  2. The point of the rule of thumb is that then the discrete distribution of the chi-squared statistic should be reasonably close to the continuous chi-squared distribution you're looking up when you do a chi-squared test.

    Here's the null distribution (i.e. cdf) of the (discrete) chi-square statistic in the above situation (n=10,p=$\frac{_1}{^2}$) (blue) and the continuous chi-square distribution that approximates it:

    enter image description here

    [If combined with Yates' continuity correction that would give a better approximation.]

  3. With the exact binomial test you're looking up what will be* the exact discrete distribution of the count in one cell, so there's no minimum sample size at which it applies, since you're not dealing with an approximation.

* under the assumptions of independence and constant probability per trial

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  • $\begingroup$ I meant to say 'expected frequency'. When I am doing a two-tailed exact binomial test of goodness of fit with p_0 =1/2, I need observed frequencies of 6 and 0, respectively, to obtain a p value below 0.05. Given this threshold, does that mean I need at least 6 observations to make any claims on statistical significance? $\endgroup$
    – You_got_it
    Commented Dec 3, 2015 at 1:58
  • $\begingroup$ That depends on your significance level; if you don't restrict yourself to a type I error rate no larger than 0.05, you can make the claim at smaller sample sizes. For example with n=5 you can achieve significance at the 0.0625 level. There's nothing sacrosanct about 0.05. $\endgroup$
    – Glen_b
    Commented Dec 3, 2015 at 3:33

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