Suppose you run a chi square GOF test on a categorical variable with 2 levels and you reject the null (p= p0). If phat > p0, can you conclude that the population proportion is not only not equal to the hypothesized value, but is actually greater?

My guess is that you can say this because if the true proportion was less than p0, the chi square statistic would be even larger (and hence more unlikely to occur if H0 was true). However, I can't recall having seen this interpretation, so I'm a little wary of it.

I know this is probably a simple question, but I could not find any mention of it from an internet search, so even pointing me to a good reference would be appreciated.



  • $\begingroup$ testing difference in proportions is not the same thing as goodness of fit ! which of the two you want answered ? $\endgroup$ – Subhash C. Davar Nov 17 '17 at 3:53
  • $\begingroup$ My understanding is that when you run a chi-square goodness of fit on a single categorical variable with 2 levels, the hypotheses are: H0: p=p0 vs H1:p is not equal to p0. I am not totally sure what you mean by a "difference in proportions" (since there are 2 interpretations of this), but this is what I was referring to $\endgroup$ – Matt Brenneman Nov 17 '17 at 13:52
  • $\begingroup$ The text of the question says true / population proportion. In statistics, p-value is the probability of rejecting a null hypothesis which is true. If you edit your question and indicate sample of your data, then it it may be more convenient for someone to respond accurately. $\endgroup$ – Subhash C. Davar Nov 17 '17 at 15:07
  • $\begingroup$ I'm not understanding your objections/concerns. First, I'm not referring to any p-value in my original post, so I don't understand why you raised this issue. Second, in the context of my problem, I am in fact testing if the population proportion is equal to a specific value, so my original statement is exactly what I meant to say. Third, your comment that I "indicate sample of your data" is unclear to me. Are you suggesting I include the sample data I am testing? I could, I suppose, but my question is really about the general hypothesis test itself. $\endgroup$ – Matt Brenneman Nov 17 '17 at 23:06
  • $\begingroup$ The objective (can you conclude) you stated in para-1 of your question is different from hypothesis of equality of two proportions ? The latter hypothesis for testing requires a t-test of difference in two proportion and I do not think that a chi- square Goodness of Fit test could be useful for this purpose. I presume that there are several missing links in the question. $\endgroup$ – Subhash C. Davar Nov 18 '17 at 4:29

I see the answer now. The chi-square test for goodness of fit only tests is the counts are from a binomial distribution (it is not a test on the value of the population proportion, per se). My mistake was in thinking that any process with 2 outcomes had to be a binomial process (forgetting of course the assumption of independent Bernoulli trials)

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