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After answering How to analyze observed vs expected when expected is just a proportion? by suggesting that the OP uses a one tail Z-test for their proportion data, I got into a debate in the comments with another user to which their point is lost on me.

In my answer, I advocated testing $H_0: p=1/2 vs. H_1: p<1/2$. I advocated a left tail test because the OP's observed proportion was 0.2.

In the comments the other user and I said:

Alexis: Your alternate hypothesis does not correspond to the compliment of the null.

Statman: I'm advocating a one tail test.

Alexis: Then pose the proper one-tail null hypothesis: $H_0:p\ge1/2$.

Statman: But why @Alexis? The observed p is 0.2<0.5

I didn't receive an answer, but received a healthy downvote.

Now the left tail test is given as a valid option in textbooks, e.g. in Introduction to Probability and Statistics by Milton et. al, so my question is what point is @Alexis trying to make about left tailed z-tests?

kjetil b halvorsen
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  • Your approach to one-sided test would work if you were testing at the limits of the interval, for example H_0: p = 0 vs H_1: p > 0, or H_0: p=1 vs H_1: p<1 – Rems Mar 03 '18 at 23:59
  • @Rems That is an excellent point. Not pertinent for the particulars of this case ($p_{0} =1/2/)$, but a good example for distributions that have at least some probability massed at discrete extreme values. – Alexis Mar 04 '18 at 00:43
  • Related: https://stats.stackexchange.com/q/8196/119261, https://stats.stackexchange.com/q/7853/119261, https://stats.stackexchange.com/q/18988/119261, https://stats.stackexchange.com/q/342074/119261 – StubbornAtom Jul 04 '20 at 14:32

2 Answers2

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My comment was specifically about your articulation of the appropriate one-sample one-sided (aka one-tailed) null hypothesis (not about one-sided tests per se) which, for proportions, should either be $H_{0}: p \ge p_{0}$ with $H_{1}: p < p_{0}$, or $H_{0}: p \le p_{0}$ with $H_{1}: p > p_{0}$. Bear in mind that null hypotheses are articulated before you evaluate your data for directionality of a rejection decision.

The null hypothesis you posed in your answer was of the form $H_{0}: p = p_{0}$ which has as its proper alternative $H_{1}: p \ne p_{0}$, since by definition an alternative hypothesis corresponds to the complementary event in the null hypothesis. However, you proposed an alternative $H_{1}: p < p_{0}$, which is not the complement of your null. Indeed, it does not even correspond to the alternative hypothesis expressed on the site you linked to in your answer. The crux of the issue is that it may truly be the case that $p>p_{0}$, but this state of nature does not fit within either your $H_{0}$ or your $H_{1}$, since the sample space of $p$ is not fully represented by your null and alternate, they cannot be well formed.

To be super explicit: Nothing is wrong with one-sided one-sample inequality tests, but you articulated the null hypothesis incorrectly for such a test.

Alexis
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    complement, not compliment – Acccumulation Mar 02 '18 at 22:27
  • No, this answer is completely wrong and I find it daunting that it has received so many upvotes. For discussion, please see _Introduction to Probability and Statistics: Principles and Applications for Engineering and the Computing Sciences_ by Milton and Arnold –  May 25 '18 at 11:25
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    @user1108 Insightful comments are welcome and useful. "Go read this book" is not so useful. Perhaps you would care to answer this question: If $H_{0}: p = p_{0}$ and you propose $H_{1}: p < p_{0}$ what would a true state of nature $p > p_{0}$ mean since it is *neither* described by $H_{0}$ *nor* described by $H_{1}$? Perhaps, instead of being daunted you could learn the logic of my argument, and share your own and we would both benefit in the exchange? – Alexis May 29 '18 at 19:26
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One problem with your suggestion is that the normal approximation for proportions is only reasonable in certain circumstances. A large sample and the observed proportion well away from the boundaries of 0 and 1. Neither of those circumstances is specified in the original question. See this question: Testing equality of two binomial proportions proportion (one near 100 %)

I have previously sparked some discussion by suggesting that the normal approximation method for confidence intervals of proportions be omitted from textbooks. You can read it here: What statistical methods are archaic and should be omitted from textbooks?

Michael Lew
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  • This was not the point I was trying to make about one-sided null hypotheses. (I don't disagree with the issues you raise in your answer, but they do not answer the OP's question, which is about the intent behind several comments made by me). – Alexis Mar 02 '18 at 21:05
  • I know that it was not the issue that you have asked about, but it is an important problem with your suggestion and it should not be ignored. – Michael Lew Mar 02 '18 at 22:09
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    Ok. On that topic: Agresti, A. and Coull, B. A. (1998). Approximate is better than “exact” for interval estimation of binomial proportions. *The American Statistician*, 52(2):119–126. – Alexis Mar 02 '18 at 22:41
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    Agresti, A. and Caffo, B. (2000). Simple and effective confidence intervals for proportions and differences of proportions result from adding two successes and two failures. *The American Statistician*, 54(4):280–288. – Alexis Mar 02 '18 at 22:43