# How do you calculate an exact two-tailed P-value using binomial distribution? [closed]

First, I will preface this question with my ulterior motive: I would like more evidence that the use of 19th and 20th century approximations play little to no pedagogic advantage in modern intro stats or intro data science courses.

First, let us agree to work with the following definition of a P-value: The probability of observing your sample—or something more extreme—given that the null hypothesis is true.

We wish to conduct a two-tailed hypothesis test for a population proportion using counts and exact probabilities from the binomial distribution. The hypotheses are $$H_0 : p = 10\%$$ $$H_a : p \ne 10\%$$ The sample obtained has $$n=189$$ and there are $$k=10$$ successful observations in this sample.

¿What is the two-tailed P-value for this test? It seems that there are reasonable arguments for either $$P(X \le 10) + P(X \ge 27) = 0.053$$ or $$P(X \le 10) + P(X \ge 28) = 0.038$$ (For those of you "addicted" to the conventional significance level of $$\alpha=0.05$$, you can probably see where I might be going with this. ;-)

To keep this in a pedagogic framework, I'm most curious for answers that might indicate how you would grade a student's work who submitted either answer...and how you would justify any loss of points that might occur.

• Take the 1 sided test p-value ($P(X\le 10)$) and double it. – AdamO Jul 11 '19 at 21:38
• I would have thought $P(X \le 10) + P(X \ge 29) = 0.0285$ has some justification as the probability of "as extreme as or more extreme than $10$" – Henry Jul 11 '19 at 22:19
• @AdamO ¿can you provide a rationale for why this would be considered exact? – Gregg H Jul 12 '19 at 0:38
• $P(X \le 10) \approx 0.0150$ while $P(X \ge 28)\approx 0.0229$ so you could say that, given the hypothesis $n=189, p=0.1$, then $28$ is a less extreme observation than $10$. Meanwhile $P(X \ge 29)\approx 0.0135$ so $29$ is a more extreme observation than $10$ – Henry Jul 12 '19 at 7:02
• This test hasn't been fully defined yet: you still have to specify a critical region. Ignoring randomized tests, there are at least 5 possibilities, depending on whether you want it to be symmetric in values, symmetric in probabilities, come as close as possible to the nominal p-value, or something else. Thus, this isn't a good question to ask learners and has little if any bearing on the ultimate questions of pedagogy that motivate you. Indeed, I don't see how this question is related even remotely to "19th and 20th century approximations:" could you explain the connection? – whuber Jul 12 '19 at 16:02

• (+1) NB Henry suggests adding the probability from the upper tail that doesn't exceed $P(X \leq x)$; not the probabilities that don't exceed $P(X = x)$. These are both common methods, & agree in this case but not always. – Scortchi - Reinstate Monica Jul 15 '19 at 12:10