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Confused about P-values in two tailed hypothesis testing using Binomial Distributions,

For a two tailed test, if we did it via critical regions we would test both the lower and upper tail with half of the significance level

When using P values, how come we do not check both tails? For example, if my hypothesis was with a test statistic 5 of :

$H_0 : p = 0.3 \\ H_1 : p \ne 0.3 $

Why do we not test for $P(X\leq 5)$ and $P(X\geq5)$ to obtain two p-values

Sorry if I've got the wrong idea in my head, not really understanding two tailed at the moment. Thank you

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A $p$ value is supposed to be the probability, if the null hypothesis is correct, of seeing the observed value or a more extreme value. A two-tailed $p$ value takes this into account by considering both low and high values as extreme, and so will generally be larger than a one-tailed $p$ value; so instead of two $p$ values you get a single value taking both notions of extreme into account

Let's adapting your example of a binomial random variable with null hypothesis $H_0: p=0.3$ and alternative hypothesis $H_a: p \not = 0.3$, and you take $n=10$ attempts observing $x=5$ successes.

  • Clearly any observation greater than $5$ should be considered more extreme than $5$ given $H_0$. And $\mathbb P(X \ge 5 \mid H_0) \approx 0.1502683$

  • But an observation of $1$ or $0$ might also be considered as extreme or more extreme than $5$. And $\mathbb P(X \le 1 \mid H_0) \approx 0.1493083$

  • So for a two-tailed $p$ value you could add these together to give about $0.2995767$ for an observation of $5$

Some people and programs instead simple double the first figure to get about $0.3005367$; either way, this particular example does not look significant

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  • $\begingroup$ I see, so its a simple as doubling the initial figure! The method we learnt is via critical regions, so was a little confused. Thank you $\endgroup$
    – Ahm
    May 22 '19 at 6:58

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