Statistical hypothesis testing - Two tailed using p-values

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

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

• 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
– Ahm
May 22 '19 at 6:58