How to evaluate probability distribution given the parameter is greater than a certain value?

Question

Let X ~ Binomial(n, p)

$H_{0}$: p >= .5

$H_{1}$: p < .5

How do we evaluate the p-value?

ie. $P(X \ge x \mid p >= .5)$

I have seen settings where this is just interpreted as $P(X \ge x \mid p=.5)$, but I'm not sure how this is the same thing.

Answer 1

your system of Hypothesis is equivalent to the following

$$ \begin{cases} \mathcal{H}_0: p=\frac{1}{2}, \\ \mathcal{H}_1: p<\frac{1}{2}, \end{cases}$$

Thus the p-value is calculated as follows:

$$\mathbb{P}\Bigg[X\leq k|p=\frac{1}{2}\Bigg]$$

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Example: to test the Hypothesis System above stated, we use a random sample of size $n=15$ observing $k=4$ successes.

The p-value is $5.92\%$ as well represented in the following drawing