Lorena Beard

2022-07-10

Let $X\sim Binomial\left(n,p\right)$
$H+0:p\ge .5$
${H}_{1}:p<.5$
How do we evaluate the p-value?
ie. $P\left(X\ge x\mid p>=.5\right)$
I have seen settings where this is just interpreted as $P\left(X\ge x\mid p=.5\right)$, but I'm not sure how this is the same thing.

Marisol Morton

Expert

your system of Hypothesis is equivalent to the following
$\left\{\begin{array}{l}{\mathcal{H}}_{0}:p=\frac{1}{2},\\ {\mathcal{H}}_{1}:p<\frac{1}{2},\end{array}$
Thus the p-value is calculated as follows:
$\mathbb{P}\left[X\le k|p=\frac{1}{2}\right]$
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 $5.92\mathrm{%}$.

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