gledanju0

2022-06-24

${X}_{i}\sim Bern\left(p\right)$, $Y=\sum _{i=1}^{N}{X}_{i}$ and ${H}_{0}:p\ge 0.5$ vs ${H}_{1}:p<0.5$ with $\alpha =0.05$.
I have to find the $p$-value at $n=30$ and $y=5$.
I'm not sure if this correct, but I figured the $p$-value was simply:
$P\left(Y\le 5\right)=\sum _{i=1}^{5}\left(\genfrac{}{}{0}{}{30}{y}\right)\left(0.5{\right)}^{y}\left(0.5{\right)}^{30-y}$
which works out to be $0.00016$.
If this is correct, then why have I been given $\alpha$?

scipionhi

Expert

The $p$-value you calculated is correct. $\alpha =5\mathrm{%}$ is given to compare the $p$-value you found with it.
As ${p}_{value}<\alpha$ you reject ${H}_{0}$.

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