Finding the $p$-value from Bernoulli hypothesis test

Question

So I'm told that $X_i \sim Bern(p)$, $Y = \sum_{i=1}^N X_i$ and $H_0: p \geq 0.5$ 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:

$$P(Y \leq 5) = \sum_{i=1}^5 \binom{30}{y}(0.5)^y (0.5)^{30-y}$$ which works out to be $0.00016$.

If this is correct, then why have I been given $\alpha$?

Answer 1

The p-value you calculated is correct. $\alpha=5\%$ is given to compare the p-value you found with it.

As $p_{value}<\alpha$ you reject $H_0$