# Binomial hypothesis test - Coin flip

A coin was flipped 10 times and landed on head 9 times. We want to test $$H_0: p = \frac{1}{2} \ vs \ H_1: p > \frac{1}{2}, p = \mathbb{P}(head)$$ with a confidence level of $$\alpha = 0.05$$. The question specifically ask for:

• The test statistic
• Distribution under $$H_0$$
• The conclusion of the test and the $$p$$-value.

My attempt:

I don't know if the test statistic is given by $$\hat{p} = 9/10 = 0.9$$ or $$\mathbb{P}(X = 9) = \binom{10}{9} 0.5^9 \cdot 0.5^1 \approx 0.00977.$$

The distribution under the null hypothesis is $$X \sim binomial(10, 1/2)$$ and with a 0.05 signficance level, our conclusion would be to reject the null hypothesis. I think that the $$p$$-value would be $$p = \binom{10}{10} \frac{1}{2^{11}}$$ but i'm really confused about this question.

• The distribution is the one I pointed out, right? We have 10 flips and $p=1/2$. In this case, I think that what I wrote is correct. Dec 2, 2022 at 3:00
• Not knowing what level you are answering this at makes it difficult to answer precisely. But no, $\hat{p}$ does not follow a binomial distribution, but $X$ does - neither of which is a "test statistic". If you do a large sample test, you need to refer to or derive the test statistic. Lastly your calculation of the $p$-value is wrong, and without showing steps it's not clear what you mean. Dec 2, 2022 at 16:47