## Binomial Testing

In the last experiment, we assumed we had a fair coin. Suppose we wanted to conduct an experiment to see whether it was indeed a fair coin. First we need to state the null and alternative hypothesis and that is:

\[H_0:p=0.5 \\ H_1: p \neq 0.5\]Now we need to set the level of rejection which should be set as close as 0.05 as possible. This is a **two-tailed test** in the sense we reject the null hypothesis if our test statistitic is in *either* tail of the binomial distribution. For example, if we get 1 head out of ten flips, then we have reason to believe this is not a fair coin. If we get 10 heads, we also have reason to believe this is not a fair coin. That is why it is a two tailed test.

To calculute the rejection region, we need to calculate probabilities for both sides of the tail so that the rejection region is close to 0.05. Using the binomial probability mass function:</a>

\[P(X=0) + P(X=1) + P(X=2) + P(X=8) + P(X=9) + P(X=10) \approx 0.11\]If we leave out the $P(X=2)$ and $P(X=8), \alpha = 0.02. $ In this case, we risk the chances of a **type II error** which is failing to reject a false null hypothesis. Type II errors occur with extremely small rejection levels; that is, $ \alpha $ is so small, no matter what the outcome of the experiment, the test statistic will not be extreme enough for us to reject the null hypothesis. Since we are going to let $ \alpha $ equal 0.11, we reduce that risk, but allow a greater chance of a **type I error** which is rejecting a true null hypothesis; in other words, type I errors occurs because, the rejection level, $\alpha$, is so high, we may have test statistics from the actual experiment that we will reject when we should not have. As mentioned, ideally $ \alpha $ should be 0.05.

So now we have set the null hypothesis, alternative hypothesis, and the rejection level. Let's proceed to conducting the experiment.