\(\newcommand{\Cov}{\mathrm{Cov}}\) \(\newcommand{\Corr}{\mathrm{Corr}}\) \(\newcommand{\Sample}{X_{1},\dots,X_{n}}\)

The **null hypothesis** (\(H_{0}\)) vs the **alternative
hypothesis** (\(H_{a}\)): The null hypothesis is assumed to be true
(default). The alternative hypothesis is the one that disproves the null.

In your experiment, you’ll either *reject* the null hypothesis, or *fail
to reject it*. What you will *not* do is *accept* it.

We usually frame \(H_{0}\) as an equality (e.g. \(H_{0}:p=0.1\)) and \(H_{a}\) as an inequality (\(H_{a}: p<0.1\)). In most cases, the logic is not impacted, but the math is made easier.

## Errors in Hypothesis Testing

**Type I error**: Incorrectly rejecting the null hypothesis.**Type II error**: Incorrectly failing to reject the null hypothesis.

By convention, the probability of a type I error is denoted by \(\alpha\) and the probability of a type II error is denoted by \(\beta\).

Lowering \(\alpha\) almost always means increasing \(\beta\)
(assuming \(n\) is fixed). We usually pick the largest acceptable
\(\alpha\), at which point it is called the **significance level**.
The test is then called the **level** \(\alpha\) **test**.