# one tailed hypothesis test

## 1. a question

One question that I always had about one tailed hypothesis testing:

- Let \(H_1 := \theta > \theta_0\)
- Then we should have \(H_0 := \theta \leq \theta_0\)
- But instead, most examples have us assume that \(H_0 = \theta_0\) and then calculate the \(p\) -value from there

### 1.1. answer

The null hypothesis really is a family of null hypotheses given by \(\theta \leq \theta_0\). Of these hypotheses, the one with the largest critical region, that is the largest chance of type I error, is the one given by \(\theta_0\). If the \(p\) -value for that hypothesis is smaller than our \(\alpha\), then it will also be the case for all other \(\theta \leq \theta_0\), so we need only consider the hypothesis \(\theta = \theta_0\).