# 5.2 - Hypothesis Testing for One Sample Proportion

5.2 - Hypothesis Testing for One Sample ProportionRecall our “test” about whether Penn State students like cold weather. we have to ask about the relationship of the data we have (from our sample) relative to the hypothesized null value. In other words, is our observed sample proportion far enough away from the 0.5 to suggest that there is evidence against the null?

We can use what we know about the sampling distribution of sample proportions to help find our evidence!

## Hypothesis Testing for One Sample Proportion

Recall that under certain conditions, the sampling distribution of the sample proportion, \(\hat{p} \), is approximately normal with mean, \(p \), standard error \(\sqrt{\dfrac{p(1-p)}{n}}\), and estimated standard error \(\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}\).

**Null:**

\(H_0\colon p=p_0\)

**Conditions:**

- \(np_0 \ge 5\) and \(n(1-p_0)\ge5\)

**Test Statistic:**

\(z^{*}=\dfrac{\hat{p}-p_{0}}{\sqrt{\dfrac{p_{0}\left(1-p_{0}\right)}{n}}}\)