# 6a.4 - Hypothesis Test for One-Sample Proportion

## Overview Section

In this section, we will demonstrate how we use the sampling distribution of the sample proportion to perform the hypothesis test for one proportion.

Recall that if $$np$$ and $$n(1-p)$$ are both greater than five, then the sample proportion, $$\hat{p}$$, will have an approximate normal distribution with mean $$p$$, standard error $$\sqrt{\frac{p(1-p)}{n}}$$, and the estimated standard error $$\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$.

In hypothesis testing, we assume the null hypothesis is true. Remember, we set up the null hypothesis as $$H_0\colon p=p_0$$. This is very important! This statement says that we are assuming the unknown population proportion, $$p$$, is equal to the value $$p_0$$.

Since this is true, then we can follow the same logic above. Therefore, if $$np_0$$ and $$n(1-p_0)$$ are both greater than five, then the sampling distribution of the sample proportion will be approximately normal with mean $$p_0$$ and standard error $$\sqrt{\frac{p_0(1-p_0)}{n}}$$.

We can find probabilities associated with values of $$\hat{p}$$ by using:

$$z^*=\dfrac{\hat{p}-p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}}$$

## Example 6-4 Section Referring back to a previous example, say we take a random sample of 500 Penn State students and find that 278 are from Pennsylvania. Can we conclude that the proportion is larger than 0.5?

Is 0.556(=278/500) much bigger than 0.5? What is much bigger?

This depends on the standard deviation of $$\hat{p}$$ under the null hypothesis.

$$\hat{p}-p_0=0.556-0.5=0.056$$

The standard deviation of $$\hat{p}$$, if the null hypothesis is true (e.g. when $$p_0=0.5$$) is:

$$\sqrt{\dfrac{p_0(1-p_0)}{n}}=\sqrt{\dfrac{0.5(1-0.5)}{500}}=0.0224$$

We can compare them by taking the ratio.

$$z^*=\dfrac{\hat{p}-p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}=\dfrac{0.556-0.5}{\sqrt{\frac{0.5(1-0.5)}{500}}}=2.504$$

Therefore, assuming the true population proportion is 0.5, a sample proportion of 0.556 is 2.504 standard deviations above the mean.

The $$z^*$$ value we found in the above example is referred to as the test statistic.

Test statistic
The sample statistic one uses to either reject $$H_0$$ (and conclude $$H_a$$ ) or fail to reject $$H_0$$.