##
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 \(H_0 \).