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 $$H_0$$.