Colorado State University — Pueblo; Fall 2016
Math 156, Introduction to Statistics
Hypothesis Testing about the Population Proportion

The Context

Again, simlar to previous hypothesis tests, but now for the proportion $p$ of a population which has some characteristic of interest. So suppose we have some population about which we are interested in the proportion about which some property $P$ is true. If we pick individuals from the population at random, then we can think of this as a qualitative RV $X$ with the values true and false, and we want to know the population parameter $p=a/N$, where $a$ is the number of individuals with the $X$ value true and $N$ is the size of the entire population.

We are interested in the question of whether $p$ is equal to some specific value $p_0$ — maybe the value it used to have before some intervention, or the value someone says it should have — but since the population size $N$ is large, it is impractical simply to go out, find $p$, and directly see if the null hypothesis $$H_0:\quad \mu=\mu_0$$ is true. Here, $p_0$ is always a specific number and we are wondering if the population proportion $p$, which is out there in the universe but which we have trouble finding out exactly, is equal to that number $p_0$.

Perhaps we have reasons to think that if the null hypothesis is not true, then maybe a particular alternative hypothesis $H_a$ might instead be true. As before, the alternative hypothesis can have one of the forms $H_a:\,p<p_0$, $H_a:\,p>p_0$, or $H_a:\,p≠p_0$.

To see which hypothesis, $H_0$ or $H_a$, is supported by the data, we take a sample of size $n$ and compute the sample proportion $\widehat{p}$. From this, we make the test statistic $$z=\frac{\widehat{p}-p_0}{\sqrt{p_0(1-p_0)/n}}\ .$$

As we have seen before with confidence intervals, the distribution of $z$ will be governed by the standard Normal distribution if the null hypothesis is true. We therefore compute the standard Normal cumulative probabilty that the test statistic has values like the one we saw, or more in the direction of the alternative hypothesis. This probability is the $p$-value of the test. Here is how we compute it depending upon the particular alternative hypothesis:

where $Z$ is the standard Normal RV [so we can look up these probabilities in a table or get them with an electronic tool].

Note: do not confuse the $p$-value of the test with the parameter $p$ about we are making hypotheses.

As before, the $p$-value is the conditional probability

$p=P($one would see values of $\overline{X}$ like what we did see, or more extreme ${}|{}\ H_0$ is true${})$ .
 
[The probability here is for the repeatable experiment of taking SRSs.] Consequently, if $p$ is small, we have reason to reject the null hypothesis, while if $p$ is not small enough, we fail to reject the null hypothesis.

Also as before, with a value $\alpha$ for the significance level, if $p<\alpha$, we say We reject the null hypothesis at the $\alpha$ significance level. A preferred method again, though, is to get the $p$ value and, if it is quite small (as your particular situation determines), you say "We reject the null hypothesis ($p={}$<number>)."