Calculating a z statistic in a test about a proportion | AP Statistics | Khan Academy

The mayor of a town saw an article that claimed the national unemployment rate is eight percent. They wondered if this held true in their own town, so they took a sample of 200 residents to test the null hypothesis. The null hypothesis is that the unemployment rate is the same as the national one versus the alternative hypothesis, which is that the unemployment rate is not the same as the national, where p is the proportion of residents in the town that are unemployed. The sample included 22 residents who were unemployed.

Assuming that the conditions for inference have been met—random, normal, and independence conditions that we've talked about in previous videos—identify the correct test statistic for this significance test. So let me just... I like to rewrite everything just to make sure I've understood what's going on. We have a null hypothesis that the true proportion of unemployed people in our town—that's what this p represents—is the same as the national unemployment.

Remember, our null hypothesis tends to be the "no news here," nothing to report, so to speak. We have our alternative hypothesis that, no, the true unemployment in this town is different, is different than eight percent.

What we would do is set some type of a significance level. We would assume that the mayor of the town sets it; let's say he or she sets a significance level of 0.05. Then what we want to do is conduct the experiment. This is the entire population of the town. They take a sample of 200 people, so this is our sample: n is equal to 200. Since it met the independence condition, we'll assume that this is less than 10 percent of the population.

Next, we calculate a sample statistic. Since we care about the true population proportion, the sample statistic we would care about is the sample proportion. We figure out that 22 out of the 200 people in the sample are unemployed, so this is 0.11.

Now, the next step is, assuming the null hypothesis is true, what is the probability of getting a result this far away or further from the assumed population proportion? If that probability is lower than alpha, then we would reject the null hypothesis, which would suggest the alternative.

But how do you figure out this probability? One way to think about it is: we could say how many standard deviations away from the true proportion the assumed proportion is. Then we could say what's the probability of getting that many standard deviations or further from the true proportion. We could use a z-table to do that, and so we want to figure out the number of standard deviations.

That would be a z statistic. So how do we figure it out? We can find the difference between the sample proportion here and the assumed population proportion. So that would be 0.11 minus 0.08, divided by the standard deviation of the sampling distribution of the sample proportions.

We can figure that out. Remember, all that is... Sometimes we don't know what the population proportion is, but here we're assuming a population proportion. So we're assuming it is 0.08, and then we'll multiply that times 1 minus 0.08, so we'll multiply that times 0.9.

This comes straight from what we've seen in previous videos: the standard deviation of the sampling distribution of sample proportions. Then you divide that by n, which is 200.

We could get a calculator out to figure this out, but this will give us some value which tells us how many standard deviations away from 0.08 is 0.11. Then we could use a z-table to find the probability of getting that far or further from the true proportion.

That will give us our p-value, which we can compare to the significance level. Sometimes, you will see a formula that looks something like this: you say, "Hey, look, you have your sample proportion. You find the difference between that and the assumed proportion in the null hypothesis."

That's what this little zero says, that this is the assumed population proportion from the null hypothesis. You divide that by the standard deviation—the assumed standard deviation of the sampling distribution of the sample proportions.

So, that would be our assumed population proportion times 1 minus our assumed population proportion divided by our sample size. In future videos, we're going to go all the way, calculate this, then look it up in a z-table and see what's the probability of getting that extreme or more extreme of a result and compare it to alpha.