Making conclusions in a test about a proportion | AP Statistics | Khan Academy

A public opinion survey investigated whether a majority, more than 50 percent, of adults supported a tax increase to help fund the local school system. A random sample of 200 adults showed that 113 of those sampled supported the tax increase. Researchers used these results to test the null hypothesis that the proportion is 0.5. The alternative hypothesis is that it's greater than 0.5, where p is the true proportion of adults that support the tax increase.

They calculated a test statistic of z approximately equal to 1.84 and a corresponding p-value of approximately 0.033. Assuming the conditions for inference were met, which of these is an appropriate conclusion? We have our four conclusions here. At any point, I encourage you to pause this video and see if you can answer it for yourself. But now we will do it together and just make sure we understand what's going on.

Before we even cut to the chase and get to the answer, we will sample the population. So n is equal to 200. From that sample, we can calculate a sample proportion of adults that support the tax increase. We see 113 out of 200 supported, which is going to be equal to, let's see, that is the same thing as 56.5 percent.

The key is to figure out the p-value: what is the probability of getting a result this much above the assumed proportion or greater? At least this much above the assumed proportion, if we assume that the null hypothesis is true. If that probability—that p-value—is below a preset threshold, if it's below our significance level (they haven't told it to us yet), it looks like they're going to give some in the choices. Well, then we would reject the null hypothesis, which would suggest the alternative. If the p-value is not lower than this, then we will fail to reject the null hypothesis.

Now, to calculate that p-value, to calculate that probability, we figure out how many in our sampling distribution—how many standard deviations above the mean of the sampling distribution. The mean of the sampling distribution would be our assumed population proportion. How many standard deviations above that mean is this right over here? That is what this test statistic is. Then we can use this to look at a z table and say, "All right, well in a normal distribution, what percentage or what is the area under the normal curve that is further than 1.84 standard deviations or at least 1.84 or more standard deviations above the mean?" They did that for us as well.

So really what we just need to do is compare this p-value right over here to the significance level. If the p-value is less than our significance level, then we reject our null hypothesis, and that would suggest the alternative. If this is not true, then we would fail to reject the null hypothesis.

So let's look at these choices, and if you didn't answer it the first time, I encourage you to pause the video again.

At the alpha is equal to 0.01 significance level, they should conclude that more than 50 percent of adults support the tax increase. So if the alpha is 0.01, the p-value right over here is roughly 3.3 percent. This is a situation where our p-value is greater than or equal to alpha; in fact, it's definitely greater than alpha here. So we would fail to reject our null hypothesis, and we wouldn't conclude that more than 50 percent of adults support the tax increase because remember our null hypothesis is that 50 do, and we're failing to reject this, so that's not going to be true.

At that same significance level, they should conclude that less than 50 percent of adults support the tax increase. No, we can't say that either. We just failed to reject this null hypothesis that the true proportion is 50.

At the alpha is equal to 0.05 significance level, they should conclude that more than 50 percent of adults support the tax increase. Well, yeah, in this situation we have our p-value, which is 0.033; it is indeed less than our significance level, in which case we reject our null hypothesis. If we reject the null hypothesis, that would suggest the alternative that the true proportion is greater than 50 percent. So I would pick this choice right over here.

Then choice d: at that same significance level, they should conclude that less than 50 percent of adults support the tax increase. No, not the situation at all. If we're rejecting our null hypothesis right over here, then we should suggest this alternative.