Constructing hypotheses for two proportions | AP Statistics | Khan Academy

Derek is a political pollster tracking the approval rating of the prime minister in his country. At the end of each month, he obtains data from a random sample of adults on whether or not they currently approve of the prime minister's performance. Using a separate sample each month, Derek wants to test if the proportion of adults who approved was significantly lower in December than it was in November.

Which of the following is an appropriate set of hypotheses for Derek's significance test? Pause this video and see if you can figure it out on your own.

All right, so let's think about ways to write a null hypothesis first. So remember, your null hypothesis is assuming that there's no news here, there's no difference. One way to say it is that the true proportion in December is equal to your true proportion in November. Another way to write that exact same thing is to say that the difference between the true proportion in December.

Let's see, they say Derek wants to test if the proportion of adults who approved was significantly lower in December than it was in November. You could write it as, because he wanted to see if November is higher or not, I'll put November first. So another way to say this exact same thing is that the true proportion of November minus the true proportion in December is equal to zero.

So each of these would be legitimate null hypotheses. This one looks good, this one looks good, this one looks good; this one is not a legitimate null hypothesis for what we're trying to do, so we could rule out D. Then the other one is this: Derek wants to test if the proportion of adults who approved was significantly lower in December than it was in November. So the news here would be if this actually is the case. If we have evidence that the proportion of adults who approved was significantly lower in December than it was in November, then the alternative hypothesis could look something like this: that the proportion in December was less than the proportion in November, or it could be that the proportion in November is greater than the proportion in December.

If we look at these choices, the proportion in December is less than the proportion in November. That's what I wrote right over here, so that looks good as well. Here they swapped it; they're saying our alternative hypothesis is that the true proportion in December is larger than the true proportion of November, which is the opposite of what's saying here, so we rule that one out.

Here they're just saying that we actually have a difference in proportions, and many times you will see something like this. But here, Derek wants to test if the proportion of adults who approved was significantly lower in December than in November. He's not interested in the other way around. If he said, "Derek wants to test if the proportion of adults who approved was significantly different in December than in November," then you would pick choice C instead of choice A. But given the way it was phrased, I would pick choice A.

Let's do another example here. It says that Kylie has a dime and a nickel, and she wonders if they have the same likelihood of showing heads when they are flipped. She flips each coin 100 times to test if there is a significant difference in the proportion of flips that they land showing heads.

Which of the following is an appropriate set of hypotheses for Kylie's significance test? So once again, pause the video and try to do it on your own.

All right, well your null hypothesis would be that there is no difference, so that the proportion of getting heads with your dime is the same as the proportion of heads with your nickel. Then your alternative hypothesis. So it says here she wants to test if there's a significant difference; she's not trying to say if the proportion of dimes coming up heads is significantly lower or significantly larger. She just cares about the difference—if there's a significant difference in the proportion of flips.

So her alternative hypothesis is that there is a difference, that these two proportions are not equal to each other. If we look at the choices, this null hypothesis looks good; this null hypothesis does not look good. Remember, your null hypothesis, you're trying to assume that there's no news here. So all of these null hypotheses—these A, B, and D's null hypotheses—look good. Then the alternative hypothesis—this is exactly what we wrote before—is for choice D.

Choice A's alternative hypothesis would work if it said, "She flips each coin 100 times to test if the proportion of heads with the dime is significantly lower than the proportion of heads with the nickel," or something like that. And then if it was the reverse, then choice B would look good. But she just wants to see if there's a difference, not if one is lower than the other, and so I would pick choice D.