Example of hypotheses for paired and two-sample t tests | AP Statistics | Khan Academy

The Olympic running team of Freedonia has always used Zeppo's running shoes, but their manager suspects Harpo's shoes can produce better results, which would be lower times. The manager has six runners; each run two laps: one lap wearing Zeppo's and another lap wearing Harpo's. Each runner flips a coin to determine which shoes they wear first. The manager wants to test if their times when wearing Harpo's are significantly lower than their times when they wear Zeppo's. Assume that all conditions for inference were met.

Which of these is the most appropriate test? An alternative hypothesis? So they're just asking us about the alternative, not even the null, but we can talk about that. So pause this video and see if you can figure it out.

So before I even go into this particular example, let's just make sure we understand the difference between a two-sample t-test and a paired t-test. When we're talking about either a two-sample t-test or a two-sample t-interval for the difference between the means, what we're doing is we're considering two populations.

You take two independent samples from those populations, and what you're trying to do is you get statistics off of these samples and you're trying to estimate the difference between the means of these populations. So it might be the difference, mu 1 minus mu 2. That's what you're trying to figure out: mu 1 minus mu 2.

A paired situation is quite different. Even though they might sound the same at first, here we're looking at just one population, and that's exactly what's happening in this situation right over here. We're trying to figure out what is the mean difference between using Zeppo's and Harpo's shoes.

So this is what we're trying to get at: the mean difference. We could call that mu sub Zeppos, Zeppos minus Harpo's. The way that we go about doing that is we take a sample, and for each subject in the sample, we perform two measurements: one where they run with the Zeppos and one when they run with the Harpos.

Then, for that sample, you can calculate a mean difference between the Zeppos and the Harpos. You're going to calculate this difference for each member of your sample, and then you're going to take the mean of all of those. So hopefully, you notice that this is quite different.

As you can imagine here in this example, we are dealing with a paired t-test. We aren't looking at two independent groups or two independent samples like you would with a two-sample t-test. The manager wants to test if the times when wearing Harpo's are significantly lower than their times when wearing Zeppo's.

So our null hypothesis, even though they're not asking that, our null hypothesis would be that there's no difference. That the mean time, the mean difference between wearing Zeppo's and Harpo's, would be equal to zero.

The alternative hypothesis? If they were just saying, "Hey, is there a difference?" then we would say that this would not be equal to zero. In the alternative, the manager explicitly wants to see if Harpo's times are lower than Zeppo's times.

So what we would want to see is if the mean difference, so the mean difference of Zeppo's minus Harpo's, we're trying to find if we can have evidence to suggest that this is actually greater than zero. And so that would be this choice right over there.