Hypotheses for a two-sample t test | AP Statistics | Khan Academy

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Market researchers conducted a study comparing the salaries of managers at a large nationwide retail store. The researchers obtained salary and demographic data for a random sample of managers. The researchers calculated the average salary of the men in the sample and the average salary of the women in the sample. They want to test if managers who are men have a higher average salary than managers who are women. Assume that all conditions for inference have been met.

Which of these is the most appropriate test and alternative hypothesis? We can see they're talking about a paired t-test and a two-sample t-test, and then they talk about the alternative hypotheses. So pause this video and try to figure this out on your own.

So first, let's think about the difference between a paired t-test and a two-sample t-test. In a paired t-test, we're going to construct hypotheses around a parameter for a population; that will often be the mean difference. So we have one population, so we're talking about the paired situation right over here.

Let's say we say, "Hey, do people run faster when they wear shorts or pants?" For each member of the population, you could see what you would, if you really had perfect information, know how fast do they run with pants and how fast do they run with shorts. Then you would calculate the difference, and then across the whole population, you could actually get that mean difference.

So the mean difference of pants minus shorts, and of course, in order to estimate that or in order to do a hypothesis test around that, you would take a sample and then you would calculate the sample mean of the difference of pants minus shorts. Then you would say, "Hey, assuming the null hypothesis is true," you would construct some null hypothesis, likely that this mean is zero. You would say, "Hey, if the null hypothesis is true that this is actually equal to zero, what's the probability that I got this result?" If that's below your significance level, then you would reject your null hypothesis and it would suggest the alternative that might be that, "Hey, maybe this mean is greater than zero."

On the other hand, a two-sample t-test is where you're thinking about two different populations. For example, you could be thinking about a population of men and you could be thinking about the population of women, and you want to compare the means between these two, say the mean salary.

So you have the mean salary for men and you have the mean salary for women. What you're trying to do with the hypothesis test is try to come up with some conclusions about the mean difference between these two parameters: the mean salary for men minus the mean salary for women.

Our null hypothesis is usually the no news here hypothesis. In this situation, our null hypothesis is that there is no difference between these means, and our alternative hypothesis in the situation that we are looking at, because they want to test if managers who are men have a higher average salary, if they just wanted to test that whether managers who are men have a different salary, then our alternative hypothesis would look something like this: where the mean of men minus the mean of women is not equal to zero.

But they aren't just testing to see if the means are different; they want to see if men have a higher average salary. So instead of not equal to zero there, we would have greater than zero for our alternative hypothesis.

So which choice is that? Well, we're clearly in a two-sample t-test situation and we want to do the greater than, not the not equal to, so we are in that choice right over there.