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## AP®︎/College Statistics

### Course: AP®︎/College Statistics > Unit 11

Lesson 5: Testing for the difference of two population means- Hypotheses for a two-sample t test
- Example of hypotheses for paired and two-sample t tests
- Writing hypotheses to test the difference of means
- Two-sample t test for difference of means
- Test statistic in a two-sample t test
- P-value in a two-sample t test
- Conclusion for a two-sample t test using a P-value
- Conclusion for a two-sample t test using a confidence interval
- Making conclusions about the difference of means

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# Hypotheses for a two-sample t test

Example of how to write hypotheses for a two-sample t test, and how to distinguish between paired and two-sample comparisons between means.

## Want to join the conversation?

- Why don't we use a paired sample as a one-sample t-test rather than a two-sample t-test?

It seems the same and a simpler calculation.(6 votes)- The next couple videos will solve this confusion.(2 votes)

## Video transcript

- [Instructor] "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?" And, 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
gonna construct hypotheses around a parameter for our population that will often be the mean difference. So, we have one population, so we're talking about the
paired situation right over here, and so, let's say we say, "Hey, do people run faster when
they wear shorts or pants?", and so for each member of the population, you can see what you would, if you really had perfect information, you would know, how fast
did they run with pants and how fast did they run with shorts? And 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, and then, you would say, "Hey," assuming the null hypothesis is true, you would construct some null hypothesis, likely that there is no,
that this mean is zero, and 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 wanna 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, and 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. So, the mean salary for men minus the mean salary for women. And our null hypothesis is usually the no-news-here hypothesis, and so in this situation,
our null hypothesis is that there is no difference between these means and that our alternative hypothesis in the situation that we are looking at, because they wanna 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 wanna see if men have
a higher average salary. So, instead of not equal 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 wanna do the greater than, not the not-equal-to, so
we are in that choice, right over there.