Main content

## Testing for the difference of two population means

# Example of hypotheses for paired and two-sample t tests

AP.STATS:

VAR‑7 (EU)

, VAR‑7.B (LO)

, VAR‑7.B.2 (EK)

, VAR‑7.C (LO)

, VAR‑7.C.2 (EK)

, VAR‑7.F (LO)

, VAR‑7.F.1 (EK)

, VAR‑7.G (LO)

, VAR‑7.G.1 (EK)

## Video transcript

- [Instructor] 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 and alternative hypothesis? So they're just asking
this 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. So when were talking about
either a two-sample T test or a two-sample T,
interval for the difference between the mean, 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, 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
of Mue one minus Mue two. That's what you're trying to figure out, Mue one minus Mue two. 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. So we could just call that
Mue sub Zeppo's minus Harpo's. And the way that we go about doing that is we take a sample and then
for each subject in the sample we perform two measurements. One where they run with the Zeppo's and one where they run with the Harpo's. And then for that sample you
can calculate a mean difference between the Zeppo's and the Harpo's. 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. And so 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 the two-sample T test. And so we run a paired T test and the manager wants to test
if their times when wearing Harpo's are significantly lower than their times when wearing Zeppo's. So our null hypothesis, even
though that their not asking that, our null hypothesis would be that there's no difference. That the mean time, that
the mean difference between wearing Zeppo's, Zeppo's
and Harpo's, and Harpo's would be equal to zero. And the alternative hypothesis. So if they were just saying,
"Hey, is there a difference?" Then we would say that this
would not be equal to zero, the alternative. But the manager explicitly
want's 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, Zeppo's minus Harpo's. We're trying to find if
we can see if we can have evidence to suggest that this
is actually greater than zero. And so that would be this
choice right over there.

AP® is a registered trademark of the College Board, which has not reviewed this resource.