>>Let’s talk about the case when we are wanting to compare the means of two different
populations but those two populations are not independent of each other. Okay. In the
previous two conversations we’ve had we’ve talked about comparing means, the differences
in means or looking at the differences in means of two populations but we assume that
those two populations were entirely different from each other. The sample that we collected
from over here from population one didn’t affect the sample over here from population
two. And in fact the populations don’t affect each other at all. But in some cases we do
have to analyze situations where these populations are not independent of each other. A lot of
times this happens in before and after kinds of studies. Okay, we talked about this previously,
before and after studies. So for example: a weight loss drug. We’re wanting to determine
if a weight loss drug works. So we have to look at the population before they took the
weight loss drug and then we they we give them the drug and then we weigh them after
they took the drug. Okay? It’s the exact same people from the before as it is the after
therefore those two populations are not independent of each other. We’re testing the weight
before and after on the exact same person. Okay? So that means that those two populations
are not independent of each other we have to have an entirely different way to address
these kind of situations, okay? So how we do this is with something called match samples
or paired samples. And so let’s say we have some observations from population one, that’s
going to be the befores, okay? And let’s say we have population two, that’s going
to be the afters, okay? So here we got y-one-one. Okay that’s going to be the the before,
the population one first observation, this first person. Okay? Let’s even put a different
color over here for i. So this is going to be the observation from each population, okay?
Observation one we’ve got the observation from the first population, the observation
from the second population and those two correspond to the same person or the same thing, or the
same- whatever it is we’re studying, this is the measurement from here this is the measurement
from the same place. Okay. So these two things are not independent of each other. Okay? We’ve
got our second observation and so on for n different observations. So here’s y-one
n y-two n, okay? And so what we do here is we calculate the difference, okay? What is
the difference between these two values, okay? So that’s just y-one-one minus y-two-one.
This would be y-one-two minus y-two-two and so on down to y-one-n minus y-two-n, okay?
So these values are the differences, the before minus the after. Okay? I use before and after
doesn’t necessarily have to mean that but that’s just one easy way to think about
this, before and after. Okay? So we calculate those differences then we find d-bar, which
is the average of the differences. The sum of the d-i’s divided by n. And we calculate
the standard deviation of the differences. That’s how different is each observa- each
difference, excuse me, the difference from d-bar squared summed across all i divided
by n minus one. Okay? So these are straightforward calculations. This is the mean, this is the
standard deviation, both of those from a sample that we talked about several units ago, we’re
just applying it here to the differences, not to the actual observations. Okay? So now
we’ve got the mean, we’ve got the standard deviation and we can do inferential statistics
on these differences, okay? So we could have a confidence interval. What is a confidence
interval for the average differences? Okay? That would be d-bar plus and minus t for our
particular degrees of freedom, alpha over two times s-sub-d over root n. Okay? Really
straightforward from here, right? A very straight forward confidence interval where degrees
of freedom is n minus one. We just have one n because this population and this population
both use the exact same number of observations, okay? And in fact the observations lie in
both populations. So that’s our degrees of freedom and our test statistic is a t test
statistic mmkay? And that is… Let’s first talk about what we want to address here. Okay.
We want to know, for example, is this the, is the mean difference, is the population
mean difference equal to some value? Or is it not equal to that value? Okay. Is this
average difference four or is it not four? Okay, is this group bigger than this group
on average by d-naught or not? Okay? We could also do the same kind of other alternative,
or the other hypotheses, we could look at greater than equal to in the null versus less
than in the alternative or less than or equal to in the null versus greater than in the
alternative. Okay? Same kind of idea we’ve always been applying with respect to the three
different kinds of hypotheses that we could address in inferential statistics in hypotheses
testing, okay? So now our test statistic. How different is the sample mean difference
from the hypothesized population mean difference. Okay. How different is the sample evidence
from the hypothesized value in standard deviation units. S-sub-d over root n. Okay? Very straightforward
test statistic. Same kind of idea, critical value approach, p-value approach both work
the exact same as they always have. If this different, if this is a lot different from
this then it’s going to lie in one of the tails, it will end up rejecting the null hypothesis
if these two are different enough from each other. If we have reason enough, if we have
evidence enough to believe that for example the real mean is not equal to d-naught then
this is going to be a lot different than d-naught and we’ll reject the null. Okay? So inferential
statistics for matched pairs, paired samples, matched samples, there’s a lot of different
interchangeable words that we use for this, where the two populations are not independent
of each other.