We now have two new tools to work with. We know about samples that represent
their populations, and we know about simple probability. Now it is time to use
those two tools together. Inferential statistics allow us to determine the
probability that a sample mean represents the population from which it
was drawn. Inferential statistics, also known as null hypothesis testing, use
probability to determine whether a particular sample or test outcome is
representative of the population from which the sample was originally drawn. If
not, if the sample mean is not exactly the same as the population mean, that may
be for one of two reasons. One, the sample mean may differ from the population mean
because of chance, or B, the sample mean may differ from the population mean
because of an effect. Probably in effect from the experiment we've just done on
the sample. Remember that when we draw a sample from a population, we expect that
the mean of the sample will be the same as the mean of the population, but when
we do an experiment, we typically draw a large sample, which we then randomly
assign into two groups. The experimental group is the group that is tested or
given a treatment, aka the treatment group. The control group is a group
identical to the experimental group, except that it is not given a treatment.
Both groups are drawn from the same population, so they should have the same
mean. If our intervention did not work, if the drug was not effective, if the
education did not sink in, then we should expect that after the experiment the
sample mean of the experimental group would still be the same as the sample
mean of the control group, both of which are the same as the population mean.
In other words, nothing has changed because the intervention didn't work. But
if after the experiment, the sample mean of the experimental group is different
than the sample mean of the control group, which didn't change from before
the experiment, we then suspect that the intervention had an effect, and this is
called hypothesis testing. A hypothesis is a testable prediction about a
real-world phenomenon. I think this drug will lower blood pressure is a testable
hypothesis. Experimental hypotheses propose a starting point, that will then
be accepted or rejected by examining the evidence that supports or contradicts it.
If the evidence supports the hypothesis, we accept it as a good explanation. If
the evidence refutes the hypothesis, we reject it as a poor explanation. Now
later, I'm gonna explain how we never really accept a hypothesis. But for now,
we're going to use accept and reject for simplicity. There are two experimental
hypotheses, the null hypothesis and the alternative hypothesis. The null
hypothesis states that the sample mean is the same as the population mean, so
any differences we do observe are due to chance.
The alternative hypothesis states that the sample mean is not the same as the
population mean, so any differences are due to an effect. We will examine whether
the sample mean differs from its population mean, and of course the sample
mean will always differ from the population mean, if you go out to enough
decimal places, but anytime a sample mean differs from its population mean there
are then two possible explanations - a) the sample mean still represents the
population mean although it may be a poor representative, or 2) the sample
mean represents a different population, also known as an
effect. So imagine we have a population of people with high blood pressure. The
diastolic blood pressure of the population is 102. A normal diastolic
blood pressure is under 80 (diastolic is the bottom number of your blood pressure,
systolic is the top number). We draw a sample from that population and we
expect the sample will have the same high blood pressure as the population
from which they were drawn. We measure them and sure enough, they have high
blood pressure, just as we expected. The diastolic blood pressure of the
population is 102 the diastolic blood pressure of our sample is 102.5.
But we say that those numbers are really the same, yes there was some random
variation, but the sample mean represents the population mean plus a little random
error. Because the sample mean will always differ from the population mean
if you go out enough decimal places, the means are still pretty close, so the
variation that we see is due to chance. Now we draw a second sample from the
population. Likewise, we expect them to have high blood pressure and sure enough
they do. Their diastolic blood pressure is 101. Again, the sample mean differs
slightly from the population mean, but we conclude that the variation is just due
to chance. Functionally, the means 102 102.5 and 101 are the same, no
difference. We're going to assign this first sample to be the control group.
We're going to assign the second sample to be the experimental group, and really
for the sake of accuracy, let me add that to do an experiment properly we would
not actually draw two samples from the population. We would draw one large
sample, and then randomly assign those participants to one of two groups. In any
case we would still have an experimental and control group, but then we give our
experimental group a new drug that is supposed to lower blood
pressure. We keep them on this drug for two months, and then we measure their
blood pressure again. After treatment we find that the control group mean changed
slightly from 102.5 to 101.2, but they still have
high blood pressure. This variation was due to chance, and the
control group mean still represents the population mean, but the experimental
group sample mean is lower than it was two months ago.
Before the experiment the diastolic blood pressure of our sample was 101, now
it is 86. Now we suspect that the sample mean
represents a different population, the population of people of normal or close
to normal blood pressure, and our explanation for why the experimental
sample mean is now so much lower - the drug had an effect. It lowered blood
pressure in the sample. We suspect that the experimental group sample mean now
represents a different population, the population of people with normal blood
pressure, whose diastolic blood pressure is under 80.