Did you know that polar bears like to walk? Polar bears can walk up to 20 miles
or 30 kilometers per day, for several days in a row, following seals, their
favorite prey across Arctic ice sheets. Let's use this in an example to
illustrate the logic of hypothesis testing. So we do a literature review, and
we find that the typical polar bear will walk about 100 kilometers in a week. Now
this 100 kilometer number is a round number, and we know that round numbers
are always liars, so you are right to be skeptical, and yes this number is
realistic, but not an actual mean, but with that full disclosure we're going to
use this 100 kilometer per week parameter for purposes of this example.
We are going to say that on the average, polar bears walk 100 kilometers per week,
and this is the population mean, mu equals 100. If I draw a sample from a
population with a mean of 100, what do you predict that the sample mean will be?
100. If I draw a second sample from a population with a mean of 100, what do
you predict that sample mean will be? Also 100. If I draw a third sample from a
population with a mean of 100, what do you predict that sample mean will be? 100.
The mean of the sample should be the same as the mean of the population from
which it was drawn. Now we're going to assign the first group to be our
experimental group, and the next two groups to be control groups. Now in a
real experiment, we would have no need for two control groups, but I'm doing
this for a reason, to illustrate changes due to chance and changes due to an
effect. We do a treatment on sample 1, the experimental group. The treatment comes
from another discovery that we made during our literature review,
and that is that whales given caffeine, swim forty-eight percent further than
similar whales with no caffeine. Knowing that caffeine led to a forty-eight
percent improvement in the whales, we scour the literature and find that this
effect has not been studied in other Arctic creatures. Ah, a gap in the
literature. Let's do an experiment. Our research question will giving polar
bears caffeine make them walk farther? After treatment we measure each of our
samples again. We want to know which sample means represent the population
mean of 100, and why. Let's start with our control groups. The sample mean for
control group 1 is 100. Does that sample mean represent the population mean? Yes,
it is exactly the same as the population mean, so it represents the population
mean quite well. In fact, we couldn't ask for better. The sample mean for control
group 2 is 104. Does that sample mean represent the population mean? Well it's
not exactly the same as the population mean, but it's still pretty close. I think
that the sample mean for control group 2 is really the same as the population
mean of 100, and the 4-point difference is best explained by random chance or
measurement error. Now the sample mean for the experimental group is 150. Does
that sample mean represent the population mean? Now I'm beginning to
have my doubts! I'm not so sure I can explain a 50-point difference just by
random chance. I'm skeptical. So how do I explain this sample mean of a hundred
and fifty among the polar bears who consumed caffeine? Here are my options
for explaining the traveling bears (or the Traveling Wilbearys). First,
the sample mean for the experimental group 150 could still represent the
population mean of 100, but with a lot of random error. Two, the sample mean for the
experimental group, 150, could represent a different population mean. It could
represent the population mean we would expect with forty-eight percent caffeine
improved performance. That population mean of 148 (forty-eight percent of 100)
with two points of random error. So which is more likely? Does the sample mean of
150 more likely represent the population from which it was drawn, or the
population expected if caffeine improves performance by forty-eight percent. As I
said this is about hypothesis testing, so let's examine those hypotheses again. The
null hypothesis states that sample one represents a population with a mean of
100. We would write that H Sub Zero mu equals 100. The experimental, or
alternative hypothesis, states that sample one does not represent a
population with a mean of 100, and better represents a population with a mean of
148. We would write that alternative as H sub 1 colon mu does not equal 100. To
answer our research question, we could calculate the probability that sample
one represents each of the two populations. 100, and 148 and then go with the
higher probability, the one that is more likely. We will find that it is more
likely that sample one represents a population with a mean of 148, so
ultimately we will reject the null hypothesis that says that sample one
represents a population with a mean of 100.