The Traditional Approach to Hypothesis Testing
For the next several pages we will consider the traditional treatment of hypothesis testing.
This is the treatment that you will find in almost any statistics text and is something that
you need to fully understand. The concepts here are central to what we mean by hypothesis
testing, no matter who is speaking about it.
We have just been discussing sampling distributions, which lie at the heart of the treat-
ment of research data. We do not go around obtaining sampling distributions, either mathe-
matically or empirically, simply because they are interesting to look at. We have important
reasons for doing so. The usual reason is that we want to test some hypothesis. Let’s go
back to the sampling distribution of differences in mean times that it takes people to leave
a parking space. We want to test the hypothesis that the obtained difference between sam-
ple means could reasonably have arisen had we drawn our samples from populations with
the same mean. This is another way of saying that we want to know whether the mean de-
parture time when someone is waiting is different from the mean departure time when there
is no one waiting. One way we can test such a hypothesis is to have some idea of the prob-
ability of obtaining a difference in sample means as extreme as 6.88 seconds, for example,
if we actually sampled observations from populations with the same mean. The answer to
this question is precisely what a sampling distribution is designed to provide.
Suppose we obtained (constructed) the sampling distribution plotted in Figure 4.1.
Suppose, for example, that our sample mean difference was only 2.88 instead of 6.88 and
that we determined from our sampling distribution that the probability of a difference in
means as great as 2.88 was .092. (How we determine this probability is not important
here.). Our reasoning could then go as follows: “If we did in fact sample from populations
with the same mean, the probability of obtaining a sample mean difference as high as 2.88
seconds is .092—that is not a terribly high probability, but it certainly isn’t a low probabil-
ity event. Because a sample mean difference at least as great as 2.88 is frequently obtained
from populations with equal means, we have no reason to doubt that our two samples came
from such populations.”
In fact our sample mean difference was 6.88 seconds and we calculated from the sam-
pling distribution that the probability of a sample mean difference as large as 6.88, when
the population means are equal, was only .0006. Our argument could then go like this: If
we did obtain our samples from populations with equal means, the probability of obtaining
a sample mean difference as large as 6.88 is only .0006—an unlikely event. Because a sam-
ple mean difference that large is unlikely to be obtained from such populations, we can rea-
sonably conclude that these samples probably came from populations with different means.
People take longer to leave when there is someone waiting for their parking space.
It is important to realize the steps in this example, because the logic is typical of most
tests of hypotheses. The actual test consisted of several stages:- We wanted to test the hypothesis, often called the research hypothesis,that people
backing out of a parking space take longer when someone is waiting. - We obtained random samples of behaviors under the two conditions.
- We set up the hypothesis (called the null hypothesis, ) that the samples were in fact
drawn from populations with the same means. This hypothesis states that leaving times
do not depend on whether someone is waiting. - We then obtained the sampling distribution of the differences between means under the
assumption that (the null hypothesis) is true (i.e., we obtained the sampling distribu-
tion of the differences between means when the population means are equal). - Given the sampling distribution, we calculated the probability of a mean difference at
least as largeas the one we actually obtained between the means of our two samples.
H 0
H 0
Section 4.3 Theory of Hypothesis Testing 91research
hypothesis
null hypothesis