- Describe the logic by which it can be concluded that someone can distinguish
between two things
- State whether random assignment ensures that all uncontrolled sources
of variation will be equal
- Define precisely what the probability is that is computed to reach
the conclusion that a difference is not due to chance
- Distinguish between the probability of an event and the probability
of a state of the world
- Define "null hypothesis"
- Be able to determine the null hypothesis from a description of an experiment
- Define "alternative hypothesis"
The statistician R. Fisher explained the concept
of hypothesis testing with a story of a lady tasting tea. Here
we will present an example based on James Bond who insisted
that Martinis should be shaken rather than stirred. Let's consider
a hypothetical experiment to determine whether Mr. Bond can
tell the difference between a shaken and a stirred martini.
Suppose we gave Mr. Bond a series of 16 taste tests. In each
test, we flipped a fair coin to determine whether to stir or
shake the martini. Then we presented the martini to Mr. Bond
and asked him to decide whether it was shaken or stirred. Let's
say Mr. Bond was correct on 13 of the 16 taste tests. Does this
prove that Mr. Bond has at least some ability to tell whether
the martini was shaken or stirred?

This result does not prove that he does; it could
be he was just lucky and guessed right 13 out of 16 times. But
how plausible is the explanation that he was just lucky? To assess
its plausibility, we determine the probability that someone who
was just guessing would be correct 13/16 times or more. This probability
can be computed from the binomial distribution and the binomial
distribution calculator shows it to be 0.0106. This is a pretty
low probability, and therefore someone would have to be very lucky
to be correct 13 or more times out of 16 if they were just guessing.
So either Mr. Bond was very lucky, or he can tell whether the
drink was shaken or stirred. The hypothesis that he was guessing
is not proven false, but considerable doubt is cast on it. Therefore,
there is strong evidence that Mr. Bond can tell whether a drink
was shaken or stirred.

Binomial
Calculator

Let's consider another example. The case study Physicians'
Reactions sought to determine whether physicians spend less
time with obese patients. Physicians were sampled randomly and
each was shown a chart of a patient complaining of a migraine
headache. They were then asked to estimate how long they would
spend with the patient. The charts were identical except that
for half the charts, the patient was obese and for the other half,
the patient was of normal weight. The chart a particular physician
viewed was determined randomly. Thirty three physicians viewed
charts of average-weight patients and 38 physicians viewed charts
of obese patients.

The mean time physicians reported that they would
spend with obese patients was 24.7 minutes as compared to a mean
of 31.4 minutes for average-weight patients. How might this difference
between means have occurred? One possibility is that physicians
were influenced by the weight of the patients. On the other hand,
perhaps by chance, the physicians who viewed charts of the obese
patients tend to see patients for less time than the other physicians.
Random assignment of
charts does not ensure that the groups will be equal in all respects
other than the chart they viewed. In fact, it is certain the groups
differed in many ways by chance. The two groups could not have
exactly the same mean age (if measured precisely enough such as
in days). Perhaps a physician's age affects how long physicians
see patients. There are innumerable differences between the groups
that could affect how long they view patients. Therefore it is
possible that these chance differences are responsible for the
difference in times?

To assess the plausibility of the hypothesis that
the differences are due to chance, we compute the probability
of getting a difference as large or larger than the observed difference
(31.4 - 24.7 = 6.7 minutes) if the difference were, in fact, due
solely to chance. Using methods presented in another
section, this probability can be computed to be 0.0057. Since
this is such a low probability, we have confidence that the difference
in times is due to the patient's weight and is not due to chance.

The Probability Value

It is very important to understand precisely what
the probability values mean. In the James Bond example, the computed
probability of 0.0106 is the probability he would be correct on
13 or more taste tests (out of 16) if he were just guessing.

It is easy to mistake this probability of 0.0106
as the probability he cannot tell the difference. This is not
at all what it means.

The probability of 0.016 is the probability of
a certain outcome (13 or more out of 16) assuming a certain
state of the world (James Bond was only guessing.) It is not
the probability that a state of world is true. Although this
might seem like a distinction without a difference, consider
the following example. An animal trainer claims that a trained
bird can determine whether or not numbers are evenly divisible
by 7. In an experiment assessing this claim, the bird
is given a series of 16 test trials. On each trial, a number
is displayed on a screen and the bird pecks at one of two keys
to indicate its choice. The numbers are chosen in such
a way that the probability of any number being evenly divisible
by 7 is 0.50. The bird is correct on 9/16 choices. Using the
binomial calculator, we can compute that the probability of
being correct nine or more times out of 16 if one is only guessing
is 0.40. Since a bird who is only guessing would do this well
40% of the time, these data do not provide convincing evidence
that the bird can tell the difference between the two types
of numbers. As a scientist, you would be very skeptical that
the bird had this ability. Would you conclude that there is
a 0.40 probability that the bird can tell the difference? Certainly
not! You would think the probability is much lower than 0.0001.

To reiterate, the probability value is the probability
of an outcome (9/16 or better) and not the probability of a particular
state of the world (the bird can tell whether a number is divisible
by 7). In statistics, it is conventional to refer to possible
states of the world as hypotheses since
they are hypothesized states of the world. Using this terminology,
the probability value is the probability of an outcome given the
hypothesis. It is not the probability of the hypothesis given
the outcome.

This is not to say that we ignore the probability
of the hypothesis. If the probability of the outcome given the
hypothesis is sufficiently low, we have evidence that the hypothesis
false. However, we do not compute the probability that the hypothesis
is false. In the James Bond example, the hypothesis is that he
cannot tell the difference between shaken and stirred martinis.
The probability value is low (0.0106), thus providing evidence
that he can tell the difference. However, we have not computed
the probability that he can tell the difference. A branch of statistics
called Bayesian statistics provides
methods for computing the probabilities of hypotheses. These computations
require that one specify the probability of the hypothesis before
the data are considered and therefore are difficult to apply in
some contexts.

The Null Hypothesis

The hypothesis that an apparent effect is due
to chance is called the null hypothesis.
In the Physicians' Reactions example, the null hypothesis is that
in the population of physicians, the mean time expected to be
spent with obese patients is equal to the mean time expected to
be spent with average-weight patients. This null hypothesis can
be written as:

μobese = μaverage

or as

μobese - μaverage
= 0.

The null hypothesis in a correlational study of
the relationship between high-school grades and college grades
would typically be that the population correlation is 0. This
can be written as

ρ = 0

where ρ is the population correlation (not
to be confused with r, the correlation in the sample).

Although the null hypothesis is usually that the
value of a parameter is 0, there are occasions on which the null
hypothesis is a value other than 0. For example, if one were testing
whether a subject differed from chance in their ability to determine
whether a flipped coin would come up heads or tails, the null
hypothesis would be that π = 0.5.

Keep in mind that the null hypothesis is typically
the opposite of the researcher's hypothesis. In the Physicians'
Reactions study, the researchers hypothesized that physicians
would expect to spend less time with obese patients. The null
hypothesis that the two types of patients are treated identically
is put forward with the hope that it can be discredited and therefore
rejected. If the null hypothesis were true, a difference as large
or larger than the sample difference of 6.7 minutes would be very
unlikely to occur. Therefore, the researchers rejected the null
hypothesis of no difference and concluded that in the population,
physicians intend to spend less time with obese patients.

If the null hypothesis is rejected, then the alternative
to the null hypothesis (called the alternative
hypothesis) is accepted. The alternative hypothesis is
simply the reverse of the null hypothesis. If the null hypothesis

μobese
= μaverage

is rejected, then there are two alternatives:

μobese
< μaverage

μobese
> μaverage.

Naturally, the direction of the sample means determines
which alternative is adopted. Some textbooks have incorrectly
argued that rejecting the null hypothesis that two populations
means are equal does not justify a conclusion about which population
mean is larger.