Inferential statistics is built on the foundation of probability theory,
and has been remarkably successful in guiding opinion about the
conclusions to be drawn from data. Yet (paradoxically) the very idea
of probability has been plagued by controversy from the beginning of the
subject to the present day. In this section we provide a glimpse of the
debate about the interpretation of the probability concept.
One conception of probability is drawn from the idea of symmetrical
outcomes. For example, the two possible outcomes of tossing a fair
coin seem not to be distinguishable in any way that affects which side
will land up or down. Therefore the probability of heads is taken to be
1212,
as is the probability of tails. In general, if there
are NN symmetrical outcomes, the probability
of any given one of them occurring is taken to be 1N
1N. Thus, if a
six-sided die is rolled, the probability of any one of the six sides coming
up is 1616.
Probabilities can also be thought of in terms of relative frequencies.
If we tossed a coin millions of times, we would expect the proportion of tosses
that came up heads to be pretty close to 121
2. As the number of tosses increases, the proportion
of heads approaches 1212
. Therefore, we can say that the probability of a head is
1212.
If it has rained in Seattle on 6262% of the
last 100,000100,000 days, then the probability of
it raining tomorrow might be taken to be 0.62
0.62. This is a natural idea but nonetheless unreasonable if we
have further information relevant to whether will rain tomorrow. For
example, if tomorrow is August 1, a day of the year on which it seldom
rains in Seattle, we should only consider the percentage of the time it
rained on August 1. But even this is not enough since the probability of
rain on the next August 1 depends on the humidity. (The chances are
higher in the presence of high humidity.) So, we should consult only
the prior occurrences of August 1 that had the same humidity as the
next occurrence of August 1. Of course, wind direction also affects
probability ... You can see that our sample of prior cases will soon be
reduced to the empty set. Anyway, past meteorological history is
misleading if the climate is changing?
For some purposes, probability is best thought of as
subjective. Questions such as "What is the probability
that Ms. Jones will defeat Mr. Smith in an upcoming congressional
election?" do not conveniently fit into either the symmetry or frequency
approaches to probability. Rather, assigning probability 0.7
0.7
(say) to this event seems to reflect the speaker's personal
opinion --- perhaps his willingness to bet according to certain odds. Such
an approach to probability, however, seems to lose the objective content of
the idea of chance; probability becomes mere opinion.
Two people might attach different probabilities to the election outcome, yet
there would be no criterion for calling one "right" and the other "wrong."
We cannot call one of the two people right simply because she assigned higher
probability to the outcome that actually transpires. After all, you would be
right to attribute probability 161
6 to throwing a six with a fair die, and your
friend who attributes 2323
to this event would be wrong. And you are still right (and
your friend is still wrong) even if the die ends up showing a six! The lack
of objective criteria for adjudicating claims about probabilities in the subjective
perspective is an unattractive feature of it for many scholars.
Like most work in the field, the present text adopts the frequentist
approach to probability. Moreover, almost all the probabilities we shall
encounter will be nondogmatic, that is, neither zero nor
one. An event with probability 00 has no
chance of occurring; an event of probability 1
1 is certain to occur. It is hard to think of any examples of
interest to statistics in which the probability is either
00 or 11. (Even the
probability that the Sun will come up tomorrow is less than
11.)
The following example illustrates our attitude about probabilities.
Suppose you wish to know what the weather will be like next Saturday
because you are planning a picnic. You turn on your radio, and the
weather person says, "There is a 1010%
chance of rain." You decide to have the picnic outdoors and, lo and
behold, it rains. You are furious with the weather person. But was
she wrong? No, she did not say it would not rain, only that rain was
unlikely. She would have been flatly wrong only if she said that the
probability is 00 and it subsequently
rained. However, if you kept track of her weather predictions over a
long periods of time and found that it rained on 50
50% of the days that the weather person said the probability
was 0.100.10, you could say her probability
assessments are wrong.
So when is it accurate to say that the probability of rain is
0.100.10? According to our frequency interpretation,
it means that it will rain 1010% of the
days on which rain is forecast with this probability.
- Inferential Statistics:
The branch of staistics concerned with drawing conclusions about a population from a smaller sample. This is generally done through random sampling, followed by inferences made about central tendency, or any of a number of other aspects of a distribution.
