Statistical Hypothesis
In attempting to reach decisions, it is useful to make assumptions or
guesses about the populations involved. Such assumptions, which may
or may not be true, are called statistical hypotheses and in general are
statements about the probability distributions of the populations.
For example, if we want to decide whether
a given coin is loaded, we formulate the
hypothesis that the coin is fair, i.e., p = 0.5,
where p is the probability of heads. Similarly, if
we want to decide whether one procedure is
better than another, we formulate the hypothesis
that there is no difference between the two
procedures (i.e., any observed differences are
merely due to fluctuations in sampling from the
same population). Such hypotheses are often
called null hypotheses, denoted by H0.
Any hypothesis that differs from a given null hypothesis is called
an alternative hypothesis. For example, if the null hypothesis is p = 0.5,
possible alternative hypotheses are p = 0.7, p ≠ 0.5, or p > 0.5. Ahypothesis
alternative to the null hypothesis is denoted by H1.
Tests of Hypothesis and Significance
If on the supposition that a particular hypothesis is true we find that
results observed in a random sample differ markedly from those expected
under the hypothesis on the basis of pure chance using sampling theory,
we would say that the observed differences are significant and we
86 PROBABILITY AND STATISTICS
would be inclined to reject the hypothesis (or at least not accept it on the
basis of the evidence obtained). For example, if 20 tosses of a coin yield
16 heads, we would be inclined to reject the hypothesis that the coin is
fair, although it is conceivable that we might be wrong.
Type I and Type II Errors
If we reject a hypothesis when it happens to be true, we say that a Type
I error has been made. If, on the other hand, we accept a hypothesis
when it should be rejected, we say that a Type II error has been made.
In either case a wrong decision or error in judgment has occurred.
In order for any tests of hypotheses or decision rules to be good,
they must be designed so as to minimize errors of decision. This is not
a simple matter since, for a given sample size, an attempt to decrease
one type of error is accompanied in general by an increase in the other
type of error. In practice one type of error may be more serious than the
other, and so a compromise should be reached in favor of a limitation of
the more serious error. The only way to reduce both types of errors is to
increase the sample size, which may or may not be possible.
Level of Significance
In testing a given hypothesis, the maximum probability with which we
would be willing to risk a Type I error is called the level of significance
of the test. This probability is often specified before any samples are
drawn so that results obtained will not influence our decision.
CHAPTER 8: Test of Hypothesis and Significance 87
You Need to Know
Procedures that enable us to decide whether to accept or
reject hypothesis or to determine whether observed samples
differ significantly from expected results are called tests of
hypotheses, tests of significance, or decision rules.
In practice a level of significance of 0.05 or 0.01 is customary,
although other values are used. If for example a 0.05 or 5% level of significance
is chosen in designing a test of a hypothesis, then there are
about 5 chances in 100 that we would reject the hypothesis when it
should be accepted; i.e., whenever the null hypothesis is true, we are
about 95% confident that we would make the right decision. In such
cases we say that the hypothesis has been rejected at a 0.05 level of significance,
which means that we could be wrong with probability 0.05.
Test Involving the Normal Distribution
To illustrate the ideas presented above, suppose that under a given
hypothesis, the sampling distribution of a statistic S is a normal distribution
with mean μS and standard deviation σS. The distribution of that
standard variable Z = (S − μS)/σS is the standard normal distribution
(mean 0, variance 1) shown in Figure 8-1, and extreme values of Z
would lead to the rejection of the hypothesis.
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