SECTION 1.5 A REVIEW OF STATISTICAL INFERENCE 29

SIGN ;
enter sample mean had been 102.5 instead of 102, the decision would have been
cance, to reject H. This can be demonstrated as follows:
iii ; H z - X-u (105-100
i the : — on
bution : » oh/n 3
d, the i According to the normal distribution table in Appendix D.3, the probability
: of obtaining a sample mean of 102.5 if the true mean is really 100 is less
on an than .05. Thus, either a rare and improbable event has occurred or the
evel of true parameter is not 100. This example was fabricated to illustrate the
teps in steps involved in testing a statistical hypothesis. In real life most hypotheses
le that are concerned not with a single population but with differences among two
nd test or more populations. The steps that have been described in connection
z — 100 with a single population are also applicable to tests involving two or more
e mean populations. Procedures and assumptions associated with testing statistical
je mean hypotheses with respect to two or more populations are described in
enter to Chapters 2 and 3. An excellent survey of hypothesis testing and statistical
nining a inference is given by Clark (1963).
ermined
TYPE | AND II ERRORS
In carrying out the decision process outlined above, the experimenter
ording to may make a correct decision or he may commit an error. If he decides to
. mean of reject H, when the population mean is really egual to 100, he has committed

. decision a type I error. On the other hand, if he decides not to reject H, when the
09 > .05, population mean is really egual to, say, 103, he has committed a type ll

,e regions H error. In summary, the two possible errors an experimenter may make are

5-2. IE the s | H
Type | error.  Reject H, (tested hypothesis) when it is true. The probability

a is the risk of making a type 1 error.

Type Il error. Fail to reject H, when it is false. The risk of making a type Il
error is designated as B.

The regions corresponding to the probability of making a type I error (a)
and a type II error (6) are shown in Figure 1.5-3.

It is apparent from Figure 1.5-3 that the probability of making a
type I error is determined by an experimenter when he specifies a. This
probability can be made as small as an experimenter wishes. It should
be noted from the figure that as the area corresponding to a is made

Mo o — smaller, the area designated as B becomes larger. Thus the two types of
errors are interrelated. The probability of committing a type II error
- Reject Ho — is determined by x, magnitude of diference between the true parameter
and parameter under H,, size of population error variance, and size of
sample (n). If, in the hypothesis-testing example described previously,
the statistic is egual to 102, a decision is made not to reject H,. This decision
may be correct or incorrect, depending on the value of the parameter. If