SECTION 1.5 A REVIEW OF STATISTICAL INFERENCE 27

V RE ARACNEEEEEINENA NEEIE MA INENENINEMNEENEEEEEWWEEREED

of a z ratio is a constant for any sample of size n because the population
parameter o is a constant. By comparison, the denominator of a t ratio for
any sample of size n is a random variable because of sampling variation
in estimating the parameter o. The numerators of both test statistics are
subject to sampling variation and hence are random variables.
In practice, the population variance reguired to compute z is

rarely ever known. The ratio

X-m

6//n
can be treated as a z variable provided that the sample size is large, say
around 100, and that the population has a normal distribution. An exami-
nation of Appendix Tables D.3 and D.4 reveals that the probabilities
associated with values of z and t are guite similar even for samples as
small as 30. As the sample size is reduced below 30, the correspondence
becomes poorer. Thus, for small samples, the t distribution should be
employed if a sample standard deviation is used to estimate o. The t
distribution and other useful sampling distributions are discussed in Section
2.1.

region
btatistic
is true

EXAMPLE ILLUSTRATING STEPS
IN HYPOTHESIS TESTING

An example may help to clarify the concepts and conventions
involved in hypothesis testing. Assume that we wish to test the hypothesis
that the average performance of some population on a psychological test
is greater than 100. An arithmetic mean is chosen as the appropriate
measure of central tendency. Two test statistics can be suggested for this
experiment: »

X- wo X- wo
si s d < —MO,
ra ri

Let us assume in our example that the population is normally distributed
and that o is known to egual 15. Under these conditions, the appropriate
test statistic is z. The statistical hypotheses can be stated as follows:

H,:u< 100
H,:u>100.

We will reject the null hypothesis in favor of the alternative hypothesis
only if an observed sample mean is so much larger than 100 that it has a
probability of .05 or less of occurring if the population mean really is
egual to 100. As written, the null hypothesis is inexact because it states a
whole region of possible values for the population mean. However, one
exact value is specified, u — 100. Actually, the hypothesis tested is z — 100

eviation
sample.
ibution,

given in
ominator