26

INTRODUCTION TO BASIC CONCEPTS IN EXPERIMENTAL DESIGN

These four steps and the conventions they summarize reguire some
amplification. First, the selection of an appropriate sample statistic is
determined by the experimenter's interest in a particular parameter of
characteristic of a population. If only one population is involved, an experi-

" menter in the behavioral sciences is generally interested in testing a hypoth-

esis with respect to the central tendency of the population. lf, as is
freguently the case, more than one population is involved, hypotheses
concerning differences among the populations in terms either of central
tendency or of dispersion may be of interest. The measures most often
adopted to describe central tendency and dispersion are the mean and
standard deviation, respectively.

Test statistics are similar to sample statistics in that both have
sampling distributions; however, unlike sample statistics, test statistics
are not used to estimate population parameters. Instead, test statistics
provide information in the form of a probability statement, which is used
by an experimenter in deciding whether or not to reject a null hypothesis.
Conventionally, an experimenter specifies a region of the sampling dis-
tribution of the test statistic based on a and H, that will lead to rejection
of the nuli hypothesis prior to computation of the test statistic. This region
is specified in such a way as to contain those values of the test statistic
that have a small probability of occurring if the null hypothesis is true
but a high probability of occurring if the alternative hypothesis is true.
Jf the test statistic falls in the region for rejection, either an improbable
event has occurred or the null hypothesis is false and should be rejected.

Two commonly used test statistics are z and z ratios. For testing a
hypothesis concerning a single population mean, z and t ratios have the
following form:

zožoe NE Sai]
o/./n 6h/no

where X — sample mean used to estimate the population mean ji, Ho —
value of population mean specified by null hypothesis, o — population
standard deviation, d < unbiased estimate of population standard deviation
calculated from a sample, and n < number of observations in the sample.
Jf it can be assumed that the population sampled has a normal distribution,
z and t are distributed as the normal curve and t distribution, respectively.
That is, a z or ft ratio can be computed for each conceivable sample of n
independent observations drawn from a normal population with mean —
u. The value of z or t will vary over the different samples from the popula-
tion. If a plot of the probability-density of each z or t value is made, the
resulting distributions will be distributed as the normal distribution and t
distribution, respectively. The t distribution, unlike the z distribution,
actually is a family of distributions. The exact shape of the t distribution
varies, depending on the number of observations in the sample. Prob-
abilities associated with obtaining various values of z or t are given in
Tables D.3 and D.4, respectively. It should be noted that the denominator

and