Statistical Methods for Psychology

For expository purposes we will assume for the moment that we are interested in test-

ing one sample mean against a specified population mean, although the approach will

immediately generalize to testing other hypotheses.

8.1 Factors Affecting the Power of a Test

As might be expected, power is a function of several variables. It is a function of (1) a, the

probability of a Type I error, (2) the true alternative hypothesis ( ), (3) the sample size,

and (4) the particular test to be employed. With the exception of the relative power of inde-

pendent versus matched samples, we will avoid this last relationship on the grounds that

when the test assumptions are met, the majority of the procedures discussed in this book

can be shown to be the uniformly most powerful tests of those available to answer the ques-

tion at hand. It is important to keep in mind, however, that when the underlying assump-

tions of a test are violated, the nonparametric tests discussed in Chapter 18, and especially

the resampling tests, are often more powerful.

The Basic Concept

First we need a quick review of the material covered in Chapter 4. Consider the two distri-

butions in Figure 8.1. The distribution to the left (labeled ) represents the sampling dis-

tribution of the mean when the null hypothesis is true and m5m 0. The distribution on the

right represents the sampling distribution of the mean that we would have if were false

and the true population mean were equal to m 1. The placement of this distribution depends

entirely on what the value of m 1 happens to be.

The heavily shaded right tail of the H 0 distribution represents a, the probability of a

Type I error, assuming that we are using a one-tailed test (otherwise it represents a/2). This

area contains the sample means that would result in significant values of t. The second dis-

tribution ( ) represents the sampling distribution of the statistic when is false and the

true mean is m 1. It is readily apparent that even when is false, many of the sample means

(and therefore the corresponding values of t) will nonetheless fall to the left of the critical

value, causing us to fail to reject a false , thus committing a Type II error. The probabil-

ity of this error is indicated by the lightly shaded area in Figure 8.1 and is labeled b. When

is false and the test statistic falls to the right of the critical value, we will correctly

reject a false. The probability of doing this is what we mean by power, and is shown in

the unshaded area of the distribution.

Power as a Function of a

With the aid of Figure 8.1, it is easy to see why we say that power is a function of a. If

we are willing to increase a, our cutoff point moves to the left, thus simultaneously

H 1

H 0

H 0

H 0

H 0

H 1 H 0

H 0

H 0

H 1

Section 8.1 Factors Affecting the Power of a Test 227

H 0 H 1

Power

Critical value

01

Figure 8.1 Sampling distribution of under X H 0 and H 1