Statistical Methods for Psychology

function of n[i.e., f(n)] will be defined differently for each individual test. The convenient

thing about this system is that it will allow us to use the same table of dfor power calcula-

tions for all the statistical procedures to be considered.

8.3 Power Calculations for the One-Sample t

We will first examine power calculations for the one-sample t test. In the preceding section

we saw that dis based on dand some function of n. For the one-sample t, that function will

be , and dwill then be defined as. Given das defined here, we can immedi-

ately determine the power of our test from the table of power in Appendix Power.

Assume that a clinical psychologist wants to test the hypothesis that people who seek

treatment for psychological problems have higher IQs than the general population. She

wants to use the IQs of 25 randomly selected clients and is interested in finding the power

of detecting a difference of 5 points between the mean of the general population and the

mean of the population from which her clients are drawn. Thus, , , and

s515.

then

Although the clinician expects the sample means to be above average, she plans to use

a two-tailed test at a5.05 to protect against unexpected events. From Appendix Power,

for d51.65 with a5.05 (two-tailed), power is between .36 and .40. By crude linear in-

terpolation, we will say that power 5 .38. This means that, if is false and is really

105, only 38% of the time can our clinician expect to find a “statistically significant” dif-

ference between her samplemean and that specified by. This is a rather discouraging

result, since it means that if the true mean really is 105, 62% of the time our clinician will

make a Type II error. (The more accurate calculation by G*Power computes the power as

.35, which illustrates that our approximation procedure is remarkably close.)

Since our experimenter was intelligent enough to examine the question of power

before she began her experiment, all is not lost. She still has the chance to make changes

that will lead to an increase in power. She could, for example, set aat .10, thus increasing

power to approximately .50, but this is probably unsatisfactory. (Journal reviewers, for

example, generally hate to see aset at any value greater than .05.)

Estimating Required Sample Size

Alternatively, the investigator could increase her sample size, thereby increasing power.

How large an ndoes she need? The answer depends on what level of power she desires.

Suppose she wishes to set power at .80. From Appendix Power, for power 5 .80, and a5

0.05, dmust equal 2.80. Thus, we have dand can simply solve for n:

=71.91

n=a

d

d

b

2

=a

2.80

0.33

b

2

=8.48^2

d=d 1 n

H 0

H 0 m 1

=1.65

d=d 1 n=0.33 125 =0.33(5)

d=

1052100

15

=0.33

m 1 = 105 m 0 = 100

1 n d=d 1 n

Section 8.3 Power Calculations for the One-Sample t 231