Statistical Methods for Psychology

and if her estimate of dis correct, then she has a probability of .43 of actually rejecting

if it is false to the extent she expects (and a probability of .57 of making a Type II

error).

We next turn the question around and ask how many subjects would be needed for

power 5 .80. From Appendix Power, this would require d52.80.

nrefers to the number of subjects per sample, so for power 5 .80, we need 63 subjects per

sample for a total of 126 subjects.

Unequal Sample Sizes

We just dealt with the case in which However, experiments often have two

samples of different sizes. This obviously presents difficulties when we try to solve for d,

since we need one value for n. What value can we use?

With reasonably large and nearly equal samples, a conservative approximation can be

obtained by letting nequal the smaller of and. This is not satisfactory, however, if the

sample sizes are small or if the two ns are quite different. For those cases we need a more

exact solution.

One seemingly reasonable (but incorrect) procedure would be to set nequal to the

arithmetic mean of and. This method would weight the two samples equally, how-

ever, when in fact we know that the variance of means is proportional not to n, but to 1/n.

The measure that takes this relationship into account is not the arithmetic mean but the har-

monic mean. The harmonic mean ( )of knumbers ( ) is defined as

Thus for two samples sizes ( and ),

we can then use in our calculation of d.

In Chapter 7 we saw an example from Aronson et al. (1998) in which they showed that

they could produce a substantial decrement in the math scores of white males just by re-

minding them that Asian students tend to do better on math exams. This is an interesting

nh

nh=

2

1

n 1

1

1

n 2

=

2 n 1 n 2

n 11 n 2

n 1 n 2

Xh=

k

a

1

Xi

Xh X 1 , X 2 ,... , Xk

n 1 n 2

n 1 n 2

n 1 =n 2 =n.

=62.72

= 2 a

2.80

0.50

b

2

=2(5.6)^2

n= 2 a

d

d

b

2

a

d

d

b

2

=

n

2

d

d

=

A

n

2

d=d

A

n

2

H 0

234 Chapter 8 Power

harmonic
mean ( )Xh