Statistical Methods for Psychology

decreasing band increasing power, although with a corresponding rise in the probability

of a Type I error.

Power as a Function of

The fact that power is a function of the true alternative hypothesis [more precisely ( 2 ),

the difference between (the mean under ) and (the mean under )] is illustrated by

comparing Figures 8.1 and 8.2. In Figure 8.2 the distance between and has been

increased, and this has resulted in a substantial increase in power, though there is still size-

able probability of a Type II error. This is not particularly surprising, since all that we are say-

ing is that the chances of finding a difference depend on how large the difference actually is.

Power as a Function of nand s^2

The relationship between power and sample size (and between power and ) is only a little

subtler. Since we are interested in means or differences between means, we are interested in

the sampling distribution of the mean. We know that the variance of the sampling

distribution of the mean decreases as either nincreases or decreases, since.

Figure 8.3 illustrates what happens to the two sampling distributions ( and ) as we

increase nor decrease , relative to Figure 8.2. Figure 8.3 also shows that, as decreases,

the overlap between the two distributions is reduced with a resulting increase in power.

Notice that the two means (m 0 and m 1 ) remain unchanged from Figure 8.2.

s^2 sX^2

H 0 H 1

s^2 sX^2 =s^2 >n

s^2

m 0 m 1

m 0 H 0 m 1 H 1

m 0 m 1

H 1

228 Chapter 8 Power

Power

Critical value

H 0 H 1

0 1

Figure 8.2 Effect on bof increasing m 0 2m 1

H 0 H 1

0 1

Figure 8.3 Effect on bof decrease in standard error of the mean