Power Curve of a Mean Test

d

0.5

n

20

α

0.05

The power of a hypothesis test is the probability of correctly rejecting the null hypothesis. Power is shown as the shaded area in the plot on the left and as a function of the noncentrality parameter,

δ

, in the plot on the right. Power depends on the effect size,

d=

μ-

μ

0

σ

, as well as the sample size,

n

, and the significance level,

α

.

The blue curve in the plot on the left is the student

t

distribution and the red curve is the noncentral

t

distribution. The blue vertical line is the critical value of

t

for rejecting the null hypothesis. The noncentrality parameter,

δ

, equals

d

n

. As the mean,

μ

, approaches

μ

0

,

d

approaches 0 and the power of the test decreases. For a typical experimental design,

α

is chosen in advance (normally 0.05), and the standard deviation,

σ

, cannot be controlled by the experimenter. Thus the sample size is chosen to achieve the desired power.