Several factors affect the power
of a statistical test. Some of the factors are under the control
of the experimenter, whereas others are not. The following example
will be used to illustrate the various factors.

Suppose
a math achievement test were known to be normally
distributed with a mean of 75 and a standard
deviation of σ.
A researcher is interested in whether a new method of teaching
results in a higher mean. Assume that although the experimenter
does not know it, the population mean μ for the new method is
larger than 75. The researcher plans to sample N subjects and
do a one-tailed test of whether the sample mean is significantly
higher than 75. In this section, we consider factors that affect
the probability that the researcher will correctly reject the
false null hypothesis that
the population mean
is 75 or lower. In other words, factors that affect power.

Sample Size

Figure 1 shows that the
larger the sample size, the higher the power. Since sample size
is typically under an experimenter's control, increasing sample
size is one way to increase power. However, it is sometimes
difficult and/or expensive to use a large sample size.

Figure 1. The relationship between sample
size and power for H_{0}: μ = 75,
real μ = 80, one-tailed α = 0.05, for σ's
of 10 and 15.

Standard Deviation

Figure 1 also shows that power is higher when
the standard deviation is small than when it is large. For all
values of N, power is higher for the standard deviation of 10
than for the standard deviation of 15 (except, of course, when
N = 0). Experimenters can sometimes control the standard deviation
by sampling from a homogeneous population of subjects, by reducing
random measurement error, and/or by making sure the experimental
procedures are applied very consistently.

Difference between Hypothesized and True Mean

Naturally, the larger the effect size, the more
likely it is that an experiment would find a significant effect.
Figure 2 shows the effect of increasing the difference between
the mean specified by the null hypothesis (75) and the population
mean μ for standard deviations of 10 and 15.

Figure 2. The relationship between μ and power for H_{0}: μ =
75, one-tailed α = 0.05, for σ's of 10 and 15.

Significance Level

There is a trade-off between the significance level and power:
the more stringent (lower) the significance level, the lower the
power. Figure 3 shows that power is lower for the 0.01 level than
it is for the 0.05 level. Naturally, the stronger the evidence
needed to reject the null hypothesis, the lower the chance that
the null hypothesis will be rejected.

Figure 3. The relationship between significance level and power
with one-tailed tests: μ = 75,
real μ =
80, and σ = 10.

One- versus Two-Tailed Tests

Power is higher with a one-tailed test
than with a two-tailed test as long as the hypothesized direction
is correct. A one-tailed test at the 0.05 level has the same power
as a two-tailed test at the 0.10 level. A one-tailed test, in
effect, raises the significance level.