Consider the case where the null hypothesis is that
the population mean, μ_{0,} is 100 with a population standard deviation, σ, of 10.
If the true population, μ_{1}, mean is 105 and we use a sample of size n=25, how
likely is it that we will be able to reject the null hypothesis
(one-tailed alpha = .05)? That is, how much statistical power do we
have?

The WISE Power Applet shown below indicates that the power is .804. The
calculations below show how we could compute the power ourselves.

Derivation of a formula for power computation

If we observe a sample mean greater than the
critical value indicated by the red dashed line, we will reject the null
hypothesis. Let us refer to this critical value as C.

The critical value C is defined on the
blue null
distribution as the value that cuts off the upper .05 (i.e., alpha) of
the blue distribution. We can find the Z score for this value by using a
standard Z table or the
WISE p-z converter applet. Let us call this score Z_{α} because it is
defined on the null distribution by alpha. In our example, Z_{α} = 1.645.
Thus, the critical value C is 1.645 standard errors greater than the
mean of the null distribution, μ_{0}.

In our example,

Similarly, we can generate a formula for the value
of C on the red sampling distribution for the alternate hypothesis,
μ_{1}
= 105. Beta error corresponds to the portion of the red curve that falls
belowC, while statistical power corresponds to the portion of the red
curve that falls aboveC.

Let us use Z_{β} as the label for the
standardized score on the red distribution corresponding to C because
the beta error is defined as the portion of the red curve that falls
below C.

, where
Z_{β} is a negative number in our
example.

We can set the two equations for C equal to each other, giving

Rearranging terms gives

or
,

where d is a common measure of effect size:
d =
(μ_{1} – μ_{0}) / σ.

The result is this elegant formula:

(Formula 1)

This formula expresses the relationship between
four concepts. If we know any three, we can compute the fourth. An easy
way to remember these four concepts is with the mnemonic BEAN:

B = beta error rate, represented by Z_{β}. Power is (1 – beta error rate).