### Power Calculation

Suppose we wish to determine the statistical power of a test of the null hypothesis that the population mean is 100 if in fact the population mean is 105 with a population standard deviation of 10, and we use a sample size of 25 cases and set alpha error to .05 for a one-tailed test.

Statistical power is equal to (1 – beta error), so to find statistical power we can solve for Z_{β}.

We can rearrange the terms in Formula 1 to solve for *Z _{β}* :

Using the BEAN acronym, we wish to solve for B because power is (1 – beta error). We need to specify the other three terms: E, A, and N.

**E**ffect size d = (*μ _{1}* –

*μ*) /

_{0}*σ*= (105 – 100) / 10 = 0.50.

**A**lpha error rate is set at .05, which corresponds to a Z score of 1.645 for a one-tailed test.

**N** is set at 25.

Thus, *Z _{β}* = 1.645 – 5 (0.50) = 1.645 – 2.500 = -0.855

*Z _{β}* corresponds to the

*Z*score for the critical value on the pink (alternate hypothesis) sampling distribution. We will reject the null hypothesis if we observe a score greater than this critical value. We consult a

*Z*table or the WISE

*p*–

*z*converter applet to find that the probability of observing a

*Z*score greater than ‑0.855 is .804. This is the value given by the WISE power applet.

Thus, in this scenario our statistical power is about 80%. If we collect data and conduct a test of statistical significance, there is an 80% chance that the test will attain statistical significance, and a 20% chance that the test will fail to detect statistical significance.

### Sample Size Calculation

We can rearrange the terms in Formula 1 to solve for *n*.

Suppose we wish to have a sample large enough to have power of 90% to detect a difference between means of 4.0 where the standard deviation is 10.0, using one-tailed alpha error rate of .01. Recalling BEAN, we need to specify B, E, and A to solve for N.

**B**:*Z*corresponds to the Z score on the pink distribution where 90% of the distribution falls above that score. We can use the WISE_{β}*p*–*z*converter or a*Z*table to find*Z*= -1.282._{β}**E**: The effect size*d*= (*μ*–_{1}*μ*) /_{0}*σ*= (4.0 / 10.0) = 0.40.**A**: The alpha error rate of .01 corresponds to*Z*= 2.326._{α}

Applying the formula for *n*:

Thus, we need a sample size of about 82 to attain the desired level of power in this scenario.

It is important to note carefully that the sign on *Z*_{β} is often negative. Subtracting a negative value is equivalent to adding a positive value.