How To Find The Beta With An Alpha Hypothesis

In all statistical hypothesis tests, there are two especially important statistics — alpha and beta. These values represent, respectively, the probability of a type I error and the probability of a type II error. A type I error is a false positive, or conclusion that states there is a significant relationship in the data when in fact there is no significant relationship. A type II error is a false negative, or conclusion that states there is no relationship in the data when in fact there is a significant relationship. Usually, beta is difficult to find. However, if you already have an alpha hypothesis, you can use mathematical techniques to calculate beta. These techniques require additional information: an alpha value, a sample size and an effect size. The alpha value comes from your alpha hypothesis; it is the probability of type I error. The sample size is the number of data points in your data set. The effect size is usually estimated from past data.

Step 1

List the values that are needed in the beta calculation. These values include alpha, the effect size and the sample size. If you do not have past data that states a clear effect size, use the value 0.3 to be conservative. Essentially, the effect size is the strength of the relationship in the data; thus 0.3 is usually taken as it is a "moderate" effect size.

Step 2

Find the Z-score for the value 1 – alpha/2. This Z-score will be used in the beta calculation. After calculating the numerical value for 1 – alpha/2, look up the Z-score corresponding to that value. This is the Z-score needed to calculate beta.

Step 3

Calculate the Z-score for the value 1 – beta. Divide the effect size by 2 and take the square root. Multiply this result by the effect size. Subtract the Z-score found in the last step from this value to arrive at the Z-score for the value 1 – beta.

Step 4

Convert the Z-score to 1 – beta as a number. "Reverse" look up the Z-score for 1 – beta by first looking up the Z-score in the Z-table. Trace this Z-score back to the column (or row) to find a number. This number is equal to 1 – beta.

Step 5

Subtract the number just found from 1. This result is beta.