How to Find the Beta With an Alpha Hypothesis

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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.

    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.

    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.

    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.

    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.

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


    • Virtually every introduction to statistics textbook has a Z-table in the appendix. If you do not have a Z-table on hand, consult a statistics book from your library.