# Statistical Power for the Binomial Distribution

### Power of one-tailed test

Example 1: What is the power of the test in Example 3 of Hypothesis Testing for the Binomial Distribution?

For this example we found 13 successes in a sample of size 24 and used a one-tailed test with α = .05 based on the binomial distribution with null and alternative hypotheses:

H0p ≤ .35
H1p > .35

As in Statistical Power of a Sample, to find the power of this test we must first calculate the critical value. This is done using the formula

xcrit = CRITBINOM(24, .35, 1−.05) = 12

This means that at least 95% of the distribution occurs for values x ≤ 12.

Figure 1 – Histogram of the distribution

In fact, 95.8% of the distribution is to the left of the critical value (inclusive) since

BINOMDIST(xcrit, n, p, TRUE) = BINOMDIST(12, 24, .35, TRUE) = .9577

The power of the test is calculated using the following formula where pobs = 13/24 = .54167:

1 – BINOMDIST(xcrit, n, pobs, TRUE) = 1 – BINOMDIST(12, 24, .54167, TRUE) = 58.30%

We can chart the power of the test for various values of pobs as shown in Figure 2.

Figure 2 – Power Curve, one-tailed test

Here cell N11 contains the formula =BINOMDIST(\$O\$8,\$O\$6,M11,TRUE) and O11 contains the formula =1−N11. The rest of the table is created by highlighting the range N11:O23 and pressing Ctrl-D.

### Power of two-tailed test

Example 2: Repeat Example 1 for a two-tailed test.

This time there are two critical values: one on the right (x+crit) and one on the left (x-crit).

These are calculated as follows:

x+crit = CRITBINOM(24, .35, 1−.05/2) = 13
x-crit = CRITBINOM(24, .35, .05/2) = 4

The power of the test is calculated using the following formula where pobs= 13/24 = .54167:

= 1 + BINOMDIST(x-crit−1, npobs, TRUE) – BINOMDIST(x+crit, n, pobs, TRUE)
= 1 + BINOMDIST(3, 24, .54167, TRUE) – BINOMDIST(13, 24, .54167, TRUE)            = 42.15%

Real Statistics Function: The Real Statistics Resource Pack provides the following function to calculate statistical powert automatically.

BINOM_POWER(p0, p1, n, tails, α) = the power of a one sample binomial test when p0 = probability of success on a single trial based on the null hypothesis, p1 = expected probability of success on a single trial, n = the sample size, tails = # of tails: 1 or 2 (default) and α = alpha (default .05).

Referring to Figure 2, we see that BINOM_POWER(.35, .45, 24, 1, .05) = .242. Referring to Figure 3, we see that BINOM_POWER(.35, .45, 24, 2, .05) = .134

Statistics for Beginners in Excel – Confidence Intervals for Sampling Distributions

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