(Basic Statistics for Citizen Data Scientist)
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:
H0: p ≤ .35
H1: p > .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, n, pobs, 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
Statistics for Beginners in Excel – Statistical Power for the Binomial Distribution
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