The R Book

Sample Size for Count Data

How many samples do you need before you have any chance of detecting a significant departure from equality? Suppose you are studying sex ratios in families. How many female children would you need to discover in a family with no males before you could conclude that a father's sex-determining chromosomes were behaving oddly? What about five females and no males? This is not significant because it can occur by chance when p = 0.5 with probability 2 × 0.5⁵ = 0.0625 (note that this is a two-tailed test). The smallest sample that gives significance is a family of six children, all of one sex: 2 × 0.5⁶ = 0.03125. How big would the sample need to be to reject the null hypothesis if one of the children was of the opposite sex? One out of seven is no good, as is one out of eight. You need a sample of at least nine children before you can reject the hypothesis that p = 0.5 when one of the children is of the opposite sex. Here is that calculation using the binom.test function:

binom.test(1,9)

           Exact binomial test
data: 1 and 9
number of successes = 1, number of trials = 9, p-value = 0.03906
alternative hypothesis: true probability of success is not equal to 0.5
95 percent confidence interval:
0.002809137   0.482496515
sample estimates:
probability of success
             0.1111111