Binomial test

A binomial test compares the number of successes observed in a given number of trials with a hypothesised probability of success. The test has the null hypothesis that the real probability of success is equal to some value denoted p, and the alternative hypothesis that it is not equal to p.  The test can also be performed with a one-sided alternative hypothesis that the real probability of success is either greater than p or that it is less than p.

binom.test(x = 3, n = 4, p=.5, alternative="greater")
## 
##  Exact binomial test
## 
## data:  3 and 4
## number of successes = 3, number of trials = 4, p-value = 0.3125
## alternative hypothesis: true probability of success is greater than 0.5
## 95 percent confidence interval:
##  0.2486046 1.0000000
## sample estimates:
## probability of success 
##                   0.75

Example

Under (the assumption of) simple Mendelian inheritance, a cross between plants of two particular genotypes produces progeny 1/4 of which are “dwarf” and 3/4 of which are “giant”, respectively.

In an experiment to determine if this assumption is reasonable, a cross results in progeny having 243 dwarf and 682 giant plants. If “giant” is taken as success, the null hypothesis is that p = 3/4 and the alternative that p != 3/4. * Conover (1971), p. 97f.

binom.test(c(682, 243), p = 3/4)
## 
##  Exact binomial test
## 
## data:  c(682, 243)
## number of successes = 682, number of trials = 925, p-value = 0.3825
## alternative hypothesis: true probability of success is not equal to 0.75
## 95 percent confidence interval:
##  0.7076683 0.7654066
## sample estimates:
## probability of success 
##              0.7372973
binom.test(682, 682 + 243, p = 3/4)   # The same.
## 
##  Exact binomial test
## 
## data:  682 and 682 + 243
## number of successes = 682, number of trials = 925, p-value = 0.3825
## alternative hypothesis: true probability of success is not equal to 0.75
## 95 percent confidence interval:
##  0.7076683 0.7654066
## sample estimates:
## probability of success 
##              0.7372973

Conclusion: Data are in agreement with the null hypothesis.