If the mean and variance of a binomial distribution are 24 and 8 respectively, find the value of \(n\).
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 36
For a binomial distribution with n trials and probability of success p (and \(q=1-p\)), the mean is np and the variance is npq. Dividing the variance by the mean leaves q.
Mean: \(np=24\) (1)
Variance: \(npq=8\) (2)
Dividing (2) by (1), np cancels: \(\displaystyle q=\frac{8}{24}=\frac{1}{3}\)
\(\displaystyle p=1-\frac{1}{3}=\frac{2}{3}\)
From (1): \(\displaystyle n=\frac{24}{p}=24\times\frac{3}{2}=36\)