X follows a binomial distribution with 16 trials, and the probability of success p can be any number between 0 and 1. The largest possible value of the standard deviation of X is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 2
For a binomial distribution, the variance is \(npq = 16\,p(1 - p)\).
The product \(p(1 - p)\) is largest when \(\displaystyle p = q = \frac{1}{2}\), because
\( \displaystyle p(1 - p) = \frac{1}{4} - \left(p - \frac{1}{2}\right)^{2} \le \frac{1}{4} \)
So the largest variance is
\( \displaystyle 16 \times \frac{1}{2} \times \frac{1}{2} = 4 \)
and the largest standard deviation is \(\sqrt{4} = 2\).
Why the other options are incorrect
Option (2) 4 — Incorrect. This is the largest variance; the standard deviation is its square root.
Option (3) 8 — Incorrect. This is the mean np when \(\displaystyle p = \frac{1}{2}\), not the standard deviation.
Option (4) 1 — Incorrect. With \(\displaystyle p = \frac{1}{2}\) the standard deviation is 2, which is larger than 1.