When can you use normal approximation to a Binomial(n,p)?

When can you use normal approximation to a Binomial(n,p)?

• A. When np ≥ 5 and n(1-p) ≥ 5.
• B. Only when n is prime.
• C. When p = 0.5 exactly.
• D. When n is less than 10.

Answer: (A)

Normal approximation works well when the binomial distribution is roughly bell-shaped, which happens when you have many trials and neither outcomes are too rare. Since a binomial with parameters n and p has mean μ = np and variance σ^2 = np(1−p), using a normal with that mean and variance makes sense only if both the expected number of successes and the expected number of failures are reasonably large. A practical rule of thumb is that both np and n(1−p) should be at least about 5. This ensures the distribution isn’t too skewed and can be approximated well by a continuous normal (often with a continuity correction for better accuracy). If either np or n(1−p) is small, the normal approximation can be poor, and you’d turn to exact binomial probabilities or, in cases where p is small and np is modest, a Poisson approximation. The conditions do not depend on n being prime, nor require p to be exactly 0.5, and a small n (like n < 10) isn’t a universal barrier—it’s just that the rule of thumb may not be met in those cases.

Normal approximation works well when the binomial distribution is roughly bell-shaped, which happens when you have many trials and neither outcomes are too rare. Since a binomial with parameters n and p has mean μ = np and variance σ^2 = np(1−p), using a normal with that mean and variance makes sense only if both the expected number of successes and the expected number of failures are reasonably large. A practical rule of thumb is that both np and n(1−p) should be at least about 5. This ensures the distribution isn’t too skewed and can be approximated well by a continuous normal (often with a continuity correction for better accuracy). If either np or n(1−p) is small, the normal approximation can be poor, and you’d turn to exact binomial probabilities or, in cases where p is small and np is modest, a Poisson approximation. The conditions do not depend on n being prime, nor require p to be exactly 0.5, and a small n (like n < 10) isn’t a universal barrier—it’s just that the rule of thumb may not be met in those cases.