Which statement about the normal approximation to the binomial is correct?

Which statement about the normal approximation to the binomial is correct?

• A. When np ≥ 5 and n(1-p) ≥ 5.
• B. It is always appropriate regardless of np and n(1-p).
• C. It should be used only when p = 0.5.
• D. It is never used for large n.

Answer: (B)

The main idea tested here is how the normal distribution can be used to approximate a binomial distribution. A binomial random variable counts the number of successes in n independent Bernoulli trials with success probability p. It is the sum of many tiny, independent switches, and the central limit theorem tells us that, as n grows, this sum tends to look like a normal distribution with mean np and variance np(1-p). That’s why the normal approximation is a natural tool for large n and for any fixed p in (0,1). In practice, though, the accuracy of this approximation depends on how large both the expected number of successes and the expected number of failures are. The common rule of thumb is to require np ≥ 5 and n(1-p) ≥ 5 (some use 10). When these conditions hold, the binomial is not too skewed and the normal curve with continuity correction provides a reasonable approximation. If these quantities are small, the normal approximation can be misleading, and other approaches (like the exact binomial calculation or a Poisson approximation when np is small) are preferable. So, conceptually, the normal approximation is rooted in the idea that the binomial converges to a normal distribution as n becomes large for any p in (0,1). The practical guidance about np and n(1-p) gives you a handle on how good that approximation is for a finite sample.

The main idea tested here is how the normal distribution can be used to approximate a binomial distribution. A binomial random variable counts the number of successes in n independent Bernoulli trials with success probability p. It is the sum of many tiny, independent switches, and the central limit theorem tells us that, as n grows, this sum tends to look like a normal distribution with mean np and variance np(1-p). That’s why the normal approximation is a natural tool for large n and for any fixed p in (0,1).

In practice, though, the accuracy of this approximation depends on how large both the expected number of successes and the expected number of failures are. The common rule of thumb is to require np ≥ 5 and n(1-p) ≥ 5 (some use 10). When these conditions hold, the binomial is not too skewed and the normal curve with continuity correction provides a reasonable approximation. If these quantities are small, the normal approximation can be misleading, and other approaches (like the exact binomial calculation or a Poisson approximation when np is small) are preferable.

So, conceptually, the normal approximation is rooted in the idea that the binomial converges to a normal distribution as n becomes large for any p in (0,1). The practical guidance about np and n(1-p) gives you a handle on how good that approximation is for a finite sample.