Under which set of conditions is a binomial distribution approximated by a normal distribution?

Question

BlackTom AI · Accepted Answer

The question asks about when a binomial distribution is approximated by a normal distribution.
Option present: 'when np>=5 and n(1-p)>=5'. This condition is a common rule of thumb: it ensures the binomial distribution has a reasonably large number of expected successes (np) and failures (n(1-p)), which makes the normal approximation reasonably accurate due to the central limit tendencies.
To justify why this condition is used, imagine the binomial distribution with parameters n and p. When both np and n(1-p) are large, the distribution becomes more symmetric and bell-shaped, reducing skewness and improving the fit to a normal curve. The intuition is that the sum of many independent Bernoulli trials tends toward normality, and these thresholds help ensure there are enough expected trials in both success and failure categories to justify that approximation.
What about the alternative interpretations or potential pitfalls? Some texts use slightly different thresholds (for example np>5 and n(1-p)>5 as a minimum, or np and n(1-p) ≥ 10 for better accuracy). The core idea remains: when both expected counts are sufficiently large, the normal distribution provides a good approximation.
If you encounter a case where np or n(1-p) is small, the normal approximation may be poor due to skewness or discrete effects, and exact binomial probabilities or a continuity correction should be preferred.
Note: In this prompt, there are no other answer choices provided to compare against, so we focus on explaining why the single given statement aligns with the standard normal approximation criterion.

EH 206 Homework 1

Under which set of conditions is a binomial distribution approximated by a normal distribution?

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