Under what condition is the normal approximation to the binomial distribution applicable?

Under what condition is the normal approximation to the binomial distribution applicable?

• A. When sample size is less than 10
• B. When both np and n(1-p) are greater than 5
• C. When p is equal to 0.5
• D. When n is a prime number

Answer: (B)

The condition under which the normal approximation to the binomial distribution is most suitable is when both np (the expected number of successes) and n(1-p) (the expected number of failures) are greater than 5. This ensures that the distribution of the binomial random variable is sufficiently spread out and resembles a normal distribution, thus allowing for the application of normal approximation techniques. When both of these parameters exceed 5, the binomial distribution meets the criteria for normality, as the Central Limit Theorem supports the approximation of the distribution of sums (or averages) of a large number of independent, identically distributed variables. Specifically, this criterion helps to assure that the approximated normal distribution captures the shape and characteristics of the original binomial distribution, leading to more accurate probability calculations. In practical terms, if either np or n(1-p) is too small, the shape of the binomial distribution may be skewed, and the normal approximation may not provide reliable results. Thus, ensuring that both parameters exceed 5 is critical for a valid application of the normal approximation in evaluating probabilities of binomially distributed events.

The condition under which the normal approximation to the binomial distribution is most suitable is when both np (the expected number of successes) and n(1-p) (the expected number of failures) are greater than 5. This ensures that the distribution of the binomial random variable is sufficiently spread out and resembles a normal distribution, thus allowing for the application of normal approximation techniques.

When both of these parameters exceed 5, the binomial distribution meets the criteria for normality, as the Central Limit Theorem supports the approximation of the distribution of sums (or averages) of a large number of independent, identically distributed variables. Specifically, this criterion helps to assure that the approximated normal distribution captures the shape and characteristics of the original binomial distribution, leading to more accurate probability calculations.

In practical terms, if either np or n(1-p) is too small, the shape of the binomial distribution may be skewed, and the normal approximation may not provide reliable results. Thus, ensuring that both parameters exceed 5 is critical for a valid application of the normal approximation in evaluating probabilities of binomially distributed events.