Question

The normal approximation of the binomial distribution is appropriate when:

a) np ³5
b) n(1 -p) ³5
c) np £5
d) n(1 -p) £.5 and np £5
e) np ³5 and n(1 -p) ³5

Answer 1

The normal approximation of the binomial distribution is appropriate when both np and n(1 - p) are greater than or equal to 5. This condition allows the binomial distribution to approximate to a normal distribution, which simplifies calculations for large sample sizes.

Step-by-step explanation:

The normal approximation of the binomial distribution is considered appropriate when the product of the number of trials (n) and the probability of success (p) is at least 5 (np ≥ 5), and the product of the number of trials and the probability of failure (nq), where q = 1 - p, is also at least 5 (nq ≥ 5). These conditions ensure that the shape of the binomial distribution is sufficiently similar to the shape of the normal distribution. Thus, the answer to the question is option e) np ≥ 5 and n(1 - p) ≥ 5.

Remember that when dealing with binomial distributions, the approximation improves as these products increase. For evaluation purposes such as hypothesis testing or confidence interval estimation, this rule of thumb is crucial for applying the normal approximation effectively. When the number of successes (np) and the number of failures (nq) are both greater than five, the binomial distribution can be approximated by a normal distribution with mean μ = np and standard deviation σ = √npq.