Final answer:
To use a normal distribution to approximate a binomial distribution, ensure np and nq are both greater than five. For large n and small p, the Poisson distribution may be used instead. With modern tools, direct calculations often eliminate the need for approximations.
Step-by-step explanation:
To determine whether a normal distribution can be used to approximate a binomial distribution, certain conditions must be met. First, there must be a fixed number of independent trials, n, and each trial must have the same probability of success, p, with only two possible outcomes: success or failure. To use the normal approximation, the product of the number of trials and the probability of success (np) and the product of the number of trials and the probability of failure (nq, where q = 1 - p) must both be greater than five. If np > 5 and nq > 5, the shape of the binomial distribution is sufficiently close to a normal distribution to justify the approximation. Sometimes the rule is stricter, preferring both np and nq to be greater than or equal to 10 for a better approximation.
Additionally, for certain cases where n is large and p is small, the Poisson distribution may be a better fit. However, with modern calculators and computer software available, the necessity to use normal approximation can often be bypassed, as these tools can calculate binomial probabilities directly, even for large n values.