Final answer:
For the normal approximation to the binomial distribution to be valid, both np and nq must be greater than five. This ensures that the distribution shape is close enough to the normal curve for the approximation to be reliable.
Step-by-step explanation:
To use the normal approximation for the binomial distribution, the conditions for np (the mean number of successes) and nq (the mean number of failures) must be such that both are greater than five. This is to ensure that the shape of the binomial distribution is sufficiently similar to that of the normal distribution. The formulas used to approximate the binomial distribution with a normal distribution are μ = np for the mean, and σ = √npq for the standard deviation, where q = 1 - p. If np > 5 and nq > 5, the approximation becomes more reliable, and it is considered even better if both values are greater than or equal to 10. This is because the normal distribution approximates the binomial distribution more closely when the number of trials is large and the probability of success is neither very high nor very low.
When performing a hypothesis test of a single population proportion, these conditions must also be met in order to use the normal distribution. In summary, np and nq must both be greater than five to apply the normal approximation to the binomial distribution.