What does np ≥ 5 guarantee in the context of p̅ sampling distribution?

The criterion np ≥ 5 is a guideline used in statistics to justify the use of the normal approximation to the binomial distribution when dealing with sample proportions. Specifically, this condition ensures that there are enough expected successes (np) and failures (n(1-p)) in the sample, which supports the central limit theorem's application. When this condition is met, the sampling distribution of the sample proportion (p̅) can be approximated by a normal distribution, allowing for more straightforward analysis and inference using normal distribution techniques.

This guideline helps to ensure that the shape of the sampling distribution is reasonably close to normal, which is a fundamental assumption for many inferential statistical methods. When the sample size is not adequate—to the point that np < 5—there is a greater risk that the distribution of sample proportions could be skewed or not sufficiently approximated by a normal curve, leading to inaccurate conclusions. Thus, meeting the np ≥ 5 condition signifies that the normal approximation for the sampling distribution is valid, allowing statisticians to proceed confidently with analyses that rely on this approximation.