Which two conditions must be satisfied for the sampling distribution of p̅ to be approximately normal?

The correct conditions for the sampling distribution of the sample proportion (p̅) to be approximately normal are that the product of the sample size (n) and the probability of success (p) must be sufficiently large, specifically, both np and n(1-p) need to be at least 10. This ensures that there are enough successes and failures in the sample for the distribution to resemble a normal distribution.

When np ≥ 10, it indicates that there are enough observed successes to provide a stable estimate of the proportion of successes in the population. Similarly, n(1-p) ≥ 10 ensures that there are also enough failures in the sample. Both conditions together assure that the variability in the sampling distribution is low enough that it can be well-approximated by a normal distribution.

Choosing a lower threshold, such as 5, (as suggested in the incorrect option) may not provide a sufficiently normal approximation when the sample size is small or the true population proportion is close to 0 or 1. Therefore, requiring both conditions to meet the threshold of 10 is essential for reliable statistical inference.