The sampling distribution can be approximated by a normal distribution as long as which conditions are met?

The correct answer involves understanding the conditions necessary for the Central Limit Theorem to apply, particularly in the context of constructing confidence intervals or conducting hypothesis tests for proportions.

When you have a sample proportion, the sampling distribution of that proportion can be approximated by a normal distribution under certain conditions. These conditions are specifically related to the sample size (n) and the estimated probability of success (p). The conditions (np > 5) and (n(1-p) > 5) ensure that both the number of successes and the number of failures in the sample are sufficiently large for the normal approximation to be valid.

This requirement is based on the idea that as the sample size increases, the distribution of the sample proportion approaches a normal distribution regardless of the shape of the population distribution, provided that these conditions hold. If the number of successes (np) or the number of failures (n(1-p)) is too small, the sampling distribution may not resemble a normal distribution, which could lead to inaccurate conclusions.

Thus, the significance of these conditions lies in their ability to identify when it is appropriate to assume a normal distribution when working with sample proportions, ensuring the validity of statistical inference methods used in these scenarios