To determine the value of p (proportion of successes) for which it is appropriate to use the normal approximation for the sampling distribution of p-hat, we need to consider the conditions required for the normal approximation to hold.
The normal approximation can be used when the sample size is sufficiently large and the sampling distribution is approximately symmetric. Specifically, there are two general guidelines:
1. The sampling distribution of p-hat will be approximately normal if np ≥ 10 and n(1-p) ≥ 10.
- Here, n is the sample size and p is the population proportion.
2. Additionally, if the population is extremely skewed or the sampling distribution is not approximately symmetric, a larger sample size is required for the normal approximation to hold.
Therefore, for a given value of p, it is appropriate to use the normal approximation for the sampling distribution of p-hat when both np ≥ 10 and n(1-p) ≥ 10 conditions are satisfied. If either condition is not met, the normal approximation may not be appropriate, and alternative methods like the binomial distribution or other approximations should be considered.
Note that the specific value of p does not impact the use of the normal approximation; instead, it is determined by the sample size and the conditions mentioned above.