To describe the sampling distribution of $\hat{p}$, we need to consider two main conditions:

1. Sample Size Condition ($n \leq 0.05N$): This condition checks whether the sample size $n$ is no more than 5% of the population size $N$. It ensures that the sample size is small enough relative to the population to treat the sampling as independent.

   • Here, $n = 500$ and $N = 15,000$.
   • Calculate $0.05N = 0.05 \times 15,000 = 750$.
   • Since $n = 500$ is less than 750, this condition is satisfied.

2. Normality Condition ($np(1-p) \geq 10$): This condition checks whether the product of the sample size, the probability of success $p$, and the probability of failure $(1-p)$ is at least 10. It ensures that the sampling distribution of $\hat{p}$ is approximately normal.

   • Here, $p = 0.2$.
   • Calculate $np(1-p) = 500 \times 0.2 \times 0.8 = 80$.
   • Since 80 is greater than or equal to 10, this condition is satisfied.

Since both conditions are satisfied, the sampling distribution of $\hat{p}$ is approximately normal.

Correct Answer:

C. Approximately normal because $n \leq 0.05N$ and $np(1-p) \geq 10$.

Would you like more details or have any questions on this?

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Here are 5 related questions to explore further:

1. What is the mean of the sampling distribution of $\hat{p}$?
2. How do you calculate the standard error of $\hat{p}$?
3. Why is the $np(1-p) \geq 10$ condition important for normality?
4. How would the sampling distribution change if $p$ were closer to 0 or 1?
5. What role does the population size play in determining the independence condition?

Tip: Always check both conditions for sample size and normality to ensure you are using the right approximation method.