Describe the sampling distribution of p̂. Assume the size of the population is 15,000 . n=700, p=0.1. Choose the phrase that best describes the shape of the sampling distribution of p̂ below. A. Not normal because n <= 0.05 N and np(1-p) < 10. B. Approximately normal because n<=0.05N and np(1-p) < 10. C. Approximately normal because n<=0.05N and np(1-p) >= 10. D. Not normal because n <= 0.05N and np(1-p) < 10.

The sampling distribution of \( \hat{p} \) can be assessed using the conditions for normality. Since your population size is 15,000 and sample size \( n = 700 \) is less than 5% of the population, we can proceed with the normal approximation. To check the condition for normality, we calculate \( np = 700 \times 0.1 = 70 \) and \( n(1 - p) = 700 \times 0.9 = 630 \). Since both values are greater than 10, the correct answer is C: Approximately normal because \( n \leq 0.05 N \) and \( np(1-p) \geq 10 \). But did you know that the Central Limit Theorem plays a crucial role here? It states that as the sample size increases, the sampling distribution of the sample proportion \( \hat{p} \) will take on a normal shape, regardless of the population's distribution, as long as certain conditions are met! Another fun fact: In real-world applications, understanding the sampling distribution of proportions helps in various fields such as market research, opinion polls, and medical studies. By ensuring you have a large enough sample and that your counts meet the conditions, you can make reliable inferences about the larger population with confidence!