Suppose a simple random sample of size $n=150$ is obtained from a population whose size is $N=25,000$ and whose population proportion with a specified characteristic is $p=0.2$
Click here to view the standard normal distribution table (page 1).
Click here to view the standard normal distribution table (page 2).
(a) Describe the sampling distribution of $\hat{p}$.

Choose the phrase that best describes the shape of the sampling distribution below.
A. Approximately normal because $n \leq 0.05 \mathrm{~N}$ and $n p(1-p) \geq 10$
B. Approximately normal because $n \leq 0.05 \mathrm{~N}$ and $n p(1-p)< 10$
C. Not normal because $n \leq 005 \mathrm{~N}$ and $n p(1-p)< 10$
D. Not normal because $n \leq 0.05 \mathrm{~N}$ and $n p(1-p) \geq 10$
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Step 1 ：The problem is asking to describe the sampling distribution of the proportion \(\hat{p}\). The shape of the sampling distribution can be approximated as normal if two conditions are met: \(n \leq 0.05N\) and \(np(1-p) \geq 10\). We need to check if these conditions are met for the given values of \(n\), \(N\), and \(p\).

Step 2 ：First, we need to check if \(n \leq 0.05N\). This condition is to ensure that the sample size is less than or equal to 5% of the population size.

Step 3 ：Second, we need to check if \(np(1-p) \geq 10\). This condition is to ensure that the sample size is large enough for the Central Limit Theorem to apply, which allows us to approximate the sampling distribution as normal.

Step 4 ：Given that \(n = 150\), \(N = 25000\), and \(p = 0.2\), we can check these conditions.

Step 5 ：Both conditions are met. The sample size is less than or equal to 5% of the population size and the sample size is large enough for the Central Limit Theorem to apply. Therefore, the sampling distribution of \(\hat{p}\) can be approximated as normal.

Step 6 ：Final Answer: The best description for the shape of the sampling distribution is \(\boxed{\text{A. Approximately normal because } n \leq 0.05N \text{ and } np(1-p) \geq 10}\).