Question 10, 8.2.11
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Part 3 of 5
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Suppose a simple random sample of size $n=75$ is obtained from a population whose size is $\mathrm{N}=15,000$ and whose population proportion with a specified characteristic is $p=0.4$.
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. Not normal because $n \leq 0.05 \mathrm{~N}$ and $n p(1-p)< 10$.
C. Not normal because $n \leq 0.05 \mathrm{~N}$ and $n p(1-p) \geq 10$.
D. Approximately normal because $n \leq 0.05 \mathrm{~N}$ and $\mathrm{np}(1-\mathrm{p})< 10$.
Determine the mean of the sampling distribution of $\hat{p}$.
$\mu_{\hat{p}}=0.4$ (Round to one decimal place as needed.)
Determine the standard deviation of the sampling distribution of $\hat{p}$.
$\sigma_{\hat{p}}=\square$ (Round to six decimal places as needed.)
Answer
Final Answer: The shape of the sampling distribution of \(\hat{p}\) is approximately normal. The mean of the sampling distribution of \(\hat{p}\) is \(\boxed{0.4}\). The standard deviation of the sampling distribution of \(\hat{p}\) is \(\boxed{0.056568}\).
Step 1 ：Given that the sample size is \(n = 75\), the population size is \(N = 15000\), and the population proportion with a specified characteristic is \(p = 0.4\).
Step 2 ：The sampling distribution of \(\hat{p}\) is approximately normal if \(n \leq 0.05 N\) and \(n p(1-p) \geq 10\).
Step 3 ：The mean of the sampling distribution of \(\hat{p}\) is equal to the population proportion \(p\), which is \(0.4\).
Step 4 ：The standard deviation of the sampling distribution of \(\hat{p}\) can be calculated using the formula \(\sqrt{\frac{p(1-p)}{n}}\).
Step 5 ：Substituting the given values into the formula, we get the standard deviation as approximately \(0.056568\).
Step 6 ：Final Answer: The shape of the sampling distribution of \(\hat{p}\) is approximately normal. The mean of the sampling distribution of \(\hat{p}\) is \(\boxed{0.4}\). The standard deviation of the sampling distribution of \(\hat{p}\) is \(\boxed{0.056568}\).
