Questions: Describe the sampling distribution of ( hatp ). Assume the size of the population is 30,000. [ n=900, p=0.8 ] Choose the phrase that best describes the shape of the sampling distribution of ( hatp ) below. A. Approximately normal because ( n leq 0.05 N ) and ( np(1-p)<10 ). B. Not normal because ( n leq 0.05 N ) and ( np(1-p)<10 ). C. Approximately normal because ( n leq 0.05 N ) and ( np(1-p) geq 10 ). D. Not normal because ( n leq 0.05 N ) and ( np(1-p) geq 10 ). Determine the mean of the sampling distribution of ( hatp ). ( muhatp= ) (Round to one decimal place as needed.) Determine the standard deviation of the sampling distribution of ( hatp ). ( sigmahatp= ) (Round to three decimal places as needed.)

Solution Steps

To assess the shape of the sampling distribution of $\hat{p}$, we check the following conditions:

1. Condition 1: $n \leq 0.05N$

   • Here, $n = 900$ and $N = 30000$.
   • Calculation: $0.05 \times 30000 = 1500$, thus $900 \leq 1500$ is satisfied.

2. Condition 2: $np(1-p) \geq 10$

   • Calculation: $$np(1-p) = 900 \times 0.8 \times (1 - 0.8) = 900 \times 0.8 \times 0.2 = 144 \geq 10$$
   • This condition is also satisfied.

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

The mean of the sampling distribution of $\hat{p}$ is given by: $$\mu_{\hat{p}} = p = 0.8$$

The standard deviation of the sampling distribution of $\hat{p}$ is calculated using the formula: $$\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.8 \times (1 - 0.8)}{900}} = \sqrt{\frac{0.8 \times 0.2}{900}} = \sqrt{\frac{0.16}{900}} \approx 0.013$$

Final Answer