Questions: Describe the sampling distribution of p̂. Assume the size of the population is 30,000. n=300, p=0.4 Choose the phrase that best describes the shape of the sampling distribution of p̂ below. A. Approximately normal because n ≤ 0.05 N and np(1-p)<10. B. Not normal because n ≤ 0.05 N and np(1-p) ≥ 10. C. Approximately normal because n ≤ 0.05 N and np(1-p) ≥ 10. D. Not normal because n ≤ 0.05 N and np(1-p)<10. Determine the mean of the sampling distribution of p̂. μp̂=0.4 (Round to one decimal place as needed.) Determine the standard deviation of the sampling distribution of p̂. σp̂= (Round to three decimal places as needed.)

Describe the sampling distribution of p̂. Assume the size of the population is 30,000.
n=300, p=0.4

Choose the phrase that best describes the shape of the sampling distribution of p̂ below.
A. Approximately normal because n ≤ 0.05 N and np(1-p)<10.
B. Not normal because n ≤ 0.05 N and np(1-p) ≥ 10.
C. Approximately normal because n ≤ 0.05 N and np(1-p) ≥ 10.
D. Not normal because n ≤ 0.05 N and np(1-p)<10.

Determine the mean of the sampling distribution of p̂.
μp̂=0.4 (Round to one decimal place as needed.)
Determine the standard deviation of the sampling distribution of p̂.
σp̂= (Round to three decimal places as needed.)

Transcript text: Describe the sampling distribution of $\hat{p}$. Assume the size of the population is 30,000 . \[ n=300, p=0.4 \] Choose the phrase that best describes the shape of the sampling distribution of $\hat{p}$ below. A. Approximately normal because $\mathrm{n} \leq 0.05 \mathrm{~N}$ and $\mathrm{np}(1-\mathrm{p})<10$. B. Not normal because $\mathrm{n} \leq 0.05 \mathrm{~N}$ and $\mathrm{np}(1-\mathrm{p}) \geq 10$. C. Approximately normal because $\mathrm{n} \leq 0.05 \mathrm{~N}$ and $\mathrm{np}(1-\mathrm{p}) \geq 10$. D. Not normal because $\mathrm{n} \leq 0.05 \mathrm{~N}$ 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 three decimal places as needed.)

Solution

Solution Steps

Step 1: Determine the Shape of the Sampling Distribution

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

Condition 1: $n \leq 0.05N$
- Given $n = 300$ and $N = 30000$: \[ 0.05N = 0.05 \times 30000 = 1500 \] Since $300 \leq 1500$, this condition is satisfied.
Condition 2: $np(1-p) \geq 10$
- Given $p = 0.4$: \[ np(1-p) = 300 \times 0.4 \times (1 - 0.4) = 300 \times 0.4 \times 0.6 = 72 \] Since $72 \geq 10$, this condition is also satisfied.

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

Step 2: Calculate the Mean of the Sampling Distribution

The mean of the sampling distribution of $\hat{p}$ is given by: \[ \mu_{\hat{p}} = p = 0.4 \]

Step 3: Calculate the Standard Deviation of the Sampling Distribution

The standard deviation of the sampling distribution of $\hat{p}$ is calculated using the formula: \[ \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} \] Substituting the values: \[ \sigma_{\hat{p}} = \sqrt{\frac{0.4 \times (1 - 0.4)}{300}} = \sqrt{\frac{0.4 \times 0.6}{300}} = \sqrt{\frac{0.24}{300}} \approx 0.028 \]

Final Answer

The shape of the sampling distribution is approximately normal because $n \leq 0.05N$ and $np(1-p) \geq 10$.
The mean of the sampling distribution is $\mu_{\hat{p}} = 0.4$.
The standard deviation of the sampling distribution is $\sigma_{\hat{p}} \approx 0.028$.

Thus, the final answers are: \[ \text{Shape: Approximately normal} \] \[ \mu_{\hat{p}} = 0.4 \] \[ \sigma_{\hat{p}} \approx 0.028 \]

$\boxed{\text{Shape: Approximately normal, } \mu_{\hat{p}} = 0.4, \sigma_{\hat{p}} \approx 0.028}$

Was this solution helpful?