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.)

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.)

Transcript text: Describe the sampling distribution of $\hat{p}$. 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 $\hat{p}$ below. A. Approximately normal because $n \leq 0.05 \mathrm{~N}$ and $n p(1-p)<10$. B. Not normal because $\mathrm{n} \leq 0.05 \mathrm{~N}$ and $\mathrm{np}(1-\mathrm{p})<10$. C. Approximately normal because $n \leq 0.05 \mathrm{~N}$ and $\mathrm{np}(1-\mathrm{p}) \geq 10$. D. Not normal because $n \leq 0.05 \mathrm{~N}$ and $\mathrm{np}(1-\mathrm{p}) \geq 10$. Determine the mean of the sampling distribution of $\hat{p}$. $\mu_{\hat{p}}=$ $\square$ (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 assess the shape of the sampling distribution of $\hat{p}$, we check the following conditions:

Condition 1: $n \leq 0.05N$
- Here, $n = 900$ and $N = 30000$.
- Calculation: $0.05 \times 30000 = 1500$, thus $900 \leq 1500$ is satisfied.
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.

Step 2: Calculate the Mean of the Sampling Distribution

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

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: \[ \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.8 \times (1 - 0.8)}{900}} = \sqrt{\frac{0.8 \times 0.2}{900}} = \sqrt{\frac{0.16}{900}} \approx 0.013 \]

Final Answer

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 $\mu_{\hat{p}} = 0.8$.
The standard deviation of the sampling distribution is $\sigma_{\hat{p}} \approx 0.013$.

Thus, the final answers are: \[ \text{Shape: } \boxed{\text{C. Approximately normal because } n \leq 0.05N \text{ and } np(1-p) \geq 10} \] \[ \mu_{\hat{p}} = \boxed{0.8} \] \[ \sigma_{\hat{p}} = \boxed{0.013} \]

Was this solution helpful?

Unhelpful

Helpful

Questions asked by other users

< Previous Next >

Thank you for your feedback.

Copied!