Questions: Describe the sampling distribution of p̂ Choose the phrase that best describes the shape of the sampling distribution below. A. Approximately normal because n ≤ 0.05 N and n p(1-p) ≥ 10 B. Approximately normal because n ≤ 0.05 N and np(1-p)<10. C. Not normal because n ≤ 005 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. μp=0.4 (Round to one decimal place as needed) Determine the standard deviation of the sampling distribution of p̂ σp= (Round to six decimal places as needed)

Describe the sampling distribution of p̂

Choose the phrase that best describes the shape of the sampling distribution below.

A. Approximately normal because n ≤ 0.05 N and n p(1-p) ≥ 10

B. Approximately normal because n ≤ 0.05 N and np(1-p)<10.

C. Not normal because n ≤ 005 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.
μp=0.4 (Round to one decimal place as needed)

Determine the standard deviation of the sampling distribution of p̂
σp= (Round to six decimal places as needed)

Transcript text: Describe the sampling distribution of $\hat{p}$ Choose the phrase that best describes the shape of the sampling distribution below. A. Approximately normal because $\mathrm{n} \leq 0.05 \mathrm{~N}$ and $n p(1-\mathrm{p}) \geq 10$ B. Approximately normal because $\mathrm{n} \leq 0.05 \mathrm{~N}$ and $\mathrm{np}(1-\mathrm{p})<10$. C. Not normal because $\mathrm{n} \leq 005 \mathrm{~N}$ 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 $\hat{p}$. $\mu_{p}=0.4$ (Round to one decimal place as needed) Determine the standard deviation of the sampling distribution of $\hat{p}$ $\sigma_{p}=$ $\square$ (Round to six decimal places as needed)

Solution

Solution Steps

Step 1: Determining the Shape of the Sampling Distribution

The sampling distribution of $\hat{p}$ is approximately normal because:

The sample size $n$ is {'less' if condition1 else 'not less'} than 5% of the population size $N$ ($n \leq 0.05N$).
The product $np(1-p)$ is {'greater' if condition2 else 'not greater'} than or equal to 10 ($np(1-p) \geq 10$).

Step 2: Calculating the Mean of the Sampling Distribution

The mean of the sampling distribution of $\hat{p}$ is equal to the population proportion $p$, which is $\mu_{\hat{p}} = 0.4$.

Step 3: Calculating the Standard Deviation of the Sampling Distribution

The standard deviation of the sampling distribution of $\hat{p}$ is calculated using the formula: \[ \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} \times \sqrt{\frac{N-n}{N-1}} \] Substituting the given values, we get $\sigma_{\hat{p}} = 0.0565$.

Final Answer:

The sampling distribution of $\hat{p}$ is approximately normal, with a mean of $\mu_{\hat{p}} = 0.4$ and a standard deviation of $\sigma_{\hat{p}} = 0.0565$.

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