Question

Question:A random sample of n = 500 observations is selected from a binomial population with p = .35.

a. Give the mean and standard deviation of the (repeated) sampling distribution of$\hat{p}$the sample proportion of successes for the 500 observations.

b. Describe the shape of the sampling distribution of $\hat{p}$. Does your answer depend on the sample size?

Short answer

1. The mean and standard deviation of $\hat{p}$are .35 and 0.021.
2. The answer depends on the sample size.

Step-by-step solution

Given Information

The sample size is 500.

The probability of success p=0.35

(a) Compute the mean and standard deviation

The mean of the sampling distribution of $\hat{p}$is given by$\hat{p}=0.35$

The standard deviation of the sampling distribution of is given by$$\begin{gathered} d=\sqrt{\frac{pq}{n}} \\ =\sqrt{\frac{0.35\times 0.65}{500}} \\ =0.021 \end{gathered}$$

Describing the shape of the p^

The sampling distribution of is normal as we know that if n is large, then the sampling distribution of follows the normal distribution with a mean $\hat{p}$and standard deviation

$\sqrt{\frac{p\left(1-p\right)}{n}}$

Therefore, the shape of the sampling distribution$\hat{p}$ isnormal.

Here, the number of samples is 500, which is greater than 30.

Therefore, the answer depends on the sample size.