Question

a survey was conducted to study the proportion of households in a community that recycle regularly. the population proportion of households that recycle is p = 0.34. a simple random sample of 100 households is selected. a. find the expected value of the sample proportion p̅. b. determine the standard error of p̅. (round your answer to four decimal places.) c. describe the sampling distribution of p̅ by specifying its shape and parameters. (round your answer for σp̅ to four decimal places.) e(p̅)= σp̅ = since np = and n(1 - p)=, the sampling distribution? be approximated with a normal distribution. d. what does the sampling distribution of p̅ describe?

Explanation:

Step1: Find the expected value of the sample proportion

The expected value of the sample proportion $E(\hat{p})$ is equal to the population proportion $p$. Given $p = 0.34$, so $E(\hat{p})=0.34$.

Step2: Calculate the standard error of the sample proportion

The formula for the standard error of the sample proportion $\sigma_{\hat{p}}=\sqrt{\frac{p(1 - p)}{n}}$. Here, $p = 0.34$, $1-p=1 - 0.34 = 0.66$ and $n = 100$. Then $\sigma_{\hat{p}}=\sqrt{\frac{0.34\times0.66}{100}}=\sqrt{\frac{0.2244}{100}}=\sqrt{0.002244}\approx0.0474$.

Step3: Check the conditions for normal - approximation

We need to check $np$ and $n(1 - p)$. $np=100\times0.34 = 34$ and $n(1 - p)=100\times0.66 = 66$. Since $np\geq10$ and $n(1 - p)\geq10$, the sampling distribution of $\hat{p}$ can be approximated with a normal distribution.

Step4: Describe the sampling distribution

The sampling distribution of $\hat{p}$ is approximately normal. The mean $\mu_{\hat{p}}=E(\hat{p}) = 0.34$ and the standard deviation $\sigma_{\hat{p}}\approx0.0474$.

Answer:

a. $E(\hat{p}) = 0.34$
b. $\sigma_{\hat{p}}\approx0.0474$
c. The sampling distribution of $\hat{p}$ is approximately normal with mean $\mu_{\hat{p}} = 0.34$ and standard deviation $\sigma_{\hat{p}}\approx0.0474$.
d. Since $np = 34\geq10$ and $n(1 - p)=66\geq10$, the sampling distribution of $\hat{p}$ can be approximated with a normal distribution.