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 = 0.25. a simple random sample of 100 households is selected. a. find the expected value of the sample proportion p̂. b. determine the standard error of p̂. (round your answer to four decimal places.) c. describe the sampling distribution of p̂ by specifying its shape and parameters. (round your answer for σp̂ to four decimal places.) d. what does the sampling distribution of p̂ describe? since np = and n(1 - p) =, the sampling distribution can be approximated with a normal distribution.

Explanation:

Step1: Find expected value of sample proportion

The expected value of the sample proportion $E(\hat{p})$ is equal to the population proportion $p$. Given $p = 0.25$, so $E(\hat{p})=0.25$.

Step2: Calculate standard - error of $\hat{p}$

The formula for the standard error of the sample proportion $\sigma_{\hat{p}}=\sqrt{\frac{p(1 - p)}{n}}$. Here, $p = 0.25$, $1-p=0.75$ and $n = 100$. Then $\sigma_{\hat{p}}=\sqrt{\frac{0.25\times0.75}{100}}=\sqrt{\frac{0.1875}{100}}=\sqrt{0.001875}\approx0.0433$.

Step3: Check normal - approximation conditions

We check $np$ and $n(1 - p)$. Given $n = 100$ and $p=0.25$, then $np=100\times0.25 = 25$ and $n(1 - p)=100\times0.75 = 75$. Since $np\geq10$ and $n(1 - p)\geq10$, the sampling distribution of $\hat{p}$ can be approximated with a normal distribution.

Step4: Describe sampling distribution

The sampling distribution of $\hat{p}$ is approximately normal. The mean (parameter) is $\mu_{\hat{p}}=E(\hat{p}) = 0.25$ and the standard deviation (parameter) is $\sigma_{\hat{p}}\approx0.0433$.

Answer:

a. $E(\hat{p}) = 0.25$
b. $\sigma_{\hat{p}}\approx0.0433$
c. The sampling distribution of $\hat{p}$ is approximately normal with mean $\mu_{\hat{p}} = 0.25$ and standard deviation $\sigma_{\hat{p}}\approx0.0433$.
d. $np = 25$, $n(1 - p)=75$, the sampling distribution can be approximated with a normal distribution.