Question

suppose the population proportion is 0.38. a. compute the standard error of the sample proportion, $sigma_{hat{p}}$, for the following sample sizes. (round your answers to five decimal places). sample size n standard error 100 200 400 800 b. what happens to the size of the standard error of the proportion as the sample size increases? a. the standard error increases as the sample size increases. b. the standard error approaches the sample proportion as the sample size increases. c. the standard error approaches the population proportion as the sample size increases. d. the standard error decreases as the sample size increases.

Explanation:

Step1: Recall formula for standard error of proportion

The formula for the standard error of a sample proportion $\sigma_{\hat{p}}$ is $\sigma_{\hat{p}}=\sqrt{\frac{p(1 - p)}{n}}$, where $p$ is the population proportion and $n$ is the sample size. Given $p = 0.38$, so $1-p=1 - 0.38=0.62$.

Step2: Calculate for $n = 100$

Substitute $p = 0.38$, $1 - p=0.62$ and $n = 100$ into the formula: $\sigma_{\hat{p}}=\sqrt{\frac{0.38\times0.62}{100}}=\sqrt{\frac{0.2356}{100}}=\sqrt{0.002356}\approx0.04854$.

Step3: Calculate for $n = 200$

Substitute into the formula: $\sigma_{\hat{p}}=\sqrt{\frac{0.38\times0.62}{200}}=\sqrt{\frac{0.2356}{200}}=\sqrt{0.001178}\approx0.03432$.

Step4: Calculate for $n = 400$

Substitute into the formula: $\sigma_{\hat{p}}=\sqrt{\frac{0.38\times0.62}{400}}=\sqrt{\frac{0.2356}{400}}=\sqrt{0.000589}\approx0.02427$.

Step5: Calculate for $n = 800$

Substitute into the formula: $\sigma_{\hat{p}}=\sqrt{\frac{0.38\times0.62}{800}}=\sqrt{\frac{0.2356}{800}}=\sqrt{0.0002945}\approx0.01716$.

For part b, as we can see from the calculations above and the formula $\sigma_{\hat{p}}=\sqrt{\frac{p(1 - p)}{n}}$, when the population - proportion $p$ is fixed, as the sample size $n$ increases, the standard error $\sigma_{\hat{p}}$ decreases.

Answer:

Sample Size $n$	Standard Error
200	0.03432
400	0.02427
800	0.01716

b. D. The standard error decreases as the sample size increases.