Question

We conduct a simulation to mimic randomly sampling from a population with of college graduates. In the population 62% had student loans. Each sample has 50 graduates in it. According to theory, what will be the standard deviation of the distribution of sample proportions?
Enter a number in decimal form. For example, enter 0.50, not 50 or 50%.
Round your answer to three decimal places, e.g. 0.125, not 0.1254

Answer 1

Answer:  0.069

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Work Shown:

p = 0.62 = given population proportion

n = 50 = given sample size

$\sigma_{\hat{p}}$ = standard deviation of the sample proportions $\hat{p}$

$\sigma_{\hat{p}}$ =  standard error of $\hat{p}$

$\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}\\\\\sigma_{\hat{p}} = \sqrt{\frac{0.62(1-0.62)}{50}}\\\\\sigma_{\hat{p}} = \sqrt{\frac{0.62*0.38}{50}}\\\\\sigma_{\hat{p}} = \sqrt{\frac{0.2356}{50}}\\\\\sigma_{\hat{p}} = \sqrt{0.004712}\\\\\sigma_{\hat{p}} \approx 0.06864400920692\\\\\sigma_{\hat{p}} \approx 0.069$

This value tells us how spread out the distribution of sample proportions $\hat{p}$ will be.