You intend to draw a random sample of size $n=583$ from a population whose parameter is $p=0.59$ What is the mean of the distribution of sample proportions? \[ \mu_{\hat{p}}= \] What is the standard deviation of the distribution of sample proportions? (Report answer accurate to 2 decimal places.) \[ \sigma_{\hat{p}}= \] Check Answer
Solution
Step 1 ：The mean of the distribution of sample proportions is equal to the population proportion. Given that the population proportion, \(p=0.59\), we have \(\mu_{\hat{p}}=p=0.59\).
Step 2 ：The standard deviation of the distribution of sample proportions is given by the formula \(\sigma_{\hat{p}}=\sqrt{\frac{p(1-p)}{n}}\), where \(p\) is the population proportion and \(n\) is the sample size. In this case, \(p=0.59\) and \(n=583\).
Step 3 ：Substituting the given values into the formula, we get \(\sigma_{\hat{p}}=\sqrt{\frac{0.59(1-0.59)}{583}}\).
Step 4 ：Solving the above expression, we get \(\sigma_{\hat{p}}=0.020369654219844296\).
Step 5 ：Rounding off the above value to two decimal places, we get \(\sigma_{\hat{p}}=0.02\).
Step 6 ：Final Answer: The mean of the distribution of sample proportions is \(\boxed{0.59}\) and the standard deviation of the distribution of sample proportions is \(\boxed{0.02}\).
