Question

A random sample is to be selected from a population that has a proportion of successes \(\pi=.65 .\) Determine the mean and standard deviation of the sampling distribution of \(p\) for each of the following sample sizes: a. \(n=10\) b. \(n=20\) c. \(n=30\) d. \(n=50\) e. \(n=100\) f. \(n=200\)

Short answer

The proportions mean is .65 for all given sample sizes. The standard deviations are: for \(n = 10\) it's 0.142, for \(n = 20\) it's 0.100, for \(n = 30\) it's 0.082, for \(n = 50\) it's 0.065, for \(n = 100\) it's 0.046 and for \(n = 200\) it's 0.033.

Step-by-step solution

Calculate Mean and Standard Deviation for \(n = 10\)

Starting with \(n = 10\), the mean \(\mu_p\) using the formula \(\mu_p = \pi\) will be .65 and the standard deviation \(\sigma_p\) using the formula \(\sigma_p = \sqrt{\pi*(1-\pi)/n}\) will be \(\sqrt{0.65*(1-0.65)/10} = 0.142\) after rounding to three decimal places.

Calculate Mean and Standard Deviation for \(n = 20\)

Now with \(n = 20\), the mean stays the same (.65) while the standard deviation is calculated as \(\sqrt{0.65*(1-0.65)/20} = 0.100\) after rounding.

Continue Calculating Mean and Standard Deviation for Remaining Samples

Following the same process for the rest of the sample sizes, the standard deviation for \(n = 30\) is 0.082, for \(n = 50\) it's 0.065, for \(n = 100\) it's 0.046 and for \(n = 200\) it's 0.033.