Question

Consider taking a random sample from a population with \(p=0.70\). a. What is the standard error of \(\hat{p}\) for random samples of size \(100 ?\) b. Would the standard error of \(\hat{p}\) be smaller for samples of size 100 or samples of size \(400 ?\) c. Does decreasing the sample size by a factor of \(4,\) from 400 to 100 , result in a standard error of \(\hat{p}\) that is four times as large?

Short answer

a. The standard error of \(\hat{p}\) for random samples of size 100 is 0.045. b. The standard error of \(\hat{p}\) would be smaller for samples of size 400. c. Decreasing the sample size by a factor of 4, from 400 to 100, doesn't result in a standard error of \(\hat{p}\) that is four times as large but rather twice as large.

Step-by-step solution

Compute Standard Error for Sample of Size 100

Let's compute the standard error for a sample size of 100. Using the standard error formula \(SE = \sqrt{ \frac{p*(1-p)}{n} }\), where \(p = 0.70\) and \(n = 100\), we get \(SE = \sqrt{ \frac{0.70*(1-0.70)}{100} } = 0.045.\)

Discuss Standard Error for Different Sample Sizes

It is known that as the sample size increases, the standard error decreases. Hence, the standard error for a sample size of 400 would be smaller than for a sample size of 100.

Effect of Reducing Sample Size on Standard Error

When we reduce the sample size, we expect the standard error to increase. But it will not increase exactly by a factor of 4. If we compute the standard error for a sample size of 400, we get \(SE = \sqrt{ \frac{0.70*(1-0.70)}{400} } = 0.0225\), which is half of the standard error when the sample size was 100, rather than four times as large.