Question

A researcher wants to estimate the proportion of students enrolled at a university who eat fast food more than three times in a typical week. Would the standard error of the sample proportion \(\hat{p}\) be smaller for random samples of size \(n=50\) or random samples of size \(n=200 ?\)

Short answer

The standard error of the sample proportion, \(\hat{p}\), would be smaller for random samples of size \(n=200\) compared to random samples of size \(n=50\).

Step-by-step solution

Understand the Standard Error of the Sample Proportion

The standard error of the sample proportion (\(\hat{p}\)) measures the variability or dispersion of the sampling distribution. It is given by the formula \(\sqrt{(\hat{p}(1-\hat{p})/n)}\), where \(\hat{p}\) is the sample proportion and n is the sample size.

Effect of Sample Size on Standard Error

In the formula for the standard error, n (the sample size) is in the denominator. This means that as n increases, the total value of the standard error decreases. That is, a larger sample size results in a smaller standard error.

Compare the Standard Errors for Different Sample Sizes

If we compare a sample size of 50 to a sample size of 200, since 200 is larger than 50, the standard error of the sample proportion will be smaller for a sample size of 200 compared to a sample size of 50.