Question

suppose you have two populations: population a—all students at illinois state university $(n = 21,000)$ and population b—all residents of homer glen, il $(n = 21,000)$. you want to estimate the mean age of each population using two separate samples each of size $n = 75$. if you construct a $95%$ confidence interval for each population mean, will the margin of error for population a be larger, the same, or smaller than the margin of error for population b? justify your reasoning

choose the correct answer below.

a. the margin of error for population a will be larger because its confidence level will be lower

b. the margin of error for population a will be the same because its confidence level, sample size, and standard deviation will be the same.

c. the margin of error for population a will be larger because its standard deviation will be larger

d. the margin of error for population a will be smaller because its sample standard deviation will be smaller

e. the margin of error for population a will be smaller because its confidence level will be higher

Explanation:

Step1: Recall the formula for margin of error

The formula for the margin of error \(E\) for a confidence interval for the population mean (when the population standard deviation \(\sigma\) is unknown and we use the sample standard deviation \(s\) as an estimate, assuming \(n\geq30\)) is \(E = t_{\alpha/2}\frac{s}{\sqrt{n}}\). For a \(95\%\) confidence interval, the critical value \(t_{\alpha/2}\) depends on the confidence level. Here, the confidence level is \(95\%\) for both populations. The sample size \(n = 75\) for both populations.

Step2: Analyze the effect of confidence level, sample size and standard deviation on margin of error

Since the confidence level (\(95\%\)) and sample size (\(n = 75\)) are the same for both populations. If we assume that the standard deviations (either population or sample, since for large \(n\) sample standard deviation is a good estimate of population standard deviation) of the age distributions for the two populations are the same (no information is given to suggest otherwise).
Using the formula \(E=t_{\alpha/2}\frac{s}{\sqrt{n}}\), when \(t_{\alpha/2}\) (determined by confidence level), \(s\) (standard deviation) and \(n\) (sample size) are the same for both populations, the margin of error will be the same.

Answer:

B. The margin of error for Population A will be the same because its confidence level, sample size, and standard deviation will be the same.