A researcher wants to estimate the average height of all students in a university. He randomly selects two samples from the student population. Sample A contains 20 students, and Sample B contains 200 students. Which of the following statements is true?(1 point)

Question

Responses

The researcher can estimate the population mean by taking the average of the sample means from Sample A and B.
The researcher can estimate the population mean by taking the average of the sample means from Sample A and B.

Sample B will have a lower standard deviation of the sample mean than Sample A.
Sample B will have a lower standard deviation of the sample mean than Sample A.

Both Sample A and Sample B will have the same standard deviation of the sample mean.
Both Sample A and Sample B will have the same standard deviation of the sample mean.

Sample A will have a lower standard deviation of the sample mean than Sample B.

Answer 1

The true statement is:

Sample B will have a lower standard deviation of the sample mean than Sample A.

This is because the standard deviation of the sample mean (also known as the standard error) decreases as the sample size increases. The formula for the standard error of the mean is given by:

\[ SE = \frac{\sigma}{\sqrt{n}} \]

where \(SE\) is the standard error, \(\sigma\) is the population standard deviation, and \(n\) is the sample size. Since Sample B (with 200 students) has a larger sample size than Sample A (with 20 students), the standard error will be smaller for Sample B, leading to a lower standard deviation of the sample mean.