To determine which statement is true regarding the standard deviation of the sample means for Sample A and Sample B, we need to understand how the standard deviation of the sample mean (also known as the standard error) is calculated.
The standard error of the sample mean (SEM) is given by the formula:
\[ SEM = \frac{s}{\sqrt{n}} \]
where \(s\) is the standard deviation of the sample, and \(n\) is the sample size.
- Sample A contains 20 students (n = 20)
- Sample B contains 200 students (n = 200)
Since Sample B has a larger sample size (n = 200) compared to Sample A (n = 20), and assuming both samples come from the same population with the same population standard deviation (s), the standard error for Sample B will be smaller:
\[ SEM_A = \frac{s}{\sqrt{20}} \quad \text{(for Sample A)} \] \[ SEM_B = \frac{s}{\sqrt{200}} \quad \text{(for Sample B)} \]
Since \(\sqrt{200} > \sqrt{20}\), it follows that \(SEM_B < SEM_A\). Therefore, Sample B will have a lower standard deviation of the sample mean than Sample A.
Given these considerations, the correct statement is:
Sample B will have a lower standard deviation of the sample mean than Sample A.