Question

Which of the following samples will have the mean closest to the population mean?
A. A sample with a standard deviation of 24 and a sample size of 12
B. A sample with a standard deviation of 53 and a sample size of 25
C. A sample with a standard deviation of 12 and a sample size of 8
D. A sample with a standard deviation of 14 and a sample size of 6.2

Answer 1

To determine which sample will have a mean closest to the population mean, we need to consider the concept of sampling distribution and the standard error of the mean (SEM). The SEM is given by the formula:

\[
\text{SEM} = \frac{\sigma}{\sqrt{n}}
\]

where \(\sigma\) is the standard deviation of the sample and \(n\) is the sample size. A smaller SEM indicates that the sample mean is likely to be closer to the population mean.

Let's calculate the SEM for each sample:

**A.** Standard deviation = 24, sample size = 12
\[
\text{SEM}_A = \frac{24}{\sqrt{12}} \approx \frac{24}{3.464} \approx 6.93
\]

**B.** Standard deviation = 53, sample size = 25
\[
\text{SEM}_B = \frac{53}{\sqrt{25}} = \frac{53}{5} = 10.6
\]

**C.** Standard deviation = 12, sample size = 8
\[
\text{SEM}_C = \frac{12}{\sqrt{8}} \approx \frac{12}{2.828} \approx 4.24
\]

**D.** Standard deviation = 14, sample size = 6.2
\[
\text{SEM}_D = \frac{14}{\sqrt{6.2}} \approx \frac{14}{2.49} \approx 5.62
\]

Now let's summarize the SEMs:
- SEM for A: 6.93
- SEM for B: 10.6
- SEM for C: 4.24
- SEM for D: 5.62

The sample with the smallest SEM is **C** with a SEM of approximately 4.24.

Therefore, the sample with the mean closest to the population mean is:

**C. A sample with a standard deviation of 12 and a sample size of 8.**