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.