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 should consider the relationship between sample size and standard deviation. Generally, larger sample sizes lead to more accurate estimates of the population mean, and lower standard deviations indicate less variability in the sample.

We can assess the effectiveness of the samples by calculating the standard error of the mean (SEM) for each option, as a smaller SEM implies a more accurate estimate of the population mean:

The formula for the standard error of the mean is:

\[
SEM = \frac{s}{\sqrt{n}}
\]

where \(s\) is the sample standard deviation and \(n\) is the sample size.

Let's calculate the SEM for each sample:

**A. Standard deviation = 24, Sample size = 12**
\[
SEM_A = \frac{24}{\sqrt{12}} = \frac{24}{3.464} \approx 6.93
\]

**B. Standard deviation = 53, Sample size = 25**
\[
SEM_B = \frac{53}{\sqrt{25}} = \frac{53}{5} = 10.6
\]

**C. Standard deviation = 12, Sample size = 8**
\[
SEM_C = \frac{12}{\sqrt{8}} = \frac{12}{2.828} \approx 4.24
\]

**D. Standard deviation = 14, Sample size = 6.2**
\[
SEM_D = \frac{14}{\sqrt{6.2}} = \frac{14}{2.49} \approx 5.62
\]

Now, let's compare the standard errors:
- \(SEM_A \approx 6.93\)
- \(SEM_B = 10.6\)
- \(SEM_C \approx 4.24\) (smallest)
- \(SEM_D \approx 5.62\)

The sample with the smallest standard error of the mean is Option C, which has a SEM of approximately 4.24. This suggests that sample C will have the mean closest to the population mean.

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