Question
Which of the following samples will have the mean closest to the population mean?(1 point)
Responses
a sample with a standard deviation of 53 and a sample size of 25
a sample with a standard deviation of 53 and a sample size of 25
a sample with a standard deviation of 24 and a sample size of 12
a sample with a standard deviation of 24 and a sample size of 12
a sample with a standard deviation of 12 and a sample size of 8
a sample with a standard deviation of 12 and a sample size of 8
a sample with a standard deviation of 14 and a sample size of 6.2
a sample with a standard deviation of 14 and a sample size of 6.2
Responses
a sample with a standard deviation of 53 and a sample size of 25
a sample with a standard deviation of 53 and a sample size of 25
a sample with a standard deviation of 24 and a sample size of 12
a sample with a standard deviation of 24 and a sample size of 12
a sample with a standard deviation of 12 and a sample size of 8
a sample with a standard deviation of 12 and a sample size of 8
a sample with a standard deviation of 14 and a sample size of 6.2
a sample with a standard deviation of 14 and a sample size of 6.2
Answers
GPT-4o mini
To determine which sample will have a mean closest to the population mean, we can use the concept of the standard error of the mean (SEM). The SEM is calculated using 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 a more accurate estimate of the population mean.
Let's calculate the SEM for each of the provided samples:
1. For the first sample:
- Standard Deviation (\(\sigma\)) = 53
- Sample Size (\(n\)) = 25
- SEM = \( \frac{53}{\sqrt{25}} = \frac{53}{5} = 10.6 \)
2. For the second sample:
- Standard Deviation (\(\sigma\)) = 24
- Sample Size (\(n\)) = 12
- SEM = \( \frac{24}{\sqrt{12}} \approx \frac{24}{3.464} \approx 6.93 \)
3. For the third sample:
- Standard Deviation (\(\sigma\)) = 12
- Sample Size (\(n\)) = 8
- SEM = \( \frac{12}{\sqrt{8}} \approx \frac{12}{2.828} \approx 4.24 \)
4. For the fourth sample:
- Standard Deviation (\(\sigma\)) = 14
- Sample Size (\(n\)) = 6.2
- SEM = \( \frac{14}{\sqrt{6.2}} \approx \frac{14}{2.49} \approx 5.62 \)
Now, comparing SEMs:
- First sample SEM = 10.6
- Second sample SEM ≈ 6.93
- Third sample SEM ≈ 4.24
- Fourth sample SEM ≈ 5.62
The third sample (with a standard deviation of 12 and a sample size of 8) has the smallest standard error of the mean, which means its mean will be the closest to the population mean.
**Answer:** A sample with a standard deviation of 12 and a sample size of 8.
\[
\text{SEM} = \frac{\sigma}{\sqrt{n}}
\]
where \(\sigma\) is the standard deviation of the sample and \(n\) is the sample size. A smaller SEM indicates that the sample mean is a more accurate estimate of the population mean.
Let's calculate the SEM for each of the provided samples:
1. For the first sample:
- Standard Deviation (\(\sigma\)) = 53
- Sample Size (\(n\)) = 25
- SEM = \( \frac{53}{\sqrt{25}} = \frac{53}{5} = 10.6 \)
2. For the second sample:
- Standard Deviation (\(\sigma\)) = 24
- Sample Size (\(n\)) = 12
- SEM = \( \frac{24}{\sqrt{12}} \approx \frac{24}{3.464} \approx 6.93 \)
3. For the third sample:
- Standard Deviation (\(\sigma\)) = 12
- Sample Size (\(n\)) = 8
- SEM = \( \frac{12}{\sqrt{8}} \approx \frac{12}{2.828} \approx 4.24 \)
4. For the fourth sample:
- Standard Deviation (\(\sigma\)) = 14
- Sample Size (\(n\)) = 6.2
- SEM = \( \frac{14}{\sqrt{6.2}} \approx \frac{14}{2.49} \approx 5.62 \)
Now, comparing SEMs:
- First sample SEM = 10.6
- Second sample SEM ≈ 6.93
- Third sample SEM ≈ 4.24
- Fourth sample SEM ≈ 5.62
The third sample (with a standard deviation of 12 and a sample size of 8) has the smallest standard error of the mean, which means its mean will be the closest to the population mean.
**Answer:** A sample with a standard deviation of 12 and a sample size of 8.