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 14 and a sample size of 6.2
a sample with a standard deviation of 14 and a sample size of 6.2
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 53 and a sample size of 25
Responses
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
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 53 and a sample size of 25
Answers
GPT-4o mini
To determine which sample will have a mean closest to the population mean, we need to consider both the standard deviation and the sample size. The larger the sample size, the more reliable the sample mean will be as an estimate of the population mean, regardless of the standard deviation.
Here's a breakdown of the options:
1. **Sample with a standard deviation of 14 and a sample size of 6**
2. **Sample with a standard deviation of 24 and a sample size of 12**
3. **Sample with a standard deviation of 12 and a sample size of 8**
4. **Sample with a standard deviation of 53 and a sample size of 25**
The reliability of a sample mean as an estimator of the population mean can be assessed by the standard error of the mean (SEM), which is calculated as:
\[ \text{SEM} = \frac{\text{standard deviation}}{\sqrt{\text{sample size}}} \]
Let's calculate the SEM for each sample:
1. **Sample 1 (SD = 14, n = 6)**:
\[ \text{SEM}_1 = \frac{14}{\sqrt{6}} \approx 5.71 \]
2. **Sample 2 (SD = 24, n = 12)**:
\[ \text{SEM}_2 = \frac{24}{\sqrt{12}} \approx 6.93 \]
3. **Sample 3 (SD = 12, n = 8)**:
\[ \text{SEM}_3 = \frac{12}{\sqrt{8}} \approx 4.24 \]
4. **Sample 4 (SD = 53, n = 25)**:
\[ \text{SEM}_4 = \frac{53}{\sqrt{25}} = 10.6 \]
Based on the SEM calculations, the sample with the smallest SEM will be the most reliable representation of the population mean.
In this case, **Sample 3** (standard deviation of 12 and a sample size of 8) has the smallest SEM (4.24), which indicates that its mean is expected to be the closest to the population mean.
Therefore, the answer is:
**a sample with a standard deviation of 12 and a sample size of 8.**
Here's a breakdown of the options:
1. **Sample with a standard deviation of 14 and a sample size of 6**
2. **Sample with a standard deviation of 24 and a sample size of 12**
3. **Sample with a standard deviation of 12 and a sample size of 8**
4. **Sample with a standard deviation of 53 and a sample size of 25**
The reliability of a sample mean as an estimator of the population mean can be assessed by the standard error of the mean (SEM), which is calculated as:
\[ \text{SEM} = \frac{\text{standard deviation}}{\sqrt{\text{sample size}}} \]
Let's calculate the SEM for each sample:
1. **Sample 1 (SD = 14, n = 6)**:
\[ \text{SEM}_1 = \frac{14}{\sqrt{6}} \approx 5.71 \]
2. **Sample 2 (SD = 24, n = 12)**:
\[ \text{SEM}_2 = \frac{24}{\sqrt{12}} \approx 6.93 \]
3. **Sample 3 (SD = 12, n = 8)**:
\[ \text{SEM}_3 = \frac{12}{\sqrt{8}} \approx 4.24 \]
4. **Sample 4 (SD = 53, n = 25)**:
\[ \text{SEM}_4 = \frac{53}{\sqrt{25}} = 10.6 \]
Based on the SEM calculations, the sample with the smallest SEM will be the most reliable representation of the population mean.
In this case, **Sample 3** (standard deviation of 12 and a sample size of 8) has the smallest SEM (4.24), which indicates that its mean is expected to be the closest to the population mean.
Therefore, the answer is:
**a sample with a standard deviation of 12 and a sample size of 8.**