To determine which sample will have the mean closest to the population mean, we can consider the concepts of sample size and standard deviation. Generally, larger sample sizes can lead to better estimates of the population mean, and smaller standard deviations generally indicate more consistency within the data.
- Sample with a standard deviation of 53 and a sample size of 25
- Sample with a standard deviation of 24 and a sample size of 12
- Sample with a standard deviation of 12 and a sample size of 8
- Sample with a standard deviation of 14 and a sample size of 6.2
To quantify the impact of sample size and standard deviation, we can calculate the standard error (SE) for each sample, which is given by the formula:
\[ SE = \frac{\sigma}{\sqrt{n}} \]
where \(\sigma\) is the standard deviation and \(n\) is the sample size.
Calculating the standard error for each sample:
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Standard error for sample 1: \[ SE_1 = \frac{53}{\sqrt{25}} = \frac{53}{5} = 10.6 \]
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Standard error for sample 2: \[ SE_2 = \frac{24}{\sqrt{12}} \approx \frac{24}{3.464} \approx 6.93 \]
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Standard error for sample 3: \[ SE_3 = \frac{12}{\sqrt{8}} \approx \frac{12}{2.828} \approx 4.24 \]
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Standard error for sample 4: \[ SE_4 = \frac{14}{\sqrt{6.2}} \approx \frac{14}{2.487} \approx 5.62 \]
Now we can compare the standard errors. A smaller standard error suggests that the sample mean is likely to be closer to the population mean.
- Sample 1: SE ≈ 10.6
- Sample 2: SE ≈ 6.93
- Sample 3: SE ≈ 4.24
- Sample 4: SE ≈ 5.62
Among these, sample 3 (standard deviation of 12 and sample size of 8) has the smallest standard error, which suggests that its mean will be closest to the population mean.
Therefore, the answer is: a sample with a standard deviation of 12 and a sample size of 8.