Question

which of the following samples will have the mean closest to the population mean? (1 point)
a sample with a standard deviation of 14 and a sample size of 62
a sample with a standard deviation of 53 and a sample size of 25
a sample with a standard deviation of 12 and a sample size of 8
a sample with a standard deviation of 24 and a sample size of 12

Explanation:

To determine which sample mean is closest to the population mean, we use the standard error of the mean formula: \( SE = \frac{s}{\sqrt{n}} \), where \( s \) is the sample standard deviation and \( n \) is the sample size. A smaller standard error indicates the sample mean is closer to the population mean.

Step 1: Calculate SE for the first sample

\( s = 14 \), \( n = 6.2 \)
\( SE_1 = \frac{14}{\sqrt{6.2}} \approx \frac{14}{2.49} \approx 5.62 \)

Step 2: Calculate SE for the second sample

\( s = 53 \), \( n = 25 \)
\( SE_2 = \frac{53}{\sqrt{25}} = \frac{53}{5} = 10.6 \)

Step 3: Calculate SE for the third sample

\( s = 12 \), \( n = 8 \)
\( SE_3 = \frac{12}{\sqrt{8}} \approx \frac{12}{2.83} \approx 4.24 \)

Step 4: Calculate SE for the fourth sample

\( s = 24 \), \( n = 12 \)
\( SE_4 = \frac{24}{\sqrt{12}} \approx \frac{24}{3.46} \approx 6.94 \)

Now we compare the standard errors: \( SE_3 \approx 4.24 \) (smallest), \( SE_1 \approx 5.62 \), \( SE_4 \approx 6.94 \), \( SE_2 = 10.6 \). The smallest standard error is from the sample with \( s = 12 \) and \( n = 8 \). Wait, wait, no—wait, let's re - check the fourth sample: \( s = 24 \), \( n = 12 \): \( \sqrt{12}\approx3.464 \), \( 24\div3.464\approx6.93 \). Third sample: \( s = 12 \), \( n = 8 \), \( \sqrt{8}\approx2.828 \), \( 12\div2.828\approx4.24 \). First sample: \( s = 14 \), \( n = 6.2 \), \( \sqrt{6.2}\approx2.49 \), \( 14\div2.49\approx5.62 \). Second sample: \( 53\div5 = 10.6 \). Wait, but wait the fourth option: "a sample with a standard deviation of 24 and a sample size of 12"? Wait, maybe I misread. Wait the options:

1. s = 14, n = 6.2: SE≈5.62
2. s = 53, n = 25: SE = 10.6
3. s = 12, n = 8: SE≈4.24
4. s = 24, n = 12: SE≈6.94

Wait, but the third sample has the smallest SE? But wait, maybe I made a mistake. Wait, no—wait the fourth option: "a sample with a standard deviation of 24 and a sample size of 12"? Wait, no, the original options: let's re - check the user's image. Wait the fourth option: "a sample with a standard deviation of 24 and a sample size of 12"? Wait, maybe the first option's sample size is 6.2? Wait, sample size should be an integer? Maybe it's a typo, maybe 62? Wait, if the first sample size is 62: \( \sqrt{62}\approx7.87 \), \( 14\div7.87\approx1.78 \). But the user wrote 6.2. Assuming the numbers are as given. Wait, but maybe the intended correct answer is the sample with the smallest standard error. Wait, the third sample (s = 12, n = 8) has SE≈4.24, first has≈5.62, fourth≈6.94, second≈10.6. So the smallest is the third? But wait, maybe the first sample size is 62 (a typo). But according to the given numbers, the third sample (s = 12, n = 8) has the smallest SE. But wait, let's re - calculate:

Wait, the key is that the standard error (SE) is \( \frac{s}{\sqrt{n}} \), and smaller SE means the sample mean is closer to the population mean.

So:

- Sample 1: \( SE=\frac{14}{\sqrt{6.2}}\approx5.62 \)
- Sample 2: \( SE = \frac{53}{\sqrt{25}} = 10.6 \)
- Sample 3: \( SE=\frac{12}{\sqrt{8}}\approx4.24 \)
- Sample 4: \( SE=\frac{24}{\sqrt{12}}\approx6.94 \)

So the smallest SE is from sample 3: "a sample with a standard deviation of 12 and a sample size of 8". But wait, maybe the first sample size is 62 (a typo). But with the given numbers, sample 3 has the smallest SE.

Wait, but maybe I misread the options. Let me check again:

Option 1: standard deviation 14, sample size 6.2

Option 2: standard deviation 53, sample size 25

Option 3: standard deviation 12, sample size 8

Option 4: standard deviation 24, sample size 12

So according to the calculations, sample 3 has the smallest SE, so its mean is closest to the population mean.

Answer:

The sample with a standard deviation of 12 and a sample size of 8 (the third option).