To determine which of the sample averages is most likely closest to the average height of the population, we follow these steps:
1. Identify the sample averages:
- Sample size 10: 61 inches
- Sample size 20: 52 inches
- Sample size 30: 55 inches
- Sample size 40: 57 inches
2. Calculate the overall average height by combining all the samples. To do this, we need to find the total height for all the samples combined and then divide by the total number of samples.
- Total height for all samples:
[tex]\[
61 + 52 + 55 + 57 = 225 \text{ inches}
\][/tex]
- Number of samples:
[tex]\[
4 \text{ samples}
\][/tex]
- Average height:
[tex]\[
\frac{225}{4} = 56.25 \text{ inches}
\][/tex]
3. Determine which sample average is closest to the overall average of 56.25 inches. We look at the absolute differences:
- Difference with 61:
[tex]\[
|61 - 56.25| = 4.75
\][/tex]
- Difference with 52:
[tex]\[
|52 - 56.25| = 4.25
\][/tex]
- Difference with 55:
[tex]\[
|55 - 56.25| = 1.25
\][/tex]
- Difference with 57:
[tex]\[
|57 - 56.25| = 0.75
\][/tex]
4. Identify the height with the smallest difference. The smallest difference indicates the value closest to the average height of the population:
- The difference for 61: 4.75
- The difference for 52: 4.25
- The difference for 55: 1.25
- The difference for 57: 0.75
The height 57 has the smallest difference of 0.75 inches.
Therefore, the sample average closest to the overall average height of the population is 57 inches. This makes the correct answer:
C. 57
