To determine which sample average is most likely closest to the average height of the entire population, we can calculate the weighted average of the heights based on the sample sizes. Here’s a detailed step-by-step solution:
1. List the sample sizes and their corresponding average heights:
- Sample size: 10, Average height: 63 inches
- Sample size: 20, Average height: 54 inches
- Sample size: 30, Average height: 57 inches
- Sample size: 40, Average height: 59 inches
2. Calculate the total height for each sample size:
- For the sample size of 10 with an average height of 63 inches: [tex]\( 10 \times 63 = 630 \)[/tex]
- For the sample size of 20 with an average height of 54 inches: [tex]\( 20 \times 54 = 1080 \)[/tex]
- For the sample size of 30 with an average height of 57 inches: [tex]\( 30 \times 57 = 1710 \)[/tex]
- For the sample size of 40 with an average height of 59 inches: [tex]\( 40 \times 59 = 2360 \)[/tex]
3. Sum up these weighted heights to get the total combined height:
[tex]\[
630 + 1080 + 1710 + 2360 = 5780
\][/tex]
4. Find the total number of students in all samples:
[tex]\[
10 + 20 + 30 + 40 = 100
\][/tex]
5. Calculate the weighted average height:
[tex]\[
\text{Weighted average} = \frac{\text{Total Height}}{\text{Total Sample Size}} = \frac{5780}{100} = 57.8 \text{ inches}
\][/tex]
6. Determine the sample average height that is closest to this weighted average:
- Compare 57.8 with the given average heights:
- 63
- 54
- 57
- 59
- The absolute differences are:
- |63 - 57.8| = 5.2
- |54 - 57.8| = 3.8
- |57 - 57.8| = 0.8
- |59 - 57.8| = 1.2
- The smallest difference is 0.8, which corresponds to the average height of 57 inches.
Therefore, the sample average height that is most likely closest to the average height of the entire population is:
C. 57
