To determine the average height of the population using the data from the samples, we need to calculate the weighted average height. This method accounts for the different sample sizes of each group.
Given data:
- Sample 1:
- Size: 10
- Average height: 63 inches
- Sample 2:
- Size: 20
- Average height: 54 inches
- Sample 3:
- Size: 30
- Average height: 57 inches
- Sample 4:
- Size: 40
- Average height: 59 inches
To find the weighted average height, follow these steps:
1. Calculate the total height contributed by each sample:
- For Sample 1: [tex]\( 63 \times 10 = 630 \)[/tex]
- For Sample 2: [tex]\( 54 \times 20 = 1080 \)[/tex]
- For Sample 3: [tex]\( 57 \times 30 = 1710 \)[/tex]
- For Sample 4: [tex]\( 59 \times 40 = 2360 \)[/tex]
2. Sum these total heights:
[tex]\[
630 + 1080 + 1710 + 2360 = 5780
\][/tex]
3. Calculate the sum of the sample sizes:
[tex]\[
10 + 20 + 30 + 40 = 100
\][/tex]
4. Divide the sum of the total heights by the sum of the sample sizes to find the weighted average:
[tex]\[
\frac{5780}{100} = 57.8
\][/tex]
Therefore, the average height closest to the population's average height is approximately 57.8 inches. So, the closest answer from the options given is:
D. 57
