To find the average height of the population based on the given sample data, we can use the concept of the weighted average. Each sample contributes to the overall estimate proportional to its size. The steps to compute the population mean are detailed below:
1. List the Sample Sizes and Average Heights:
- Sample Size 1: [tex]\(10\)[/tex], Average Height 1: [tex]\(61\)[/tex] inches
- Sample Size 2: [tex]\(20\)[/tex], Average Height 2: [tex]\(52\)[/tex] inches
- Sample Size 3: [tex]\(30\)[/tex], Average Height 3: [tex]\(55\)[/tex] inches
- Sample Size 4: [tex]\(40\)[/tex], Average Height 4: [tex]\(57\)[/tex] inches
2. Calculate the Total Weighted Height:
The weighted height is computed by multiplying each sample's size by its respective average height and then summing these products:
[tex]\[
\text{Total Weighted Height} = (10 \times 61) + (20 \times 52) + (30 \times 55) + (40 \times 57)
\][/tex]
Breaking it down:
[tex]\[
(10 \times 61) = 610
\][/tex]
[tex]\[
(20 \times 52) = 1040
\][/tex]
[tex]\[
(30 \times 55) = 1650
\][/tex]
[tex]\[
(40 \times 57) = 2280
\][/tex]
Adding these together:
[tex]\[
610 + 1040 + 1650 + 2280 = 5580
\][/tex]
3. Determine the Total Sample Size:
Sum the sizes of all the samples:
[tex]\[
10 + 20 + 30 + 40 = 100
\][/tex]
4. Calculate the Population Mean Estimate:
The population mean estimate is the total weighted height divided by the total sample size:
[tex]\[
\text{Population Mean Estimate} = \frac{5580}{100} = 55.8 \text{ inches}
\][/tex]
Therefore, the most likely average height of the population is 55.8 inches.
