To determine the closest average height of the population given data from four samples, we can calculate the weighted average of the sample heights. This weighted average takes into account the size of each sample, giving more weight to the larger samples.
Here are the steps involved:
1. Calculate the total sample size:
Add up the sizes of all the samples.
[tex]\[
\text{Total sample size} = 10 + 20 + 30 + 40 = 100
\][/tex]
2. Calculate the weighted sum of the heights:
Multiply each sample size by its respective average height and then sum these values.
[tex]\[
\text{Weighted sum} = (10 \times 63) + (20 \times 54) + (30 \times 57) + (40 \times 59)
\][/tex]
Calculate each term:
[tex]\[
10 \times 63 = 630
\][/tex]
[tex]\[
20 \times 54 = 1080
\][/tex]
[tex]\[
30 \times 57 = 1710
\][/tex]
[tex]\[
40 \times 59 = 2360
\][/tex]
Now, sum these values:
[tex]\[
\text{Weighted sum} = 630 + 1080 + 1710 + 2360 = 5780
\][/tex]
3. Calculate the weighted average height:
Divide the weighted sum by the total sample size.
[tex]\[
\text{Weighted average height} = \frac{5780}{100} = 57.8
\][/tex]
4. Determine the closest average height from the given options:
Compare the weighted average height (57.8) to each of the sample averages:
[tex]\[
\begin{align*}
\text{Absolute difference with 63:} & \quad |63 - 57.8| = 5.2 \\
\text{Absolute difference with 54:} & \quad |54 - 57.8| = 3.8 \\
\text{Absolute difference with 57:} & \quad |57 - 57.8| = 0.8 \\
\text{Absolute difference with 59:} & \quad |59 - 57.8| = 1.2 \\
\end{align*}
\][/tex]
The smallest absolute difference is with 57.
Thus, the answer is C. 57, which is most likely the closest to the average height of the population.
