The average heights of four samples taken from a population of students are shown in the table. Which of these is most likely closest to the average height of the population?
| Sample size | Average height (inches) |
|---|---|
| 10 | 63 |
| 20 | 54 |
| 30 | 57 |
| 40 | 59 |
A. 63
B. 54
C. 59
D. 57
Answer (1)
Calculate the weighted average of the sample heights using the formula: ∑ s am pl e s i ze ∑ ( s am pl e s i ze × a v er a g e h e i g h t ) .
Substitute the given values into the formula: 10 + 20 + 30 + 40 ( 10 × 63 ) + ( 20 × 54 ) + ( 30 × 57 ) + ( 40 × 59 ) .
Simplify the expression: 100 630 + 1080 + 1710 + 2360 = 100 5780 = 57.8 .
Choose the answer option closest to 57.8, which is 57. Therefore, the final answer is 57 .
Explanation
Understanding the Problem We are given the average heights of four samples of students, each with a different sample size. To estimate the average height of the entire population, we should calculate a weighted average, giving more weight to the samples with larger sizes.
Setting up the Calculation The weighted average is calculated as follows: 10 + 20 + 30 + 40 ( 10 × 63 ) + ( 20 × 54 ) + ( 30 × 57 ) + ( 40 × 59 ) This formula takes into account the sample size and the average height of each sample to provide a better estimate of the population average.
Calculating the Weighted Average Now, let's perform the calculation: 10 + 20 + 30 + 40 ( 10 × 63 ) + ( 20 × 54 ) + ( 30 × 57 ) + ( 40 × 59 ) = 100 630 + 1080 + 1710 + 2360 = 100 5780 = 57.8 So, the weighted average height is 57.8 inches.
Finding the Closest Option We need to choose the answer option that is closest to the weighted average of 57.8 inches. Looking at the options: A. 63 B. 54 C. 59 D. 57 The closest value to 57.8 is 57.
Final Answer Therefore, the average height of the population is most likely closest to 57 inches.
Examples
Imagine you're trying to estimate the average grade of all students in a school. You have data from a few classrooms, but the classrooms have different numbers of students. To get the best estimate, you'd use a weighted average, giving more importance to the classrooms with more students. This is similar to estimating the average height of students based on different sample sizes. Weighted averages are useful in many real-world scenarios where you want to combine data from different sources with varying levels of reliability.
