To find which of the given sample averages is most likely closest to the average height of the overall population, we can use the concept of weighted average. Let's break down the calculations step by step.
### 1. Gather the data:
The table provides us with the following information:
- Sample sizes: 10, 20, 30, 40
- Corresponding average heights (in inches): 63, 54, 57, 59
### 2. Calculate the total height for each sample:
To find the total height for each sample, multiply the sample size by its average height:
- For the first sample: [tex]\(10 \times 63 = 630\)[/tex]
- For the second sample: [tex]\(20 \times 54 = 1080\)[/tex]
- For the third sample: [tex]\(30 \times 57 = 1710\)[/tex]
- For the fourth sample: [tex]\(40 \times 59 = 2360\)[/tex]
### 3. Calculate the overall total height:
Add all the total heights calculated in the previous step:
[tex]\[630 + 1080 + 1710 + 2360 = 5780\][/tex]
### 4. Calculate the overall total sample size:
Add all the sample sizes together:
[tex]\[10 + 20 + 30 + 40 = 100\][/tex]
### 5. Calculate the weighted average height:
The weighted average height is found by dividing the overall total height by the overall total sample size:
[tex]\[\frac{5780}{100} = 57.8\][/tex]
### Conclusion:
Based on the weighted average calculations, the most likely value closest to the average height of the entire population is approximately 57.8 inches.
