Final answer:
To find the mean height of all students, calculate the weighted mean by multiplying each group's mean height by their population, summing these products, and dividing by the total population. The mean height of all students is 1.74 meters.
Step-by-step explanation:
The question asks us to calculate the mean height of all the students in a school given the mean heights of the senior and junior students along with their respective populations. To calculate the combined mean height, we will use a weighted mean, which is essentially the sum of the products of the group means and the number of students in each group, divided by the total number of students.
The weighted mean height (M) is calculated as follows:
- Multiply the mean height of senior students (1.78m) by the number of senior students (640).
- Multiply the mean height of junior students (1.66m) by the number of junior students (360).
- Add the results of steps 1 and 2 to get the combined height total for all students.
- Divide the combined height total by the total number of students (640 + 360).
Applying the above steps:
1. 640 seniors × 1.78m = 1139.2m
2. 360 juniors × 1.66m = 597.6m
3. 1139.2m + 597.6m = 1736.8m (Combined height total)
4. 1736.8m ÷ 1000 students = 1.7368m, rounded to two decimal places is 1.74m.
Therefore, the mean height of all the students in the school is 1.74 meters.