Answer:
To find the mean height of the boys, you can use the concept of weighted averages. The weighted average is calculated by taking into account the number of items in each category (in this case, the number of girls and boys) and their respective means (heights).
Let's denote:
- N_g as the number of girls (150).
- N_b as the number of boys (total students - number of girls = 400 - 150 = 250).
- H_g as the mean height of the girls (61.0 inches).
- H_b as the mean height of the boys (the value we want to find).
- H_t as the overall mean height of all students (68.8 inches).
The weighted average formula is:
H_t = (N_g * H_g + N_b * H_b) / (N_g + N_b)
Now, plug in the known values:
68.8 = (150 * 61.0 + 250 * H_b) / (150 + 250)
Now, solve for H_b:
68.8 = (9150 + 250 * H_b) / 400
Multiply both sides by 400:
400 * 68.8 = 9150 + 250 * H_b
Now, subtract 9150 from both sides:
400 * 68.8 - 9150 = 250 * H_b
Simplify the left side:
27520 - 9150 = 250 * H_b
18370 = 250 * H_b
Now, divide by 250 to find H_b:
H_b = 18370 / 250
H_b = 73.48
So, the mean height of the boys is approximately 73.48 inches.