Question

In an, all boys school, the heights of the student body are normally distributed with a mean of 68 inches and a standard deviation of 2 inches. Out of the 499 boys who go to that school, how many would be expected to be between 67 and 70 inches tall, to the nearest whole number?

Answer 1

266

Explanation:

You want to know the expected number of boys out of 499 that would have a height between 67 and 70 inches, when the height distribution is normal with mean 68 and standard deviation 2 inches.

Probability

The probability that a randomly chosen boy will have a height between 67 and 70 inches is the area under the normal PDF curve, scaled to have a mean of 68 and a standard deviation of 2. Spreadsheets and appropriate calculators can tell you that area using the normalCDF function. The calculator shown in the attachment finds it to be 0.53280720734.

Expected value

The expected number of boys in the given height range is this probability multiplied by the number of boys at the school.

499 · 0.53280720734 ≈ 265.9 ≈ 266

About 266 boys would be expected to be between 67 and 70 inches tall.