Maths Probability

Question

According to the US Army medical disqualification list, men must be between 150cm and 200cm tall. In the US, the average height of males is 176cm with a standard deviation of 7.5cm. What proportion of US males would meet the US Army height regulations? Assume that heights are Normally distributed.

Answer 1

Let X = height of randomly selected US male in centimeters

We want P(150 < X < 200).

We'll need the number of standard deviations between 176 cm (the mean) and 150 cm (the lower limit of acceptable height): (176 - 150)/7.5 ≈ 3.47

Also the number of standard deviations between 176 cm and 200 cm (the upper limit of acceptable height): (200 - 176)/7.5 ≈ 3.2

Using this information, P(150 < X < 200) can be converted to a z-value lookup:

P(150 < X < 200) = P(-3.47 < z < 3.2), where z is standard normal. The result, rounded to 3 decimal places, is 0.999. This is according to RStudio