Question

The heights (measured in inches) of men aged 20 to 29 follow approximately the normal distribution with mean 69.4 and standard deviation 2.8. Between what two values does the middle 94% of all heights fall? (Please give responses to at least one decimal place)

Answer 1

Let X be the random variable denoting the heights of men aged between 20 and 29.

Thus, X ~ N(69.4, 2.8) i.e. Z = (X - 69.4)/2.8 ~ N(0,1).

Let, P(-a < Z < a) = 0.94 i.e. (a) - (-a) = 0.94 i.e. 2(a) - 1 = 0.94 i.e. (a) = 0.97 i.e. a = (0.97) = 1.881.

Thus, P(-1.881 < Z < 1.881) = 0.94 i.e. P[-1.881 < (X - 69.4)/2.8 < 1.881] = 0.94 i.e. P[64.1332 < (X - 69.4)/2.8 < 74.6668] = 0.94.

Hence, 94% of all heights fall between 64.1332 and 74.6668. (Ans).