a) and b)

code


x = rnorm(100000,mean = 185,sd = 15)
sum(x>190)/100000

sum((x < 185 + 2 * 15)& (x > 185 - 2 * 15))/100000

> sum(x>190)/100000
[1] 0.37044
>
> sum((x <  185 + 2 * 15)& (x > 185 - 2 * 15))/100000
[1] 0.954
a)

the proportion from the simulated data that a man is longer than 190 cm. = 0.3704

theoretical probability= P(X > 190 ) = P(Z > (190-185)/15) = P(Z > 1/3) = 0.3694

b)

the proportion of man between two standard deviation above and two standard deviation below the mean.

= 0.954

theoretical probability = 0.95

both are very close