Question

Heights X of 18 years old man in U.S. are normally distributed with mean μ = 68 inches and standard deviation σ = 3 inches.

(a) What is the probability that the height of a randomly selected 18 years old man in U.S. will be less than 67 inches?

(b) Whats the probability that the mean height X ̄ of 9 randomly selected 18 years old U.S. men will be less than 67 inches?

Answer 1

(a) The wanted probability is

$P(x < 67) = \Phi \left( {\frac{{\beta - \mu }}{\sigma }} \right) - \Phi \left( { - \infty } \right) = \Phi \left( {\frac{{67 - 68}}{3}} \right) - \Phi \left( { - \infty } \right) = 0,5 - \Phi \left( {0.33} \right) = 0.5 - 0.1293 = {\rm{0}}{\rm{.3707}}$

Answer: 0.3707

(b) Replace thepopulation standard deviation by the standard error:

${\sigma _x} = \frac{\sigma }{{\sqrt n }} = \frac{3}{{\sqrt 9 }} = 1$

then the wanted probability is

$P(x < 67) = \Phi \left( {\frac{{\beta - \mu }}{{{\sigma _x}}}} \right) - \Phi \left( { - \infty } \right) = \Phi \left( {\frac{{67 - 68}}{1}} \right) - \Phi \left( { - \infty } \right) = 0,5 - \Phi \left( 1 \right) = 0.5 - 0.3413 = {\rm{0}}{\rm{.1587}}$

Answer: 0.1587