4 14 points The height of males in the United States is norm

4. (14 points) The height of males in the United States is normally distributed with a standard deviation of 2.8 inches. If 90% of males are shorter than 72.7 inches, (a) What is the average height of males in the United States? (b) What percentage of males are between 70.0 and 72.0 inches tall?

Solution

a)
Standard Deviation ( sd )=2.8
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
P ( Z < X ) = 0.9
Value of z to the cumulative probability of 0.9 from normal table is 1.282
P( X-u/s.d < X - U /2.8 ) = 0.9
That is, ( 72.7 - U /2.8 ) = 1.2816
--> ( 72.7 - U ) = 1.2816*2.8
--> ( 72.7 - U ) = 3.588 ( 2.8)
--> U = 72.7 - 3.588 ( 2.8 ) = 62.6536  

b)

To find P(a < = Z < = b) = F(b) - F(a)
P(X < 70) = (70-62.654)/2.8
= 7.346/2.8 = 2.6236
= P ( Z <2.6236) From Standard Normal Table
= 0.99565
P(X < 72) = (72-62.654)/2.8
= 9.346/2.8 = 3.3379
= P ( Z <3.3379) From Standard Normal Table
= 0.99958
P(70 < X < 72) = 0.99958-0.99565 = 0.0039