If X is normally distributed with a mean of 20 and a standar

If X is normally distributed with a mean of 20 and a standard deviation of 5, find
a. P(X 35)
b. P(X 30)
c. P(10 X 40)
d. P(|X 20| > 10

Solution

Normal Distribution
Mean ( u ) =20
Standard Deviation ( sd )=5
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
a)
P(X < 35) = (35-20)/5
= 15/5= 3
= P ( Z <3) From Standard Normal Table
= 0.9987                  
P(X > = 35) = (1 - P(X < 35)
= 1 - 0.9987 = 0.0013                  
b)
P(X > 30) = (30-20)/5
= 10/5 = 2
= P ( Z >2) From Standard Normal Table
= 0.0228                  
P(X < = 30) = (1 - P(X > 30)
= 1 - 0.0228 = 0.9772                  
c)
To find P(a < = Z < = b) = F(b) - F(a)
P(X < 10) = (10-20)/5
= -10/5 = -2
= P ( Z <-2) From Standard Normal Table
= 0.02275
P(X < 40) = (40-20)/5
= 20/5 = 4
= P ( Z <4) From Standard Normal Table
= 0.99997
P(10 < X < 40) = 0.99997-0.02275 = 0.9772