The random variable x has a normal distribution with 75 and

The random variable x has a normal distribution with = 75 and -10. Find the following probabilities. (** Please draw the graph) Show your work step by step Show your work Please draw graph a. P (xS 80) b. P (x2 85) C. P (70 sxs 75) ed. P (x> 80) P(x=78)

Solution

Normal Distribution
Mean ( u ) =75
Standard Deviation ( sd )=10
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
a)
P(X > 80) = (80-75)/10
= 5/10 = 0.5
= P ( Z >0.5) From Standard Normal Table
= 0.3085                  
P(X < = 80) = (1 - P(X > 80)
= 1 - 0.3085 = 0.6915                  
b)
P(X < 85) = (85-75)/10
= 10/10= 1
= P ( Z <1) From Standard Normal Table
= 0.8413                  
P(X > = 85) = (1 - P(X < 85)
= 1 - 0.8413 = 0.1587                  
c)
To find P(a < = Z < = b) = F(b) - F(a)
P(X < 70) = (70-75)/10
= -5/10 = -0.5
= P ( Z <-0.5) From Standard Normal Table
= 0.30854
P(X < 75) = (75-75)/10
= 0/10 = 0
= P ( Z <0) From Standard Normal Table
= 0.5
P(70 < X < 75) = 0.5-0.30854 = 0.1915                  
d)
P(X > 80) = (80-75)/10
= 5/10 = 0.5
= P ( Z >0.5) From Standard Normal Table
= 0.3085                  
e)
To find P(a < = Z < = b) = F(b) - F(a)
P(X < 77.5) = (77.5-75)/10
= 2.5/10 = 0.25
= P ( Z <0.25) From Standard Normal Table
= 0.59871
P(X < 78.5) = (78.5-75)/10
= 3.5/10 = 0.35
= P ( Z <0.35) From Standard Normal Table
= 0.63683
P(77.5 < X < 78.5) = 0.63683-0.59871 = 0.0381                  
f)
P(X > 110) = (110-75)/10
= 35/10 = 3.5
= P ( Z >3.5) From Standard Normal Table
= 0.0002                  
P(X < = 110) = (1 - P(X > 110)
= 1 - 0.0002 = 0.9998