Consider the population of all 1gal cans of dusty rose paint

Consider the population of all 1-gal cans of dusty rose paint manufactured by a particular paint company. Suppose that a normal distribution with mean = 5 ml and standard deviation = 0.2 ml is a reasonable model for the distribution of the variable x = amount of red dye in the paint mixture. Use the normal distribution model to calculate the probabilities below. (Round all answers to four decimal places.)

Solution

Normal Distribution
Mean ( u ) =5
Standard Deviation ( sd )=0.2
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
a)                  
P(X > 5) = (5-5)/0.2
= 0/0.2 = 0
= P ( Z >0) From Standard Normal Table
= 0.5                  
b)
P(X < 5.4) = (5.4-5)/0.2
= 0.4/0.2= 2
= P ( Z <2) From Standard Normal Table
= 0.9772
c)
P(X > 5.4) = (5.4-5)/0.2
= 0.4/0.2 = 2
= P ( Z >2) From Standard Normal Table
= 0.0228                  
P(X < = 5.4) = (1 - P(X > 5.4)
= 1 - 0.0228 = 0.9772                  
d)
To find P(a < = Z < = b) = F(b) - F(a)
P(X < 4.6) = (4.6-5)/0.2
= -0.4/0.2 = -2
= P ( Z <-2) From Standard Normal Table
= 0.02275
P(X < 5.2) = (5.2-5)/0.2
= 0.2/0.2 = 1
= P ( Z <1) From Standard Normal Table
= 0.84134
P(4.6 < X < 5.2) = 0.84134-0.02275 = 0.8186