It was found that the mean length of 100 dragon flies was 20

It was found that the mean length of 100 dragon flies was 20.05cm with a standard deviation of 0.02cm. Find the probability that a fly selected at random would have a length

a. Between 20.03cm and 20.08cm

b. Between 20.06cm and 20.07cm

c. Less than 20.01cm

d. Greater than 20.09cm

Solution

Mean ( u ) =20.05
Standard Deviation ( sd )=0.02
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
a)
To find P(a < = Z < = b) = F(b) - F(a)
P(X < 20.03) = (20.03-20.05)/0.02
= -0.02/0.02 = -1
= P ( Z <-1) From Standard Normal Table
= 0.15866
P(X < 20.08) = (20.08-20.05)/0.02
= 0.03/0.02 = 1.5
= P ( Z <1.5) From Standard Normal Table
= 0.93319
P(20.03 < X < 20.08) = 0.93319-0.15866 = 0.7745                  
b)
To find P(a < = Z < = b) = F(b) - F(a)
P(X < 20.06) = (20.06-20.05)/0.02
= 0.01/0.02 = 0.5
= P ( Z <0.5) From Standard Normal Table
= 0.69146
P(X < 20.07) = (20.07-20.05)/0.02
= 0.02/0.02 = 1
= P ( Z <1) From Standard Normal Table
= 0.84134
P(20.06 < X < 20.07) = 0.84134-0.69146 = 0.1499                  
c)
P(X < 20.01) = (20.01-20.05)/0.02
= -0.04/0.02= -2
= P ( Z <-2) From Standard Normal Table
= 0.0228                  
d)
P(X > 20.09) = (20.09-20.05)/0.02
= 0.04/0.02 = 2
= P ( Z >2) From Standard Normal Table
= 0.0228              

[ANSWERS]
a. P(20.03 < X < 20.08) = 0.7745
b. P(20.06 < X < 20.07) = 0.1499
c.    P(X < 20.01) = 0.0228  
d. P(X > 20.09) = 0.0228