Suppose the lengths of the pregnancies of a certain animal a

Suppose the lengths of the pregnancies of a certain animal are approximately normally distributed with mean mu= 207 and standard deviation sigma = 17 days. What is the probability that a randomly selected pregnancy lasts less than 201 days? What is the probability that a random sample of 16 pregnancies has a mean gestation period of 201 days or less? What is the probability that a random sample of 37 pregnancies has a mean gestation period of 201 days or less?

Solution

Normal Distribution
Mean ( u ) =207
Standard Deviation ( sd )=17
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
a)
P(X < 201) = (201-207)/17
= -6/17= -0.3529
= P ( Z <-0.3529) From Standard Normal Table
= 0.3621                  
b)
P(X > 201) = (201-207)/17/ Sqrt ( 16 )
= -6/4.25= -1.4118
= P ( Z >-1.4118) From Standard Normal Table
= 0.921                  
P(X < = 201) = (1 - P(X > 201)
= 1 - 0.921 = 0.079                  
c)
P(X > 201) = (201-207)/17/ Sqrt ( 37 )
= -6/2.795= -2.1469
= P ( Z >-2.1469) From Standard Normal Table
= 0.9841                  

P(X < = 201) = (1 - P(X > 201)
= 1 - 0.9841 = 0.0159