Although most of us buy milk by the quart or gallon farmers

Although most of us buy milk by the quart or gallon, farmers measure daily production in pounds. Ayrshire cows average 55 pounds of milk a day, with a standard deviation of 6 pounds. For Jersey cows, the mean daily production is 52 pounds, with a standard deviation of 5 pounds. Assume that Normal models describe milk production for these breeds.

a) we select an Ayrshire at random. What\'s the probability that she averages more than 58 pounds of milk a day?

b) What\'s the probability that a randomly selected ayrshire gives more milke than a randomly selected jersey?

c) a farmer has 20 jerseys. whats the probability that the average production for this small herd exceeds 54 pounds of milk a day, assuming they can be treated as a random sample of all jersey cows?

d) a neighboring farmer has 10 ayrshires. what\'s the probability that his herd average is at least 5 pounds higher than the average for part c\'s jersey herd?

Solution

a)
Ayrshire Jersey
Mean ( u ) = 55
Standard Deviation ( sd )=6
Normal Distribution = Z= X- u / sd ~ N(0,1)  
P(X > 58) = (58-55)/6
= 3/6 = 0.5
= P ( Z >0.5) From Standard Normal Table
= 0.3085                  
b)
Jersey Milk production is 52 pounds
P(X > 52) = (52-55)/6
= -3/6 = -0.5
= P ( Z >-0.5) From Standard Normal Table
= 0.6915                  
c)
Normal Distribution
Mean ( u ) = 52
Standard Deviation ( sd )=5
Number ( n ) = 20
Normal Distribution = Z= X- u / (sd/Sqrt(n) ~ N(0,1)                  
P(X > 54) = (54-52)/5/ Sqrt ( 20 )
= 2/1.118= 1.7889
= P ( Z >1.7889) From Standard Normal Table
= 0.0368