A machine at a juice packaging plant is designed to dispense on average 20 ounce

Question

A machine at a juice packaging plant is designed to dispense on average 20 ounces per bottle of juice with a Standard Deviation of 2 ounces. In case nothing is known about the Shape of the Distribution of the juice amounts dispensed, answer the following questions. What percentage of juice bottles is expected to contain between 16 and 24 ounces of juice? At least 75% At least 89% None of the above Which range contains at least 89% of juice bottles? 16-24 ounces 14-26 ounces None of the above Referring to the filling of juice bottles example above, in case it is known that the Shape of the Distribution of the juice amounts dispensed is Unimodal and relatively Symmetric, answer the following questions. What percentage of juice bottles is expected to contain between 18 and 22 ounces? Approximately 68% Approximately 95% None of the above Which range contains almost all (99.7%) juice bottles? 18-22 ounces 14-26 ounces None of the above

Explanation / Answer

Q1.
Mean ( u ) =20
Standard Deviation ( sd )=2
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
a.
To find P(a < = Z < = b) = F(b) - F(a)
P(X < 16) = (16-20)/2
= -4/2 = -2
= P ( Z <-2) From Standard Normal Table
= 0.02275
P(X < 24) = (24-20)/2
= 4/2 = 2
= P ( Z <2) From Standard Normal Table
= 0.97725
P(16 < X < 24) = 0.97725-0.02275 = 0.9545
[ANSWER] atleast 89%
Q2.              
[ANSWER] 16-24
Q3.
To find P(a < = Z < = b) = F(b) - F(a)
P(X < 18) = (18-20)/2
= -2/2 = -1
= P ( Z <-1) From Standard Normal Table
= 0.15866
P(X < 22) = (22-20)/2
= 2/2 = 1
= P ( Z <1) From Standard Normal Table
= 0.84134
P(18 < X < 22) = 0.84134-0.15866 = 0.6827                  
[ANSWER]
Approximately 68%
Q4.
About 99.7% of the area under the normal curve is within two standard deviation of the mean. i.e. (u ± 3s.d)
So to the given normal distribution about 95% of the observations lie in between
= [20 ± 3 * 2]
= [ 20 - 3 * 2 , 20 + 3* 2]
= [ 14 , 26 ]                  

[ANSWER] 14-26 Ounces

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