Question

Bottles of 2L milk produced by a certain manufacturer are known to be normally distributed with a mean of 2009mL and a standard deviation of 20mL.
Q1) What proportion of bottles contain less than 2L of milk?

Q2) What is the probability that any given milk bottle contains a volume within one standard deviation of 2L?

Q3) The manufacturer employs an automatic system that rejects milk bottles that contain less than a specified volume. To what volume should they set this system so that they only discard 5% of bottles?

Answer 1

This is a normal distribution question with


a) P(x < 2000.0)=?
The z-score at x = 2000.0 is,

z = -0.45
This implies that


b) P(-1.0 < z < 1.0) = P(z < z2) - P(z < z1)
P(-1.0 < z < 1.0) = 0.8413447460685429 - 0.8413447460685429
P(-1.0 < z < 1.0) = 0.6827

c) Given in the question
P(X < x) = 0.05
This implies that
P(Z < -1.6449) = 0.05
With the help of formula for z, we can say that

x = 1976.102
PS: you have to refer z score table to find the final probabilities.
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