Question

Boxes are labeled as containing 400 g of cereal. The machine filling the boxes produces weights that are normally distributed with standard deviation 12 g.

(a) If the target weight is 400 g, what is the probability that the machine produces a box with less than 375 g of cereal? (Round your answer to three decimal places.)


(b) Suppose a law states that no more than 10% of a manufacturer's cereal boxes can contain less than the stated weight of 400 g. At what target weight should the manufacturer set its filling machine? (Round your answer to the nearest gram.)

Answer 1

a)

X ~ N ( µ = 400 , σ = 12 )
We convert this to standard normal as
P ( X < x ) = P ( Z < ( X - µ ) / σ )
P ( ( X < 375 ) = P ( Z < 375 - 400 ) / 12 )
= P ( Z < -2.08 )
P ( X < 375 ) = 0.019 (From Z table)

b)

X ~ N ( µ = 400 , σ = 12 )
P ( X <= x ) = 10% = 0.1
To find the value of x
Looking for the probability 0.1 in standard normal table to calculate critical value Z = -1.2816
Z = ( X - µ ) / σ
-1.2816 = ( X - 400 ) / 12
X = 384.6
X = 385 (Rounded to nearest gram)