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Posts: 186
2 weeks ago
Cereal A box of Raspberry Crunch cereal contains a mean of 13 ounces with a standard deviation of 0.5 ounce. The distribution of the contents of cereal boxes is approximately Normal. What is the probability that a case of 12 cereal boxes contains a total of more than 160 ounces?
Textbook 

Stats: Modeling the World


Edition: 4th
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Answer verified by a subject expert
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Posts: 264
2 weeks ago
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Two methods can be used to solve this problem:

Method 1:
Let B = weight of one box of cereal and T = weight of 12 boxes of cereal. We are told that the contents of the boxes are approximately Normal, and we can assume that the content amounts are independent from box to box.
E(T) = E(B1 + B2 +...+ B12 ) = E(B1) + E(B2) + ... + E(B12 ) = 156 ounces
Since the content amounts are independent,
Var(T) = Var(B1 + B2 +...+ B12  ) = Var(B1) + Var(B2) + ... + Var(B12 ) = 3
SD(T) = = = 1.73 oz.
We model T with N(156, 1.73).
z = = 2.31 and P(T > 160) = P(z > 2.31) = 0.0104
There is a 1.04% chance that a case of 12 cereal boxes will weigh more than 160 ounces.

Method 2:
Using the Central Limit Theorem approach, let = average content of boxes in the case. Since the contents are Normally distributed, is modeled by N.
P = P( > 13.33) = P = P(z > 2.31) = 0.0104.
There is a 1.04% chance that a case of 12 cereal boxes will weigh more than 160 ounces.
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