A cereal manufacturer claims that the gross weight (including packaging) of a box of cereal labeled as weighing 12 oz has a mean of 12.2 oz and a standard deviation of 0.1 oz. You gather 75 boxes and weigh them all together. Let S denote the total weight of the 75 boxes of cereal.
a. If the claim is true, what is P(S ? 914.8)?
b. Based on the answer to part (a), if the claim is true, is 914.8 oz an unusually small total weight for a sample of 75 boxes?
c. If the total weight of the boxes were 914.8 oz, would you be convinced that the claim was false? Explain.
d. If the claim is true, what is P(S ? 910.3)?
e. Based on the answer to part (d), if the claim is true, is 910.3 oz an unusually small total weight for a sample of 75 boxes?
f. If the total weight of the boxes were 910.3 oz, would you be convinced that the claim was false? Explain.
(a) S is approximately normal with mean 
= 75(12.2) = 915 and 
The z-score of 914.8 is (914.8 ? 915) / 0.86603 = ?0.23.
The area to the left of z = ?0.23 is 0.4090.
P(S < 914.8) = 0.4090.
(b) No. More than 40% of the samples will have a total weight of 914.8 ounces or less if the claim is true.
(c) No, because a total weight of 914.8 ounces is not unusually small if the claim is true.
(d) S is approximately normal with mean 
The ????-score of 910.3 is (910.3 ? 915) / 0.86603 = ?5.43.
The area to the left of z = ?5.43 is negligible.
P(S < 910.3) ? 0.
(e) Yes. Almost none of the samples will have a total weight of 910.3 ounces or less if the claim is true.
(f) Yes, because a total weight of 910.3 ounces is unusually small if the claim is true.
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