Let X equal the weight of a fat-free Fig New-ton cookie. Assume that the distrib
ID: 2960435 • Letter: L
Question
Let X equal the weight of a fat-free Fig New-ton cookie. Assume that the distribution of X is N(14.22,0.0854). These cookies are sold in packages that have a label weight of 340 grams. The number of cookies in a package is usually 24, 25, or 26. Assume that a package is filled with a random sample of cookies, how many cookies should be put into a package to be quite certain (say, wuth a probability of at least 0.95) that the total weight of the cookies exceeds 340 grams(Keep in mind that extra cookies in a package decrease profit.)
Explanation / Answer
Let X_n be the distribution of the weight of n cookies.
X_n = the sum of n random variables with distribution N(14.22,0.0854)
So X_n is N(14.22*n,0.0854/sqrt(n))
To have 95% confidence we need a less than 5% chance that x drawn from X_n is less than 340.
The left tail of N(14.22*n, 0.0854/sqrt(n)) has area 5% when we use a cutoff of
14.22*n - (0.0854/sqrt(n))*1.6449
(find the value 1.6449 by looking up probability .95 in a table of the normal distribution)
So we need 340 < 14.22*n - (0.0854/sqrt(n))*1.6449
14.22*24 - (0.0854/sqrt(24))*1.6449 = 341.25
14.22*23 - (0.0854/sqrt(23))*1.6449 = 327.03
So it's sufficient to include 24 cookies. 23 wouldn't be enough.
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