8. A company that manufactures and bottles apple juice uses a machine that autom

Question

8. A company that manufactures and bottles apple juice uses a machine that automatically fills 16- ounce bottles. There is some variation, however, in the amounts of liquid dispensed into the bottles that are filled. The amount dispensed has been observed to be approximately normally distributed with mean 16 ounces and standard deviation 1 ounce (a) Determine the proportion of bottles that will have more than 17 ounces dispensed into them. (b) Suppose bottles are rejected if they have more than 18 ounces or less than 14 ounces. How many bottles would you expect to be rejected in a sample of 10,000 bottles?

Explanation / Answer

Since the amount dispensed (X) are normally distributed, the solution can be found easily using the MS Excel function NORM.DIST

(a) To find the proportion of bottles having more than 17 ounces dispensed, we find the cumulative probability of the amount dispensed being upto 17 ounces and then subtract this from 1.

P(X<17) = (=NORM.DIST(17,16,1,TRUE) = 0.8413

P (X>17) = 1 - P(X<17) = 1-0.8413 = 0.1587

(b) We need to find the probability: P(X<14) + P(X>18)

P(X<14) = (=NORM.DIST(14,16,1,TRUE) = 0.0228

P(X<18) = (=NORM.DIST(18,16,1,TRUE) = 0.9772

P(X>18) = 1- P(X<18) = 1-0.9772 = 0.0228

Required proportion expected to be rejected = P(X<14) + P(X>18) = 0.0228 + 0.0228 = 0.0456

In a sample of 10,000 bottles, number of bottles expected to be rejected = 10000 x 0.0456 = 456

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