An underweight package

Question

A filling machine produces packages whose weights are normally distributed with a mean of 500 grams and a standard deviation of 20 grams.

What is the probability that a package weighs less than 480 grams?

Accepted Answer

Convert the weight into a z-score, counting standard deviations away from the mean: z = \frac{x - \mu}{\sigma}. For x = 480, mean \mu = 500, spread \sigma = 20, z = \frac{480 - 500}{20} = -1. The value is 1 standard deviation below average, so the z-score is negative. This time we want the area to the left of the z-score directly, because "less than 480" is a lower tail. The standard normal table gives the area below z = -1 as about 0.1587: P(X < 480) = 0.1587. So about 15.9% of packages come in under 480 grams. By the symmetry of the bell curve, this equals the area above z = +1, a handy che…

An underweight package

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