A package heavier than expected

Question

A shipped package is an item plus its box. The item weight is normally distributed with mean 500 g and standard deviation 3 g. The box weight is normally distributed with mean 100 g and standard deviation 4 g. The two are independent.

What is the probability that the total package weight exceeds 610 g?

Accepted Answer

When you add two independent normal quantities, the result is again normal. The means simply add. The catch is the spread: variances (the squares of the standard deviations) add, not the standard deviations themselves. Mean of the total: 500 + 100 = 600 	ext{ g}. Variance of the total (add the squares): 3^2 + 4^2 = 9 + 16 = 25, so the total's standard deviation is \sqrt{25} = 5 g. The total weight is normal with mean 600 g and standard deviation 5 g. Now it is an ordinary normal question. Convert 610 g to a z-score: z = \frac{610 - 600}{5} = 2. The area beyond z = 2 is about 1 - 0.9772 = 0.0…

A package heavier than expected

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