Which factory out-produces the other?

Question

Factory A's daily output is normally distributed with mean 80 units and standard deviation 12. Factory B's daily output is normally distributed with mean 50 units and standard deviation 16. The two are independent.

On a given day, what is the probability that Factory A produces more than Factory B?

Accepted Answer

Compare the two outputs through their difference. Let D = (	ext{A's output}) - (	ext{B's output}). "A produces more" means D 0. The difference of two independent normals is normal: the mean subtracts, the variances add. Mean of D: \;80 - 50 = 30 units. Variance of D: \;12^2 + 16^2 = 144 + 256 = 400, so the standard deviation of D is \sqrt{400} = 20 units. So D is normal with mean 30 and standard deviation 20. Convert 0 to a z-score: z = \frac{0 - 30}{20} = -1.5. We want the area to the right of z = -1.5, about 0.9332: P(D 0) \approx 0.9332. A's average lead of 30 units is 1.5 standard devia…

Which factory out-produces the other?

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