Component lifetime in a window

Question

An electronic component has a lifetime that is normally distributed with a mean of 1000 hours and a standard deviation of 50 hours.

What is the probability that a component lasts between 950 and 1100 hours?

Accepted Answer

For a range, convert both endpoints to z-scores, then take the difference of the two left-hand areas. The z-score counts standard deviations from the mean: z = \frac{x - \mu}{\sigma}. Lower endpoint x = 950: \;z = \dfrac{950 - 1000}{50} = -1. Upper endpoint x = 1100: \;z = \dfrac{1100 - 1000}{50} = 2. The probability of landing between them is the area below the upper z minus the area below the lower z: P(950 \le X \le 1100) = \Phi(2) - \Phi(-1) = 0.9772 - 0.1587 = 0.8185, where \Phi just means "area to the left." So about 81.9% of components last somewhere in this window. Picture it as the a…

Component lifetime in a window

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