Question

The life of a semiconductor laser at a constant power is normally distributed with a mean of 7000 hours and a standard deviation of 600 hours. What is the probability that a laser fails before 5000 hours

Answer 1

0.04% probability that a laser fails before 5000 hours

Explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

`\mu = 7000, \sigma = 600`

What is the probability that a laser fails before 5000 hours

This is the pvalue of Z when X = 5000. So

`Z = (X - \mu)/(\sigma)`

`Z = (5000 - 7000)/(600)`

`Z = -3.33`

`Z = -3.33` has a pvalue of 0.0004.

0.04% probability that a laser fails before 5000 hours