Question

The life of light bulbs is distributed normally. The standard deviation of the lifetime is 2525 hours and the mean lifetime of a bulb is 590590 hours. Find the probability of a bulb lasting for at most 622622 hours. Round your answer to four decimal places.

Answer 1

0.8997 = 89.97% probability of a bulb lasting for at most 622 hours.

Explanation:

Normal Probability Distribution

Problems of normal distributions can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the z-score 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 p-value, we get the probability that the value of the measure is greater than X.

Mean of 590 hours, standard deviation of 25 hours.

This means that
`\mu = 590, \sigma = 25`

Find the probability of a bulb lasting for at most 622 hours.

This is the p-value of Z when X = 622.

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

`Z = (622 - 590)/(25)`

`Z = 1.28`

`Z = 1.28` has a p-value of 0.8997.

0.8997 = 89.97% probability of a bulb lasting for at most 622 hours.