Question

The light bulbs used to provide exterior lighting for a large office building have an average lifetime of 700 hours. If lifetime is approximately normally distributed with a standard deviation of 50 hours, how often should all the bulbs be replaced so that no more than \(20 \%\) of the bulbs will have already burned out?

Short answer

In order to ensure no more than 20% of bulbs have burned out, all bulbs should be replaced every 658 hours.

Step-by-step solution

Identify the percentiles

Given that no more than 20% of the bulbs should burn out before replacement, it essentially means that we need to find out the 20th percentile of the normal distribution as at that point, 20% of the bulbs would have burned out.

Calculate the Z-Score for the 20th percentile

The z-score related to the 20th percentile (from a standard normal distribution table or calculator) typically equates to -0.84. Note, the negative value indicates a percentile below the mean (50th percentile in this case).

Calculate the corresponding lifetime

Use the formula of z-score to calculate the corresponding value in hours for this z-score. The formula is: \( Z = \frac{X - \mu}{\sigma} \), where X is the value we want to find, \( \mu \) is the mean (700 hours), and \( \sigma \) is the standard deviation (50 hours). Rearranging the formula to find X gives \( X = Z\sigma + \mu \), substituting the known values into the formula results in \( X = -0.84*50 + 700 = 658 \) hours.