normal distribution

Question

Laboratory tests show that the lives of light bulbs are normally distributed with a mean of 750 hours and a standard deviation of 75 hours. Find the probability that a randomly selected light bulb will last between 900 and 975 hours. NEED HELP

Answer 1

A normal distribution is shaped like a bell curve with the average (mean) at the center.

A light bulb in your problem is most likely to last about 750 hours. One standard deviation up from that would be 750 + 75, which is 825 hours, and one standard deviation down is 750 - 75, which is 675 hours.

According to the empirical rule, the interval from the average to +1 standard deviation (750 to 825) and the interval from the average to -1 standard deviation (675 to 725) each have a 34.1% probability.

From +1 to +2 standard deviations and -1 to -2 standard deviations (825 to 900 and 600 to 675) the probability is 13.5% for each interval.

From +2 to +3 standard deviations and -2 to -3 standard deviations (900 to 975 and 525 to 600) the probability is 2.1% for each interval.

These percentages are the same for any normal distribution problem. If you add up the chances you can find the probability a light bulb will last an increasing range of times centered around the average:

Within one standard deviation of the average: 68.3% (chance it will last 675 to 825 hours)

Within two standard deviations of the average: 95.4% (chance it will last 600 to 900 hours)

Within three standard deviations of the average: 99.7% (chance it will last 525 to 975 hours)

And only a very few, like three out of a thousand, would be likely to last less than 525 hours or more than 975 hours.

For a visual aid to this I recommend you google "standard deviation normal distribution percentages" and you will get a good picture of the bell curve from wikipedia or mathbits to go along with this explanation.