Question

A generator manufacturer found that the smallest, least expensive generator that they manufacture lasts, on average, 2000 hours with a standard deviation of 150 hours. At least what percent of the generators last between 1550 and 2450 hours?

a) 68.27%
b) 81.87%
c) 95.45%
d) 99.73%

Answer 1

At least 99.73% of the generators last between 1550 and 2450 hours according to the empirical rule because this range covers three standard deviations above and below the mean lifetime of the generators.

Step-by-step explanation:

The student's question relates to the application of the empirical rule (or the 68-95-99.7 rule) in statistics, which describes how data are distributed in a normal distribution. According to this rule, 68.27% of data fall within one standard deviation from the mean, 95.45% fall within two standard deviations, and 99.73% fall within three standard deviations

Given that the average lifespan of the generators is 2000 hours with a standard deviation of 150 hours, we see that 1550 hours is three standard deviations below the mean (2000 - 3*150), and 2450 hours is three standard deviations above the mean (2000 + 3*150). Therefore, according to the empirical rule, at least 99.73% of the generators will last between 1550 and 2450 hours.

The correct answer for the student's question is option (d) 99.73%.