Question

The physical plant at the main campus of a large state university receives daily requests to replace fluorescent lightbulbs. the distribution of the number of daily requests is approximately normal and has a mean of 53 and a standard deviation of 11. what is the approximate proportion of lightbulb replacement requests numbering between 42 and 53?

Answer 1

The Empirical Rule applies to a normal, bell-shaped curve and states that within one standard deviation of the mean (both left-side and right-side) there is about 68% of the data; within two standard deviations of the mean (both left-side and right-side) there is about 95% of the data; and within three standard deviations of the mean (both left-side and right-side) there is about 99.7% of the data. See display below from Section 3.3 Measures of Variation in the textbook.

 

Example: IQ Scores have a bell-shaped distribution with a mean of 100 and a standard deviation of 15. What percentage of IQ scores are between 70 and 130?

Solution: 130 – 100 = 30 which is 2(15). Thus, 130 is 2 standard deviations to the right of the mean. 100 – 70 = 30 which is 2(15). Thus, 70 is 2 standard deviations to the left of the mean. Since 70 to 130 is within 2 standard deviations of the mean, we know that about 95% of the IQ scores would be between 70 and 130.