Chebyshev’s Theorem and the Empirical Rule

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Chebyshev’s Theorem and The Empirical Rule

Suppose we ask 1000 people what their age is. If this is a representative sample then there will be very few people of 1-2 years old just as there will not be many 95 year olds. Most will have an age somewhere in their 30’s or 40’s. A list of the number of people of a certain age may look like this:

|Age |Number of people |
|0 |1 |
|1 |2 |
|2 |3 |
|3 |8 |
|.. |.. |
|.. |.. |
|30 |45 |
|31 |48 |
|.. |..
…show more content…
Summarizing the above we get the following table:

|Interval |k |[pic] |% |
|[pic] |2 |[pic] |75 |
|[pic] |3 |[pic] |88.9 |
|[pic] |4 |[pic] |93.75 |
|[pic] |5 |[pic] |96 |
|[pic] |6 |[pic] |97.2 |

Do we have to restrict ourselves to whole numbers as values for k? No, we may take any value for k as long as it larger than 1. For instance, for k = 2.5 we get the result that [pic] in the interval [pic] years

Example 1:
Students Who Care is a student volunteer program in which college students donate work time in community centers for homeless people. Professor Gill is the faculty sponsor for this student volunteer program. For several years Dr. Gill has kept a record of the total number of work hours volunteered by s student in the program each semester. For students in the program, for each semester the mean number of hours was 29.1 hours with a standard deviation of 1.7 hours. Find an interval for the number of hours volunteered in which at least 88.9% of the students in this program would fit.

Solution:
From the table above we see that a percentage of 88.9 will coincide with an interval of [pic] hours. This can be rewritten as an interval from 24 to 34.2 hours volunteered each semester.

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