Question

For any set of n measurements, the fraction included in the interval y − ks to y + ks is at least 1 − 1 k2 . This result is known as Tchebysheff's theorem. A personnel manager for a certain industry has records of the number of employees absent per day. The average number absent is 8.5, and the standard deviation is 3.5. Because there are many days with zero, one, or two absent and only a few with more than ten absent, the frequency distribution is highly skewed. The manager wants to publish an interval in which at least 75% of these values lie. Use Tchebysheff's theorem to find such an interval.

Answer 1

This interval is between 1.5 and 15.5.

Explanation:

Tchebysheff's Theorem

The Tchebysheff's Theorem can also be applied to non-normal distribution. It states that:

At least 75% of the measures are within 2 standard deviations of the mean.

At least 89% of the measures are within 3 standard deviations of the mean.

An in general terms, the percentage of measures within k standard deviations of the mean is given by
`100(1 - (1)/(k^(2)))`.

In this question, we have that:

Mean = 8.5

Standard deviation = 3.5

The manager wants to publish an interval in which at least 75% of these values lie.

By the Tchebysheff's Theorem, at least 75% of the measures are within 2 standard deviations of the mean.

8.5 - 2*3.5 = 1.5

8.5 + 2*3.5 = 15.5

This interval is between 1.5 and 15.5.