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• More than • Less than • Greater than • Within • At least • Between • At most

The length of long distance telephone calls 1 2 2 3 3 3 5 7 7 9 10 14 20 29 42

Chebyshev’s Theorem: the proportion of values from a data set that will
fall within k standard deviations of the mean will be at least , where *k* is a
number greater than 1 (*k* is not necessarily an integer)

The empirical (Normal Bell Shape) rule: when the distribution is *bell-
shape* the following statements, which make up the empirical rule, are
true.

• Approximately 68% of the data values will fall within one standard deviation of the mean.

• Approximately 95% of the data values will fall within two standard deviations of the mean.

• Approximately 99.7% of the data values will fall within three standard deviations of the mean

Example: The data represent the ages of 30 customers who ordered a product advertised on television. Count the number of data values that fall within two standard deviations of the mean. Compare this with the number from Chebyshev’s theorem.

42 44 62 35 20 30 56 20 23 41 55 22 31 27 66 21 18 24 42 25 32 50 31 26 36 39 40 18 36 22

**Percentile Formula
**The percentile corresponding to a given value X is computed by using the

following formula:

The following are scores on a Scientific Achievement Test given to a group of 40 students.

46 58 65 70 76 49 59 66 71 78 50 59 66 71 79 53 60 66 72 80 54 62 66 73 82 55 63 68 73 83 55 64 68 73 84 57 65 69 74 88

Verify the Empirical Rule

Use Chebyshev’s theorem to solve the following 1. The mean of a distribution is 20 and the standard deviation is 2.

• At least what percentage of values will fall between 10 and 30?

• At least what percentage of the values will fall between 12 and 28?

2. In a distribution of 200 values, the mean is 50 and the standard deviation is 5.

• At least how many values will fall between 30 and 70? • At least how many values will be less than 40 or more than

60?

3. In a sample of hourly wages of employees who work in restaurants in a large city have a mean of $5.02 and a standard deviation of $0.09. Using Chebyshev’s theorem, find the range in which at least 75% if the data values fall.

4. In a sample of the labor costs per hour to assemble a certain product have a mean of $2.60 and a standard deviation of $0.15. Using Chebyshev’s Theorem, find the values in which at least 88.89% if the data will lie.

5. In a sample of highway fatalities, 84% of the fatalities lies between 116 and 162. Find the mean and standard deviation of this distribution using Chebyshev’s Theorem.