# Empirical Rule.

A samples contains 1,000 values. The histogram for this sample is bell-shaped. Approximately 950 values in the sample are known to be within $100 of the mean. Suppose the mean is $5,000. Use Empirical Rule to perform the following:

a. calculate a value for the standard deviation.

b. determine the data value that is 97.5 percentile

c. Calculate the z-score for 116th percentile

The division manager for Northern Pipe and Steel Company decided to implement a new intensive system for the managers of Northern's three plants. The plan called for a bonus to be paid the next month to the manager whose plant had the greatest relative improvement over the average monthly production volume. The following data reflected the historical production volumes at the three plants.

Plant1: m=700, q=200

Plant2: m=2300, q=350

Plant3: m=1200, q=30

At the close of the next month.the monthly output for the three plants was

Plant1 810

Plant2 2600

Plant3 1320

Suppose the division manager awarded the bonus to the manager of Plant2 because her plant increased by over units over the mean. What was a bigger increase that that of any of the other managers. Do you agree with who received the bonus this month? Explain, using the appropriate statistical measures to support your position.

#### Solution Preview

A samples contains 1,000 values. The histogram for this sample is bell-shaped. Approximately 950 values in the sample are known to be within $100 of the mean. Suppose the mean is $5,000. Use Empirical Rule to perform the following:

a. calculate a value for the standard deviation.

Approximately 95% of the data points in a bell shaped distribution lies within +/- 2 standard deviations of the means. Since 950 of the 1000 values lies within $100 of the mean, we have 2*standard deviation = 100

Thus, standard deviation = 100/2 = 50

b. determine the data value that is 97.5 percentile

Approximately 95% of the data points in a bell shaped distribution lies within +/- 2 standard deviations of the means. That means 2.5% data lies below (Mean - 2*standard deviations) and 2.5% data lies above (Mean + 2*standard ...

#### Solution Summary

The solution uses the empirical rule to calculate the standard deviation, determine data value, and calculate the z-score.