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6.2 Normal Distribution 193
DEMONSTRATION
PROBLEM 6.8
The U.S. Environmental Protection Agency publishes figures on solid waste generation
in the United States. One year, the average number of waste generated per
person per day was 3.58 pounds. Suppose the daily amount of waste generated
per person is normally distributed, with a standard deviation of 1.04 pounds. Of
the daily amounts of waste generated per person, 67.72% would be greater than
what amount?
Solution
The mean and standard deviation are given, but x and z are unknown. The problem
is to solve for a specific x value when .6772 of the x values are greater than that
value.
If .6772 of the values are greater than x, then .1772 are between x and the mean
(.6772 - .5000). Table 6.2 shows that the probability of.1772 is associated with a
z value of 0.46. Because x is less than the mean, the z value actually is - 0.46. Whenever
an x value is less than the mean, its associated z value is negative and should be
reported that way.
.6772
.1772
.5000
Solving the z equation yields
and
x = ?
μ = 3.58
σ = 1.04
z = x - m
s
-0.46 = x - 3.58
1.04
x = 3.58 + (-0.46)(1.04) = 3.10
Thus 67.72% of the daily average amount of solid waste per person weighs more
than 3.10 pounds.
STATISTICS IN BUSINESS TODAY
Warehousing
Tompkins Associates conducted a national study of warehousing
in the United States. The study revealed many
interesting facts. Warehousing is a labor-intensive industry
that presents considerable opportunity for improvement in
productivity. What does the “average” warehouse look like?
The construction of new warehouses is restricted by prohibitive
expense. Perhaps for that reason, the average age of
a warehouse is 19 years. Warehouses vary in size, but the
average size is about 50,000 square feet. To visualize such
an “average” warehouse, picture one that is square with
about 224 feet on each side or a rectangle that is 500 feet by
100 feet. The average clear height of a warehouse in the
United States is 22 feet.
Suppose the ages of warehouses, the sizes of warehouses,
and the clear heights of warehouses are normally
distributed. Using the mean values already given and the
standard deviations, techniques presented in this section
could be used to determine, for example, the probability
that a randomly selected warehouse is less than 15 years
old, is larger than 60,000 square feet, or has a clear height
between 20 and 25 feet.