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210 Part IV: Quality Assurance
98.7%
95%
68%
m – 3s m – 2s m – 1s m m + 1s m + 2s m + 3s
Figure 18.3 Application of the empirical rule.
to compute the percentage of data that will fall within k standard deviations from
the mean (k = 1, 2, 3):
1. About 68 percent of the data will fall within one standard deviation
of the mean, that is, between m – 1s and m + 1s.
2. About 95 percent of the data will fall within two standard deviations
of the mean, that is, between m – 2s and m + 2s.
3. About 99.7 percent of the data will fall within three standard
deviations of the mean, that is, between m – 3s and m + 3s.
Part IV.A.2
Figure 18 .3 illustrates these features of the empirical rule.
EXAMPLE 18.11
A soft-drink filling machine is used to fill 16-ounce soft-drink bottles. Since the amount
of beverage varies slightly from bottle to bottle, it is believed that the actual amount of
beverage in the bottle forms a bell-shaped distribution with a mean of 15.8 oz. and standard
deviation of 0.15 oz. Use the empirical rule to find what percentage of bottles contain
between 15.5 oz. and 16.1 oz. of beverage.
Solution:
From the information provided to us in this problem, we have
m = 15.8 oz.
s = .15 oz.
We are interested in finding the percentage of bottles that contain between 15.5 oz. and
16.1 oz. of beverage. Comparing Figure 18.4 with Figure 18.3, it is obvious that approximately
95 percent of the bottles contain between 15.5 oz. and 16.1 oz. of beverage, since
15.5 and 16.1 are two standard deviations away from the mean.
Continued