The
𝑧
of 1.0 indicates that weekly income of $1,100 is
one standard deviation above the mean, and
𝑧
of -1.0
indicates that a $900 income is one standard deviation
below the mean

The mean labour cost to repair a heat pump is $90 with
a standard deviation of $22. Monte’s Plumbing
completed repairs on two heat pumps this morning. The
labour cost for the first was $75, and it was $100 for the
second. Assume the distribution of labour costs follows
the normal distribution. Compute
𝑧
values for each, and
comment on your findings.
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....

The Empirical Rule
1.
Approximately 68% of
observations will lie
within
±1
standard
deviation of the mean
2.
About 95% of
observations will lie
within
±2
standard
deviation of the mean
3.
Practically all, 99.7% of
observations will lie
within
±3
standard
deviation of the mean

To verify the Empirical Rule:
z of 1.00 = .3413 so .3413
*
2 = .6826 or about 68%
z of 2.00 = .4772 so .4772
*
2 = .9544 or about 95%
z of 3.00 = .4987 so .4987
*
2 = .9974 or about 99.7%
The Empirical Rule

Example: The Empirical Rule
As part of its quality assurance program, the Autolite
Battery Company conducts tests on battery life. For a
particular D-cell alkaline battery, the mean life is 19
hours. The useful life of the battery follows a normal
distribution with a standard deviation of 1.2 hours.
a)
About 68% of the batteries failed between what
two values?
b)
About 95% of the batteries failed between what
two values?
c)
Virtually all of the batteries failed between what
two values?

Example: The Empirical Rule
a)
About 68% of the batteries failed between what
two values?
𝜇 + 1𝜎 = 19 ± 1.2 = 17.8 and 20.2 hrs
b)
About 95% of the batteries failed between what
two values?
𝜇 + 2𝜎 = 19 ± 2(1.2) = 16.6 and 21.4 hrs
c)
Virtually all of the batteries failed between what
two values?
𝜇 + 3𝜎 = 19 ± 3(1.2) = 15.4 and 22.6 hrs

The mean of a normal probability distribution is 60; the
standard deviation is 5.
a)
About what percent of the observations lie
between 55 and 65?
b)
About what percent of the observations lie
between 50 and 70?
c)
About what percent of the observations lie
between 45 and 75?
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....