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8. 1 Population Mean:
σ
Known
•
In order to develop an interval estimate of a population mean, either the
population standard deviation,
σ
, or the sample standard deviation,
s
, must be
used to compute the margin of error
•
In most applications
σ
is not known, and
s
is used to compute the margin of error
•
In some applications, however, large amounts of relevant historical data are
available and can be used to estimate the population standard deviation prior to
sampling
•
Also, in quality control applications where a process is assumed to be operating
correctly, or “in control,” it is appropriate to treat the population standard
deviation as known
•
We refer to such cases as the
σ
known
case
Margin of Error and the Interval Estimate
•
The sampling distribution of
x
can be used to compute the probability that
x
will
be within a given distance of
μ
•
Therefore,
σ
=
σ
/
√
(n)
•
Because the sampling distribution shows how values of
x
are distributed around
the population mean
μ
, the sampling distribution of
x
provides information about
the possible differences between
x
and
μ
•
In the introduction to this chapter we said that the general form of an interval
estimate of the population mean
μ
is
x
±
margin of error
•
Any sample mean
x
that is within the darkly shaded region will provide an
interval that contains the population mean
μ
•
Confidence interval
– another name for an interval estimate
•
Confidence coefficient
– the confidence level expressed as a decimal value. For
example: .95 is the confidence coefficient for a 95% confidence level
•
Confidence level
– the confidence associated with an interval estimate. For
example: if an interval estimation procedure provides intervals such that 95% of
the intervals formed using the procedure will include the population parameter,
the interval estimate is said to be constructed at the 95% confidence level
•
With the margin of error given by
z
(
σ
/
√
(n)), the general form of an interval
estimate of a population mean for the
σ
known case follows:
o
x
±
z
(
σ
/
√
(n))
o
where,
(1
α
) is the confidence coefficient
z
is the z value providing an area of
α
/2 in the upper tail of the
standard normal probability distribution
Values of z
for the Most Commonly Used Confidence Levels
Confidence Level
α
α
/2
z
90%
.10
.05
1.645
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View Full Document95%
.05
.025
1.960
99%
.01
.005
2.576
•
Comparing the results for the 90%, 95%, and 99% confidence levels, we see that
in order to have a higher degree of confidence, the margin of error and thus the
width of the confidence interval must be larger
Practical Advice
•
If the population follows a normal distribution, the confidence interval provided
by the equation is exact
•
If the population doesn’t follow a normal distribution, the confidence interval
provided by the equation will be approximate
•
In this case, the quality of the approximation depends on both the distribution fot
he population and the sample size
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 Spring '08
 HERBERT

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