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A point estimate, by itself, does not provide enough information regarding vari-
ability encompassed into the simulation response (output measure). This variability
represents the differences between the point estimates and the population parameters.
Hence, an interval estimate in terms of a confidence interval is constructed using the
estimated average (
y
) and standard deviation (
s
y
). A confidence interval is a range of
values that has a high probability of containing the parameter being estimated. For
example, the 95% confidence interval is constructed in such a way that the probabil-
ity that the estimated parameter is contained with the lower and upper limits of the
interval is of 95%. Similarly, 99% is the probability that the 99% confidence interval
contains the parameter.
The confidence interval is symmetric about the sample mean
y
. If the parameter
being estimated is
µ
y
, for example, the 95% confidence interval (CI) constructed
around an average of
y
=
28.0% is expressed as follows:
25
.
5%
≤
µ
y
≤
30
.
5%
this means that we can be 95% confident that the unknown performance mean (
µ
y
)
falls within the interval [25.5%, 30.5%].
Three statistical assumptions must be met in a sample of data to be used in
constructing the confidence interval. That is, the data points should be normally,
independent, and identically distributed. The following formula typically is used to
compute the CI for a given significance level (
α
):
/
√
n
/
√
n
y
−
t
α/
2
,
n
−
1
s
≤
µ
≤
y
+
t
α/
2
,
n
−
1
s
(6.4)
where
y
is the average of multiple data points,
t
n
−
1
,
α
/2
is a value from the Student
t
distribution
5
level of significance.
For example, using the data in Table 6.4, Figure 6.2 shows a summary of both
graphical and descriptive statistics along with the computed 95% CI for the mean,
median, and standard deviation. The graph is created with Minitab statistical software.
The normality assumption can be met by increasing the sample size (
n
) so that the
central limit theorem (CLT) is applied. Each average performance
y
(average “Usage,”
for example) is determined by summing together individual performance values (
y
1
,
y
2
,
for an
α
,y
n
) and by dividing them by
n
. The CLT states that the variable representing
the sum of several independent and identically distributed random values tends to
be normally distributed. Because (
y
1
,y
2
,
...
,y
n
) are not independent and identically
distributed, the CLT for correlated data suggests that the average performance (
y)
will be approximately normal if the sample size (
n
) used to compute
y
is large,
n
...
≥
30. The 100%(1
−
α
) confidence interval on the true population mean is expressed
5
A probability distribution that originates in the problem of estimating the mean of a normally distributed
population when the sample size is small. It is the basis of the popular Students
t
tests for the statistical
significance of the difference between two sample means, and for confidence intervals for the difference
between two population means.
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