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Chapter 6: Monte Carlo Methods for Inferential Statistics 201

We get a p-value of 0.071. If we are doing the hypothesis test at the 0.05 sig-
nificance level, then we would not reject the null hypothesis. This is consis-
tent with the results we had previously.

Note that in each approach, knowledge of the distribution of T under the
is needed. How to tackle situations where we do not
null hypothesis H 0
know the distribution of our statistic is the focus of the rest of the chapter.

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Confiden
Confiden ccee II t Innte er eerr vvaall lss ss
ConfidenConfiden
In Chapter 3, we discussed several examples of estimators for population
parameters such as the mean, the variance, moments, and others. We call
these point estimates. It is unlikely that a point estimate obtained from a ran-
dom sample will exactly equal the true value of the population parameter.
Thus, it might be more useful to have an interval of numbers that we expect
will contain the value of the parameter. This type of estimate is called an
interval estimate. An understanding of confidence intervals is needed for the
bootstrap methods covered in Section 6.4.
θ
Let represent a population parameter that we wish to estimate, and let T
denote a statistic that we will use as a point estimate for θ. The observed
ˆ
θ
value of the statistic is denoted as θ. An interval estimate for will be of the
form
ˆ
ˆ
θ Lo < θ < θ Up , (6.3)
ˆ ˆ θ ˆ
where θ Lo and θ Up depend on the observed value and the distribution of
the statistic T.
If we know the sampling distribution of T, then we are able to determine
ˆ ˆ
values for θ Lo and θ Up such that
ˆ
P θ Lo <( ˆ θ < θ Up) = 1 – α , (6.4)
where 0 < α < 1. Equation 6.4 indicates that we have a probability of 1 – α
that we will select a random sample that produces an interval that contains
⋅
θ. This interval (Equation 6.3) is called a 1 –( α) 100% confidence interval.
The philosophy underlying confidence intervals is the following. Suppose
we repeatedly take samples of size n from the population and compute the
random interval given by Equation 6.3. Then the relative frequency of the
θ
⋅
intervals that contain the parameter would approach 1 –( α) 100% . It
should be noted that one-sided confidence intervals can be defined similarly
[Mood, Graybill and Boes, 1974].
To illustrate these concepts, we use Equation 6.4 to get a confidence interval
µ
for the population mean . Recall from Chapter 3 that we know the distribu-
⁄
( α 2)
X
tion for . We define z as the z value that has an area under the standard
© 2002 by Chapman & Hall/CRC