Stat 4502 Chap 3, Page 1 of 22
Chapter 3
One Sample Location Problem
Introduction
Remember the “
measures of location
” from your previous statistics course(s)? These are
the parameters of a population that tell where the population data are located. You should
have heard of the mean (arithmetic mean or average), median and the mode. In your
previous study of statistical inference you have seen different methods for making
inferences about the population mean using sample data. You should also remember that
for small samples the procedures you have seen required that the samples come from
normal populations. What if you know that this is not true? You may use nonparametric
(or distribution free) methods that do not require such assumptions.
In this chapter we will see some inferential methods on the median as a measure of
location. (Remember that when a population has a
symmetric distribution,
the mean and
the median are equal.)
There are two different types of population and sample data that may be used:
The first type
of data is called
paired replicates
(you probably have studied this under
one of the titles “Dependent Samples” or “Paired Samples” or “Matched Samples.” Here
we have two populations and
one random sample of n units
from each, (a total of 2
n
observations) selected in such a way that t
he samples are dependent
on each other. The
most common example of this type of data is seen in “Case – Control” Studies, or
“Before – After” (more specifically, pretreatment and posttreatment observations,
where an individual (or an animal) is observed before a certain treatment, then given a
certain “treatment” and the effect of the treatment on the same individual is observed
after a certain period has elapsed. Thus, we have two observations on each population
unit selected into the sample, which give two dependent samples. In these studies we are
interested in whether the “treatment” has any significant effect, i.e., whether there is a
shift in the location of the population as a result of “treatment.” Such an effect (or shift) is
estimated by using the
differences between the
pairs
of observations
. Thus we end up
analyzing n differences.
The second type
of data where the methods of this chapter are used is the case of one
sample from one population. We will see that the analysis of n
differences
obtained from
two dependent samples is exactly the same as the analysis of one sample from one
population.
We will study two different (but related) tests:
signed rank test
and
sign test.
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Paired Replicates Analyses by Way of Signed Ranks
Data:
We obtain 2
n observations as a result of 2 observations (say, X
i
and Y
i
)on each
of n subjects (blocks, patients, etc.). Then, we find the differences, Z
i
= X
i
– Y
i
, as
shown in Table – 1.
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 Summer '11
 YESILCAY
 Statistics, Normal Distribution, Statistical hypothesis testing, zi

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