STA 4502 Chapter 3 - Chapter 3 One Sample Location Problem...

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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, pre-treatment and post-treatment 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. Stat 4502 Chap 3, Page 1 of 26
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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. Assumptions:
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STA 4502 Chapter 3 - Chapter 3 One Sample Location Problem...

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