6 Comparison of Population Proportions
In this chapter we present basic inferential methods for categorical data, espe-
cially the analysis of two-way contingency tables. LetX 1 andX 2 denote two
categorical variables,X 1 havingIlevels andX 2 havingJlevels, thusIJcombi-
nations of classifications. We display the data in a rectangular table havingI
rows for the categories ofX 1 andJcolumns for the categories ofX 2 ; theIJcells
represent theIJcombinations of outcomes. When the cells contain frequencies
of outcomes, the table is called acontingency tableorcross-classified table, also
referred to as aIbyJorIJtable. Most topics in this chapter are devoted to
the analyses of these two-way tables; however, before we can get there, let’s
start with the simplest case: that of a one-sample problem with binary data.
6.1 ONE-SAMPLE PROBLEM WITH BINARY DATA
In this type of problem, we have a sample of binary dataðn;xÞwithnbeing an
adequately large sample size andxthe number of positive outcomes among the
nobservations, and we consider the null hypothesis
H 0 :p¼p 0wherep 0 is a fixed and known number between 0 and 1: for example,
H 0 :p¼ 0 : 25p 0 is often a standardized or referenced figure, for example, the e¤ect of a
standardized drug or therapy, or the national smoking rate (where the national
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