categorical variable measured across two or more populations (called a chi-square test for homogeneity
of proportions) and a situation in which there are two categorical variables measured across a single
population (called a chi-square test for independence).
To answer this question, we need to compare the observed values in the sample with the expected
values we would get if the sample of black Americans really had the same distribution of blood types as
white Americans. The values we need for this are summarized in the following table.
It appears that the numbers vary noticeably for types A and B, but not as much for types AB and O.
The table can be rewritten as follows.
Before working through this problem, a note on symbolism. Often in this book, and in statistics in
general, we use English letters for statistics (measurements from data) and Greek letters for parameters
(population values). Hence, is a sample mean and μ is a population mean; σ is a sample standard
deviation and σ is a population standard deviation, etc. We follow this same convention in this chapter:
we will use χ 2 when referring to a population value or to the name of a test and use X 2 when referring to
the chi-square statistic.
The chi-square statistic (X 2 ) calculates the squared difference between the observed and expected
values relative to the expected value for each category. The X 2 statistic is computed as follows:
The chi-square distribution is based on the number of degrees of freedom, which equals, for the
goodness-of-fit test, the number of categories minus 1 (df = c – 1). The X 2 statistic follows approximately
a unique chi-square distribution, assuming a random sample and a large enough sample, for each different
number of degrees of freedom. The probability that a sample has a X 2 value as large as it does can be