SECTION 6.6 χ^2 and Goodness of Fit 413
are independent. If the table hasr rows andccolumns, then we shall
denote the entries of the table bynij,where 1≤i≤rand 1≤j≤c.
The entriesnij are often referred to as thecell counts. The sum of
all the cell counts is the total numbern in the sample. We denote by
C 1 , C 2 , ...,Cc the column sums and byR 1 , R 2 , ...,Rr the row sums.
Then in analogy with the above example, the contribution to theχ^2
statistic from the (i,j) table cell is (nij−RinCj)^2 /RinCj,as under the null
hypothesis of independence of the random variables defined by the rows
and the columns, the fractionRinCj represents theexpectedcell count.
The completeχ^2 statistic is given by the sum of the above contributions:
χ^2 =∑
i,j(nij−RinCj)^2
ÅR
iCj
nã ,and has (r−1)(c−1) degrees of freedom.
Example 3. It is often contended that one’s physical health is depen-
dent upon one’s material wealth, which we’ll simply equate with one’s
salary. So suppose that a survey of 895 male adults resulted in the
following contingency table:
Salary (in thousands U.S.$)
Health 15–29 30–39 40–59 ≥60 Totals
Fair 52 35 76 63 226
Good 89 83 78 82 332
Excellent 88 83 85 81 337
Totals 229 201 239 226 895One computesχ^26 = 13.840. SinceP(χ^26 ≥ 13 .840) = 0.031, one infers a
significant deviation from what one would expect if the variables really
were independent. Therefore, we reject the independence assumption.
Of course, we still can’t say any more about the “nature” of the de-
pendency of the salary variable and the health variable. More detailed
analyses would require further samples and further studies!
We mention finally that the above can be handled relatively easily
by the TI calculatorχ^2 test. This test requires a single matrix input,A,