11.5Tests of Independence in Contingency Tables Having Fixed Marginal Totals 501
and
ˆeij=npˆiqˆj=NiMj
nwherenis the total size of the sample.
In addition, it is still true that whenH 0 is true,TSwill approximately have a chi-square
distribution with (r−1)(s−1) degrees of freedom. (The quantitiesrandsrefer, of
course, to the numbers of possible values of theX- andY-characteristic, respectively.) In
other words, the test of the independence hypothesis is unaffected by whether the marginal
totals of one characteristic are fixed in advance or result from a random sample of the entire
population.
EXAMPLE 11.5a A randomly chosen group of 20,000 nonsmokers and one of 10,000
smokers were followed over a 10-year period. The following data relate the numbers
of them that developed lung cancer during that period.
Smokers Nonsmokers Total
Lung cancer 62 14 76
No lung cancer 9,938 19,986 29,924
Total 10,000 20,000 30,000Test the hypothesis that smoking and lung cancer are independent. Use the 1 percent level
of significance.
SOLUTION The estimates of the expected number to fall in eachijcell when smoking and
lung cancer are independent are
ˆe 11 =(76)(10,000)
30,000=25.33ˆe 12 =(76)(20,000)
30,000=50.67ˆe 21 =(29,924)(10,000)
30,000=9,974.67ˆe 22 =(29,924)(20,000)
30,000=19,949.33