Introduction to Probability and Statistics for Engineers and Scientists

500 Chapter 11:Goodness of Fit Tests and Categorical Data Analysis

The Test Statistic for Independence in a Contingency Table

Start

Quit

1

2

3

ABC

10

10

13

12

24

20

D

6

9

7

7

10

10

The test statistic has value t = 1.81478

FIGURE 11.2

a random sample of people from this population and then noted their characteristics.
However, another way in which we could gather data is to fix in advance the numbers of
men and women in the sample and then choose random samples of those sizes from the
subpopulations of men and women. That is, rather than let the numbers of women and
men in the sample be determined by chance, we might decide these numbers in advance.
Because doing so would result in fixed specified values for the total numbers of men and
women in the sample, the resulting contingency table is often said to havefixed margins
(since the totals are given in the margins of the table).
It turns out that even when the data are collected in the manner prescribed above, the
same hypothesis test as given in Section 11.4 can still be used to test for the independence
of the two characteristics. The test statistic remains

TS=

∑

i

∑

j

(Nij−ˆeij)^2

ˆeij

where

Nij= number of members of sample who have bothX-characteristici

andY-characteristicj

Ni= number of members of sample who haveX-characteristici

Mj= number of members of sample who haveY-characteristicj