Applied Statistics and Probability for Engineers

9-8 CONTINGENCY TABLE TESTS 321

where uiis the probability that a randomly selected element falls in row class iand vjis the

probability that a randomly selected element falls in column class j. Now, assuming inde-

pendence, the estimators of uiand vjare

(9-40)

Therefore, the expected frequency of each cell is

(9-41)

Then, for large n, the statistic

(9-42)

has an approximate chi-square distribution with (r 1)(c 1) degrees of freedom if the null

hypothesis is true. Therefore, we would reject the hypothesis of independence if the observed

value of the test statistic ^20 exceeded ^2 ,(r 1)(c 1).

EXAMPLE 9-14 A company has to choose among three pension plans. Management wishes to know whether

the preference for plans is independent of job classification and wants to use  0.05. The

opinions of a random sample of 500 employees are shown in Table 9-3.

To find the expected frequencies, we must first compute (340500) 0.68, 

(160500)  0.32, (200500) 0.40, (200500) 0.40, and (100500) 

0.20. The expected frequencies may now be computed from Equation 9-41. For example, the

expected number of salaried workers favoring pension plan 1 is

The expected frequencies are shown in Table 9-4.

The eight-step hypothesis-testing procedure may now be applied to this problem.

1. The variable of interest is employee preference among pension plans.


2. H 0 : Preference is independent of salaried versus hourly job classification.

E 11 nuˆ 1 vˆ 1  5001 0.68 21 0.40 2  136

vˆ 1 vˆ 2 vˆ 3

uˆ 1 uˆ 2

^20  a

r

i 1

a

c

j 1

1 Oij Eij 22

Eij

Eijnuˆivˆj

1

na

c

j 1

Oija

r

i 1

Oij

vˆj

1

n^ a

r

i 1

Oij

uˆi^1

n^ a

c

j 1

Oij

Table 9-3 Observed Data for Example 9-14

Pension Plan

Job Classification 1 2 3 Totals

Salaried workers 160 140 40 340

Hourly workers 40 60 60 160

Totals 200 200 100 500

Table 9-4 Expected Frequencies for Example 9-14

Pension Plan

Job Classification 1 2 3 Totals

Salaried workers 136 136 68 340

Hourly workers 64 64 32 160

Totals 200 200 100 500

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