Applied Statistics and Probability for Engineers

322 CHAPTER 9 TESTS OF HYPOTHESES FOR A SINGLE SAMPLE

3. H 1 : Preference is not independent of salaried versus hourly job classification.


4.  0.05


5. The test statistic is


6. Since r 2 and c 3, the degrees of freedom for chi-square are (r 1)(c 1) 
   (1)(2)  2, and we would reject H 0 if ^20 ^2 0.05,2 5.99.


7. Computations:


8. Conclusions: Since , we reject the hypothesis of inde-
   pendence and conclude that the preference for pension plans is not independent of
   job classification. The P-value for is. (This value
   was computed using a hand-held calculator.) Further analysis would be necessary to
   explore the nature of the association between these factors. It might be helpful to
   examine the table of observed minus expected frequencies.

Using the two-way contingency table to test independence between two variables of

classification in a sample from a single population of interest is only one application of con-

tingency table methods. Another common situation occurs when there are rpopulations of

interest and each population is divided into the same ccategories. A sample is then taken from

the ith population, and the counts are entered in the appropriate columns of the ith row. In this

situation we want to investigate whether or not the proportions in the ccategories are the same

for all populations. The null hypothesis in this problem states that the populations are homo-

geneouswith respect to the categories. For example, when there are only two categories, such

as success and failure, defective and nondefective, and so on, the test for homogeneity is really

a test of the equality of rbinomial parameters. Calculation of expected frequencies, determi-

nation of degrees of freedom, and computation of the chi-square statistic for the test for ho-

mogeneity are identical to the test for independence.

EXERCISES FOR SECTION 9-8

^20 49.63 P1.671 10

11

^20 49.63^2 0.05,25.99

160

6422

64

160

3222

32

49.63



1160

13622

136

1140

13622

136

140

6822

68

140

6422

64

^20  a

2

i 1

a

3

j 1

1 oij Eij 22

Eij

^20  a

r

i 1

a

c

j 1

1 oij Eij 22

Eij

Machines

Shift ABCD

1 41201216

23111914

3 15171610

9-65. A company operates four machines three shifts each

day. From production records, the following data on the num-

ber of breakdowns are collected:

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