Page 374 - Applied Statistics And Probability For Engineers

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322 CHAPTER 9 TESTS OF HYPOTHESES FOR A SINGLE SAMPLE

3. H : Preference is not independent of salaried versus hourly job classification.
1
4. 0.05
5. The test statistic is

r c 1o
E 2 2
ij
ij
2
a a
0
i 1 j 1 E ij
6. Since r 2 and c 3, the degrees of freedom for chi-square are (r
1)(c
1)
2
if 2 5.99.
(1)(2) 2, and we would reject H 0 0 0.05,2
7. Computations:
2 3 1o
E 2 2
ij
ij
2
0 a a
i 1 j 1 E ij
1160
1362 2 1140
1362 2 140
682 2 140
642 2

136 136 68 64
160
642 2 160
322 2
49.63
64 32
2
2
8. Conclusions: Since 0 49.63 0.05,2 5.99 , we reject the hypothesis of inde-
pendence and conclude that the preference for pension plans is not independent of
2
job classification. The P-value for 0 49.63 is P 1.671 10
11 . (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 r populations of
interest and each population is divided into the same c categories. 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 c categories are the same
for all populations. The null hypothesis in this problem states that the populations are homo-
geneous with 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 r binomial 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
9-65. A company operates four machines three shifts each
Machines
day. From production records, the following data on the num-
ber of breakdowns are collected: Shift A B C D
1 41 20 12 16
2 31 11 9 14
3 15 17 16 10