Applied Statistics and Probability for Engineers

320 CHAPTER 9 TESTS OF HYPOTHESES FOR A SINGLE SAMPLE

9-63. Define Xas the number of underfilled bottles from a

filling operation in a carton of 24 bottles. Sixty cartons are

inspected and the following observations on Xare recorded:

Values 0 1 2 3

Frequency 39 23 12 1

(a) Based on these 75 observations, is a binomial distribution

an appropriate model? Perform a goodness-of-fit proce-

dure with 0.05.

(b) Calculate the P-value for this test.

9-64. The number of cars passing eastbound through the in-

tersection of Mill and University Avenues has been tabulated

by a group of civil engineering students. They have obtained

the data in the adjacent table:

(a) Does the assumption of a Poisson distribution seem

appropriate as a probability model for this process? Use

0.05.

(b) Calculate the P-value for this test.

Vehicles Vehicles

per Observed per Observed

Minute Frequency Minute Frequency

40 14 53 102

41 24 54 96

42 57 55 90

43 111 56 81

44 194 57 73

45 256 58 64

46 296 59 61

47 378 60 59

48 250 61 50

49 185 62 42

50 171 63 29

51 150 64 18

52 110 65 15

9-8 CONTINGENCY TABLE TESTS

Many times, the nelements of a sample from a population may be classified according to two

different criteria. It is then of interest to know whether the two methods of classification are

statistically independent; for example, we may consider the population of graduating engi-

neers, and we may wish to determine whether starting salary is independent of academic dis-

ciplines. Assume that the first method of classification has rlevels and that the second method

has clevels. We will let Oijbe the observed frequency for level iof the first classification

method and level jon the second classification method. The data would, in general, appear as

shown in Table 9-2. Such a table is usually called an r  ccontingency table.

We are interested in testing the hypothesis that the row-and-column methods of classifi-

cation are independent. If we reject this hypothesis, we conclude there is some interaction be-

tween the two criteria of classification. The exact test procedures are difficult to obtain, but an

approximate test statistic is valid for large n. Let pijbe the probability that a randomly selected

element falls in the ijth cell, given that the two classifications are independent. Then pij uivj,

Table 9-2 An r cContingency Table

Columns

12 p c

1 O 11 O 12 p O 1 c

Rows

2 O 21 O 22 p O 2 c

rOr 1 Or 2 p Orc

o o o o o

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