14.4
The Onesample Runs
Test
Chapter 14
Statistics for Management
Richard I. Levin and David S.
Rubin
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1
Concept of randomness
• Assumption: the samples were randomly
selected.
• Meaning: chosen without preference or
bias.
• Recurrent (repeated) patterns cases.
• E.g.: Applicants for advanced job training
were to be selected without regard to
gender from a large population
(W=woman, M=man).
• The first group:
W,W,W,W,M,M,M,M,W,W,W,W,M,M,M,M
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Continued
• Total number of applicants is equally
divided between the sexes.
• The order is not random.
• A random process would rarely list two
items in alternating groups of four.
• Another order: W,M, W,M, W,M, W,M,
W,M, W,M, W,M, W,M.
• A random process would not produce such
an orderly pattern of men and women.
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Theory of runs
• A run is a sequence of identical
occurrences preceded and followed
by different occurrences or by none
at all.
• Example of
- three runs: W,M,M,M,M,W
- six runs:
W,W,W,M,M,W,M,M,M,M,W,W,W,W,M
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Symbols
• n1 = number of occurrences of type 1
• n2 = number of occurrences of type 2
• R = number of runs
• Example
M,W,W,M,M,M,M,W,W,W,M,M,W,M,W,W,M
• n1= 8, n2 = 9, r = 9
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Problem
• A manufacturer of breakfast cereal uses a
machine to insert randomly one of two
types of toys in each box. The company
wants randomness so that every child in
the neighborhood does not get the same
toy. Testers choose samples of 60
successive boxes to see whether the
machine is properly mixing the two types
of toys. Using the symbols A and B to
represent the two types of toys, a tester
reported that one such batch looked like in
the next slide.
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The batch
B,A,B,B,B,A,A,A,B,B,A,B,B,B,B,A,A,A,A,B,
A,B,A,A,B,B,B,A,A,B,A,A,A,A,B,B,A,B,B,A,
A,A,A,B,B,A,B,B,B,B,A,A,B,B,A,B,A,A,B,B
• n1=29 number of boxes containing toy A
• n2=31 number of boxes containing toy B
• r = 29
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The sampling distribution of
the r statistic
• A one-sample runs test: based on the idea that
too few or too many runs show that the items
were not chosen randomly.
• The mean:
2n1n2
µr =
+1
n1 + n 2
• The standard error:
2n1n2(2n1n2 – n1 – n2)
σr =
(n1 + n2)2(n1 + n2 – 1)
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For the example
2(29)(31)
µr =
+ 1 = 30.97
29 + 31
2(29)(31)(2(29)(31) – 29 – 31)
σr =
(29 + 31)2(29 + 31 – 1)
= 3.84
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Testing the hypotheses
• The sampling distribution of r can be
closely approximated by the normal
distribution if either n1 or n2 larger
than 20.
• α = 0.20
• H0:The toys are randomly mixed.
• H1:The toys are not randomly mixed.
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Standardizing the sample r
statistic
r - µr
z=
29 – 30.97
=
σr
= -0.513
3.84
• Accept the null hypothesis.
• Tolak Ho jika Z statistik lebih besar
1.19 atau lebih kecil dari -1.19.
• Conclusion: toys are being inserted
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in boxes in random order.
11
Exercise
• Professor Ike Newton is interested in
determining whether his brightest
students (those making the best grades)
tend to turn in their test earlier (because
they can recall material faster) or later
(because they take longer to write down
all they know) than the other in the class.
For a particular physics test, he observes
that the students make the following
grades in order of turning their tests in:
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Data
Order
1-10
11-20
21-30
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Grades
94
88
69
93
50
51
70
74
90
66
55
90
85
85
57
74
47
88
89
92
98
63
86
79
72
80
59
68
63
89
13
Question
1. If Professor Newton counts those making
a grade of 90 and above as his brightest
students, then at a 5 percent level of
significance, can he conclude the
brightest students turned their tests in
randomly?
2. If 60 and above is passing in Professor
Newton’s class, then at a 5 percent level
of significance did the students passing
versus those not passing turn their tests
in randomly?
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