STAT 528
HOMEWORK #2
Due: Wednesday, 09/09/15.
1. The following data were collected from a machine that makes photocopies. The density of the
black toner on 169 copies was recorded. Although you are given SAS code to make the graphical
displayscan, you can use any software you want to answer the following. The data set has also been
sent to you as a .txt file.
(a) (2pt) Make a histogram of the density measurements and describe the distribution.
(b) (1pt) Make a scatterplot of the density measurements (y) against the sample (x).
(c) (1pt) One common assumption is that the data are independent and identically distributed.
Does that seem like a reasonable assumption? Justify your answer.
x
y
x
y
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
220.10
220.32
220.25
220.41
220.42
220.33
220.66
220.52
220.62
220.53
220.97
220.72
220.53
220.70
220.75
220.99
220.56
220.71
220.58
221.16
221.22
220.82
220.63
220.90
220.64
220.55
220.57
220.37
220.60
221.11
220.86
220.93
220.55
220.94
220.86
220.72
220.43
220.74
220.52
220.47
220.64
220.74
220.45
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
220.54
220.31
220.60
220.49
220.31
220.49
220.52
220.41
220.28
219.98
220.41
220.30
220.46
220.56
221.02
220.25
220.32
220.20
220.59
220.47
220.37
220.51
220.53
220.95
220.79
220.48
220.87
221.09
220.66
221.26
220.96
220.52
221.09
220.89
220.74
220.96
220.89
221.30
221.15
220.93
220.82
220.25
x
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
y
x
y
219.63
219.66
220.48
220.80
220.59
220.79
220.77
220.83
220.63
220.81
220.56
220.72
220.39
220.93
221.08
220.38
220.65
220.50
220.14
220.45
220.42
220.28
220.61
220.47
220.19
220.19
220.43
220.52
220.29
219.87
220.28
220.15
220.48
220.57
220.29
220.17
220.46
220.51
220.33
220.47
220.56
220.66
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
220.21
220.57
220.11
221.09
220.50
220.49
220.60
220.87
220.75
221.25
220.95
221.13
221.03
220.68
220.71
220.84
220.88
221.06
221.19
220.85
220.72
221.25
221.16
220.85
220.82
220.67
220.47
220.72
220.79
220.66
220.64
220.63
220.02
220.54
220.58
220.12
220.90
220.64
219.90
220.08
220.27
220.05
2. (2pt) Two decision rules are given below. Assume (i) that they apply to a normally distributed quality
characteristic, (ii) the control chart has 3σ control limits, and (iii) the sample size is n = 5. What is the Type
I error (in words) for each of these rules?
• Rule 1: If one or more of the next seven sample means fall outside of the control limits, then conclude
that the process is out of control.
• Rule 2: If all of the next seven means fall on the same side of the center line, then conclude that the
process is out of control.
4. (2pt) Two decision rules are given below. Assume (i) that they apply to a normally distributed
quality characteristic, (ii) the control chart has 3σ control limits with a specified centerline µ0 , and
3.(iii)
Suppose
a measurement
is to be
taken
from
each of
randomly
items rules?
produced by a process.
the sample
size is n =(Y
5. )What
is the
Type
I error
(in9words)
for selected
each of these
Assume Y ∼ N (µ, σ 2 ) with known µ and σ 2 .
• Rule 1: If one or more of the next seven sample means fall outside of the control limits, then
(a)
(1pt) Find
onecontrol.
or more of these 9 points would fall outside µ ± 3σ control limits?
conclude
thatthe
theprobability
process is that
out of
(b) (2pt) Find the probability that two or more of these 9 points would fall outside µ ± 2σ warning limits?
• Rule 2: If all of the next seven means fall on the same side of the center line, then conclude
(c)
(1pt)
the is
probability
that all 9 points fall on one side of the centerline µ?
that
theFind
process
out of control.
(d) (1pt) Find the probability that all 9 points alternate on different sides of the centerline µ?
5. Suppose a measurement (Y ) is to be taken from each of 9 randomly selected items produced by a
(e) (1pt) Find the probability
that two or more of2 these 9 points would fall outside µ ± 2σ warning limits
process.but
Assume
∼N
σ 2 ) with
known µ and σ .
withinYthe
3σ(µ,
control
limits?
(a) (1pt) Find the probability that one or more of these 9 points would fall outside µ ± 3σ control
limits?
4. Suppose
a measurement (Y ) is to be taken from each of 9 randomly selected items produced by a process.
Assume
∼ Nthe
(µ +probability
σ, σ 2 ) withthat
known
and
σ 2 . of these 9 points would fall outside µ ± 2σ warning
(b)
(2pt)YFind
twoµor
more
limits?
(a) (1pt) Find the probability that one or more of these 9 points would fall outside µ ± 3σ control limits?
(c)(b)(1pt)
Find
the
9 points
the9 same
sidefall
of outside
the centerline
(2pt)
Find
theprobability
probabilitythat
thatall
two
or morefall
of on
these
pointsside
would
µ ± 2σ µ?
warning limits?
(d)(c)(1pt)
Find
the
onside
different
of theµ?centerline µ?
(1pt)
Find
theprobability
probabilitythat
thatall
all99points
pointsalternate
fall on one
of the sides
centerline
(e)(d)(1pt)
Extra
Find thethat
probability
thatalternate
two or more
of these
9 points
fall outside
(1pt)
FindCredit:
the probability
all 9 points
on different
sides
of the would
centerline
µ?
µ
±
2σ
warning
limits
but
within
the
3σ
control
limits?
(e) (1pt) Find the probability that two or more of these 9 points would fall outside µ ± 2σ warning limits
but within the 3σ control limits?
6. (1pt) Exercise 5.33, page 225. You can answer this without having to answer Exercise 5.32.
(1pt)
Exercisedata
5.33,is page
edition). You
can answer
this without
having
to answer
Exercise
7. 5.The
following
a list225
of (6th
the numbers
of rejected
aluminum
castings
by day
and defect
type5.32.
over a three-week period.
Make
Pareto
charts
for numbers
each of the
three weeks.
Whatcastings
are the by
major
and over a
6. (a)
The(2.5pt)
following
data
is a list
of the
of rejected
aluminum
day similarities
and defect type
three-week
period.
differences
in the charts for the three weeks?
(b)(a)(1.5pt)
Make
Paretocharts
chart for
foreach
the of
3-week
totals.
What
areare
thethe
primary
causes of rejected
(2.5pt)
Makea Pareto
the three
weeks.
What
major similarities
and differences
in the charts for the three weeks?
castings?
(1.5pt)
Make
a Pareto
chart
for the 3-week
What
are the
primary
rejected any
castings?
(c)(b)(1pt)
Make
a scatter
plot
of Production
(y)totals.
vs Daily
Total
Scrap
(x). causes
Do youof observe
(c)
(1pt) Make
a scatter
plot
Production
(y)context
vs Daily
Scrap (x). Do you observe any patterns?
patterns?
If so,
describe
theofpattern
in the
of Total
the problem.