ISyE-ME512_Exam 2B
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Exam #2B: ISyE/ME 512 Inspection, Quality Control and Reliability
Spring 2023
Monday, 4/3/2023
NOTES:
1.
One-page note sheet (double sided) is allowed. Calculators are
allowed. Cell phone and
computer are not allowed during the exam.
2.
Write your answer on this exam paper. DO NOT use your own paper. If you need more
space,
write on the back side of this exam
paper.
3.
Show your steps. Answers with only final result will NOT earn full mark of that
problem.
4.
The test is worth 100 points. The assigned points for each question are given beside
it.
5.
There are
15
pages in this
exam. Please show the steps and procedures in your solution in
order to get full points.
Name(print):
Student
ID:
Signature,
Date:
===============================================================
ISyE/ME 512_Exam #2
Cover Page
Question 1:
An X-bar control chart with a sample size of
n=4
is used to monitor the process mean. Two
decision rules are given here:
Rule 1: If a single point falls outside the
3-sigma
control limits, conclude that the process is out
of control.
Rule 2: If one or more of the next five samples yield values fall between 2-sigma and 3-sigma
control limits.
a)
What is the Type I error using rule 1 alone?
b)
What is the Type I error using rule 2 alone?
c)
What is the overall Type I error based on the two rules, given that the two rules are
independent?
d)
If the process has a mean shift of 1.5
σ
, what is the Type II error using Rule 1?
ISyE/ME 512_Exam #2
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(Additional page for Question 1)
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Question 2:
Samples of n = 6 items each are taken from a process at regular intervals. A quality characteristic
is measured, and
´
x
and R values are calculated for each sample. After 50 samples, we have
∑
i
=
1
50
´
x
i
=
2000
,
∧
∑
i
=
i
50
R
i
=
200
Assume that the quality characteristics is normally distributed.
a)
Compute the control limits for the
´
x
and
R
control charts.
b)
All points on both control charts fall between the control limits computed in a). What are
the natural tolerance limits of the process?
c)
If the specification limits are 41 ± 5.0, what are your conclusions regarding the ability of
the process to produce items within these specifications?
d)
Assuming that if an item exceeds the upper specification limit it can be reworked, and if
it is below the lower specification limit it must be scrapped, what percentage scrap and
rework is the process producing?
ISyE/ME 512_Exam #2
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(Additional page for Question 2)
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Question 3:
A process is being monitored with a fraction nonconforming control chart. The process average
has been shown to be 0.081. Three-sigma control limits are used, and the procedure calls for
taking daily samples of 400 items.
a)
Calculate the upper and lower control limits.
b)
If the process average should suddenly shift to 0.1, what is the probability that the shift
would be detected on the first subsequent sample?
c)
What is the probability that the shift in b) would be detected on the first or second sample
taken after the shift?
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(Additional answer page for Question 3)
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Question 4: (
20
points
)
The manufacturer wishes to set up a control chart at the final inspection station for a gas water
heater. Defects in workmanship and visual quality features are checked in this inspection. For the
past 22 working days, 176 water heaters were inspected and a total of 924 nonconformities were
reported.
a)
What type of control chart would you recommend here, and how would you use it?
b)
Using two water heaters as the inspection unit, calculate the center line and control limits
that are consistent with the past 22 days of inspection data.
c)
What is the probability of type I error for the control chart in b)?
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Question 5. (
20 points
)
Control charts for
´
x
and s have been maintained on a process and have exhibited statistical
control. The sample size is n = 4. The control chart parameters are as follows:
´
xChart
s Chart
UCL = 710
UCL = 18.08
CL = 700
CL = 7.979
LCL = 690
LCL = 0
a)
Estimate the process parameters μ and σ.
b)
If the specifications are at 705 ± 15, and the process output is normally distributed,
estimate the fraction nonconforming.
c)
For the
´
x
chart, find the probability of a type I error, assuming σ is constant.
d)
Suppose the process mean shifts to 693 and the standard deviation simultaneously shifts
to 12. Find the probability of detecting this shift on the
´
x
chart on the first subsequent
sample, and find the average run length.
ISyE/ME 512_Exam #2
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Appendix
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