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453
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Mechanical Engineering
Date
Dec 6, 2023
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2
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1
ME 453:
Data Science in Manufacturing Quality Control
Homework 2
Assigned:
September
13,
2023
Due:
September
27,
2023
Points:
The total number of points is 100. The assigned points are given besides questions. To get full
credit you must SHOW ALL OF YOUR WORK.
Homework guidelines:
1.
The total number of points is 100 and 130 for 3-credit-hour and 4-credit-hour sections,
respectively. The assigned points are given beside questions. To get full credit you must SHOW
ALL OF YOUR WORK.
2.
Problems marked by “[4 credit only]” should be completed by students in the 4-credit-hour
section.
Students in the 3-credit-hour section are welcome to work on these problems and
the earned
points will be used to compensate for lost points in homework assignments.
3.
A complete submission features the following items: (a) a brief report including all figures
and
results, and explanations of necessary steps taken to obtain them; and (b) and the source
code
(Python is recommended).
4.
Item (a) can be scanned copies of handwritten or typeset reports. Both items shall be
submitted through Canvas.
Problem 1 (15 points)
Perform
a
simulation
study
on
confidence
interval.
Take
X
∼
N
(10
,
0
.
01).
(1)
Generate 100 random samples using software, with each sample of size
n
= 10.
Take
α
= 0
.
05,
compute CI for all samples. (5 points)
(2)
Visualize CIs with an error bar plot (Hint: check matplotlib.pyplot.errorbar()), with sample ID on
horizontal axis, and CI on vertical axis. Indicate the location of true mean
µ
. (5 points)
(3)
How many CIs failed to capture
µ
? Distinguish them in the error bar plot. (5 points)
Problem 2 (15 points + 5 points)
Suppose
we
wish
to
test
the
hypothesis:
H
0
:
µ
=
20
H
1
:
µ
≠
20
It is given that
σ
2
=
4
.
0, true mean
µ
=
16, and
α
=
0
.
05.
(1)
What sample size
n
must be used to ensure that the probability of type II error is no greater
than
0.1? (5 points)
(2)
Under the sample size calculated in (1), what is the resulting type II error? (5 points)
(3)
Use software to plot type II error vs. sample size
n
. Do you observe a monotonic trend? (5 points)
(4) [4 credit only] What is the
probability of type II error when the sample size approaches infinity, i.e.,
࠵? →
∞
? What does this imply? (5 points)
2
Problem 3 (10 points + 10 points)
(1)
What is the type I error for X-bar control charts with probability limit of 0.001 and sample size of
4? (5 points)
(2)
Assuming the mean shift is 1
.
5
σ
, what is the type II error for detecting this mean shift by using
this control chart? (5 points)
(3)
[4 credit only] Use software to generate a plot for type II error vs. mean shift. (10 points)
Problem 4 (30 points + 15 points)
An X-bar control chart with 1% probability limit is used to monitor process mean. Inspection
decision
is made based on four successive samples with sample size
n
= 9 using the following rules:
Rule
1
:
If
the
mean
of
any
sample
exceeds
the
upper
or
lower
control
limit.
Rule 2
: If the sample means of four samples in a row fall on the same side of the center line.
(1)
What is type I error rate using Rule 1 alone? (5 points)
(2)
What is type I error rate using Rule 2 alone? (5 points)
(3)
If the process has a mean shift of 1
σ
(one process standard deviation), what is type II error rate
using Rule 1? (10 points)
(4)
If the process has a mean shift of 1
σ
, what is type II error rate using Rule 2? (10 points)
(5)
[4 credit only] Use software to generate plots of type II error rate vs. mean shift vs. for both Rules 1
and 2. Please provide a figure with both plots and provide a legend. (10 points)
(6)
[4 credit only] Based on the above results, discuss the pros and cons of Rules 1 and 2. Note that
this is an open-ended question. Any answer that is well justified will receive full credit. (5 points)
Problem 5 (30 points)
For a 3-sigma
X
¯
chart with
n
=
9, it is given that UCL
=
104, LCL
=
96. If a 2
σ
shift occurs in
the
process
mean,
(1)
What is the detection power of this control chart? (10 points)
(2)
What is the probability of detecting the shift by the 3
rd
sample after the shift? (10 points)
(3)
What is expected length to detect this change? (10 points)
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