HW4 (1)
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ME 453: Data Science in Manufacturing Quality Control
Homework 4
Assigned: October 11, 2023
Due: October 25, 2023
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.
Indicates the problem which you need to use Python.
4. 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).
5. Item (a) can be scanned copies of handwritten or typeset reports.
Both items shall be
submitted through Canvas.
Problem 1 (35 points + 10 points)
A fraction nonconforming control chart with
n
= 400 has the following parameters:
UCL = 0
.
0962
Centerline = 0
.
0500
LCL = 0
.
0038
(1) Find the width of the control limits in standard deviation units. (10 points)
(2) What is Type I error of this control chart? (10 points)
(3) Suppose the process fraction nonconforming shifts to 0.15. What is the probability of de-
tecting the shift on the first subsequent sample? (15 points)
(4) [4 credit only]
Use software to generate an operating characteristic (OC) curve by varying
the fraction nonconforming between 0 and 0.6 in a step of 0.01. (10 points)
1
Problem 2 (15 points + 10 points)
A process has an in-control fraction nonconforming of
p
= 0
.
01.
(1) What sample size would be required for the fraction nonconforming control chart if it is
desired to have a probability of at least one nonconforming unit to be at least 0.9?
(10
points)
(2) Use the smallest possible sample size that satisfies the requirement in (1) to construct an
np
control chart. (5 points)
(3) [4 credit only] Calculate the Type I error rate for the control chart obtained in (2).
(10
points)
Problem 3 (25 points)
A control chart for nonconformities (
c
chart) is used to monitor a pizza bakery process.
The
quality characteristic of interest is the number of burnt spots on a pizza. At each sampling time,
two pizzas are randomly selected for inspection. The inspection unit is defined as one pizza. The
average number of nonconformities per sample when the process is in control is estimated to be
two.
(1) Find the appropriate three-sigma control limits for the
c
chart. (5 points)
(2) What is the probability of type I error for this control chart? (10 points)
(3) What is the sample size? (5 points)
(4) If a
u
chart is used, what are the three-sigma control limits? (5 points)
Problem 4 (25 points + 10 points)
It is known from Phase I data that we have 40 defects per 1000 products.
(1) One engineer suggests making a quality check every hour by selecting 100 products, how to
construct a
c
monitoring chart? (8 points)
(2) How many products should be checked at each sampling hour in order to obtain a positive
lower control limit in
c
chart? (12 points)
(3) One month later, this engineer decides to inspect 400 products at each sampling hour. Under
this new sampling strategy, would you suggest using
c
chart or
u
chart for continuously
monitoring this process? Justify your answer. (5 points)
(4) [4 credit only] For (3), if a
u
chart is suggested, what is a convenient inspection unit that
you would like to suggest? What is the sample size? What are the control limits (CL, UCL,
and LCL)? What are
c
0
and
u
0
? (10 points)
2
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