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机器学习quiz代写-ISSN 1532-0545
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I N F O R M S
Transactions on Education
Vol. 17, No. 1, September 2016, pp. 26–33
ISSN 1532-0545 (online)
http://dx.doi.org/10.1287/ited.2016.0158cs
© 2016 INFORMS
Case
ABCtronics: Manufacturing, Quality Control, and
Client Interfaces
Arnab Adhikari
Indian Institute of Management Calcutta, Joka, Kolkata 700104, India, arnaba10@email.iimcal.ac.in
Indranil Biswas
Indian Institute of Management Lucknow, Prabandh Nagar, Lucknow 226013, India, indranil@iiml.ac.in
Arnab Bisi
Johns Hopkins Carey Business School, Baltimore, Maryland 21202, abisi1@jhu.edu
Keywords : probability distributions; sampling distribution; confidence interval; hypothesis testing; linear
regression; case study
History : Received: June 2014; accepted: December 2015.
1. Introduction
“Today is going to be long,” the thought came to Phil
McDermott, as he casually checked the time on his
watch while negotiating a busy crowd in the high-
speed rail station. After arriving at the station, he
started for his office. As an intern at ABCtronics, he
never anticipated he would be asked to attend today’s
meeting. But Jim was adamant; last evening he had
said, “Look at this as a good learning opportunity.”
After that, Phil was left with little or no option.
The steel and glass-structured gigantic office build-
ing was already in sight. The location houses the man-
ufacturing facility of eight-inch fabrication of ABC-
tronics, along with other departments, such as the
quality and reliability team (QRT), the sales and mar-
keting team (SMT), and the customer interface team
(CIT). Phil directly reported to Jim Morris, the chief
operating officer. Today, all important vertical heads
of the plant were meeting for the quarterly review.
Quarterly reviews are routine processes in every man-
ufacturing company. Today was different. Complaints
from a major client site had increased manifold, and
in spite of boom time in the chip industry, ABCtron-
ics had not done well over the last couple of quarters.
“Tempers are going to run high today,” Phil thought
as he threw his finished coffee cup in the trash can
nearby and entered the office. But Phil wondered,
as an intern, how he could add value to such a
discussion?
2. Company Background
ABCtronics, a semiconductor manufacturing com-
pany, was established in 1997. It started its opera-
tions on a small scale. Over time, it had become a
medium scale enterprise. The company offers a vari-
ety of wafer product lines, such as a mixed-signal
integrated circuit, analog, and high-voltage circuit
boards. ABCtronics is dependent on one major client
(XYZsoft) for a good portion of its business. On the
other hand, the semiconductor manufacturing indus-
try is affected by a highly cyclical demand pattern.1
During upturns, semiconductor manufacturers have
to ensure they have sufficient production capacity
to meet high customer demand. During downturns,
companies must contend with excess capacity because
of weaker demand and high fixed costs associated
with manufacturing facilities. Market analysts already
predicted that the integrated chip market would face
1 According to noted market analyst Wahlstrom (2014), “Given the
cyclical nature of the chip industry, foundries tend to add too much pro-
duction capacity as they attempt to meet burgeoning demand during the
good times, yet are left with excess capacity and are on the hook for the
high fixed costs associated with their equipment during the bad times.”
26
Adhikari, Biswas, and Bisi: Case: ABCtronics
INFORMS Transactions on Education 17(1), pp. 26–33, © 2016 INFORMS 27
shrinkage in 2008–2009, and growth would resume
from 2010 onward (The Economist 2009). Despite the
increasing market size from 2010, sales revenue of
ABCtronics is not up to the mark compared to its ini-
tial year’s sales.
3. Industry Background
With a global sales figure approaching $24.70 billion,
the semiconductor industry has experienced continu-
ous sales revenue increases over the last three years
(Semiconductor Industry Association 2013). Samsung
has the highest installed wafer capacity with a pro-
duction capacity of approximately 1.9 million wafers
per month that represents 12.6% of the world’s total
capacity (CdrInfo Report 2014). Other industry lead-
ers are Taiwan Semiconductor Manufacturing Com-
pany (10%), Micron (9.3%), Toshiba/SanDisk (8%),
and SK Hynix (7%).
The industry faces an economic challenge for two
reasons: (i) cyclical nature of demand and (ii) the high
cost associated with research and development (R&D)
(McKinsey Report 2011). Ever increasing costs related
to upgrading the existing fabrication plants compli-
cates the scenario further. Demand cycles, though bad
for the entire industry, have proved to be a blessing
in disguise for underperformers. Semiconductor chip
manufacturers have invested heavily in R&D to meet
the expectations of Moore’s Law.2 As a result, com-
plexities of the chip design and costs have naturally
gone up.
A high cost of R&D has also led to substantial cap-
ital requirements for building state-of-the-art facili-
ties for wafer fabrication. A McKinsey report indi-
cates that “R&D spending amounted to approximately
17% of industry revenue for semiconductor companies (up
from 14% a decade earlier) versus 3% for automakers,”
(McKinsey Report 2011). As a result, the industry
also focuses on quality assurance processes. In the
semiconductor manufacturing lines, the uncertainty
regarding the health of processes and wafers often
leads to “major scrap events” as well as higher cost.
In case of mixed-signal IC chips, it can be as high as
50% of the manufacturing cost. One of the main ways
of tackling the quality problem is by monitoring some
key parameters for deviations. These kinds of controls
stem from statistical process control techniques.
2 Moore’s Law: The number of transistors in a dense integrated
circuit doubles approximately every two years. There is no fun-
damental obstacle to achieving device yields of 100%. At present,
packaging costs so far exceed the cost of the semiconductor struc-
ture itself that there is no incentive to improve yields, but they can
be raised as high as is economically justified (Moore 1965).
Figure 1 Wafer Shaping Process
Back seal
poly-silicon
Quality check
and packaging
Crystal growth
Ingot surface grinding
or notching
Wafer slicing by
multiple wire saw
Edge rounding
Lapping/Grinding
Donor anneal
Double/Single
side polish
Final polish
(CMP)
Edge polish
Epitaxy
Note. CMP: Chemical Mechanical Polishing.
4. Semiconductor Fabrication Process
The fabrication of integrated circuit (IC) chips is a
highly complex process that involves hundreds of
separate steps. The overall process lasts for several
weeks. At ABCtronics, hundreds of IC chips are fab-
ricated simultaneously on a six by eight-inch disc of
silicon, termed as a “wafer” (see Figure 1). The wafers
are processed in groups called “lots.” The circuit ele-
ments such as transistors, resistors, and capacitors are
manufactured in layers on the wafer, with alternate
deposition of material and exposure to light through a
mask; finally they are subjected to an etching process
that removes the unexposed material. The exposure
to light is referred as photo-lithography (see Figure 2).
This process itself contributes to the variability in the
quality of the IC chips. The features created in this
way are currently as small as 0.16 m (1 microme-
ter (m) = 1 × 10−6 meter), and therefore, fabrication
is required to be done in a virtually sterile environ-
ment, often referred to as a “clean room.” ABCtronics
Adhikari, Biswas, and Bisi: Case: ABCtronics
28 INFORMS Transactions on Education 17(1), pp. 26–33, © 2016 INFORMS
Figure 2 Photo-Lithography
Plasma etching
Metal deposition
Ion implantation
Oxidation
Photoresist
coating
Stepper exposure Optical mask
Photoresist
development
Acid etch
Spin, rinse,
and dry
Process steps
Photoresist
removal
enforces a strict quality control policy. Samples of
wafers from each lot are subjected to quality checks at
various steps during the process to assess the impact
of particular defects, the thickness of different lay-
ers, and the performance of test structures created in
the areas between the chips. At the end of the line,
each chip on every wafer is subjected to functional-
ity tests (to reduce probing time, testing of each chip
is stopped after the first failed functionality test; this
serves as an equivalent of a rejection rule).
The quantity and the complexity of the process and
associated testing have forced the quality improve-
ment efforts to focus primarily on summary statistics
such as the number of salable IC chips per lot. This
kind of figure has the advantage of suggesting sim-
ple and unambiguous screening rules (e.g., mark a
lot if less than a particular proportion of the IC chips
are usable). However, such one-dimensional analy-
sis fails to address more pressing issues such as the
possible causes of the defects, and crucial informa-
tion for process improvement. Large manufacturing
houses of IC chips are therefore always looking for
avenues to improve upon the quality checks and cus-
tomized design of such control measures. Similarly,
ABCtronics is also concerned about maintaining high
quality. From an internal investigation, the fabrication
process is currently facing two major problems: high
downtime and chemical impurities.
5. Quality Control Tests
The cost of quality assurance in manufacturing IC
chips is very high. Moreover, as the complexity of IC
chip design increases, the probability of faulty pro-
duction also goes up. Since it is practically impossi-
ble to attain 100% yield on any IC chip manufactur-
ing, quality checks are incorporated at several stages
of the manufacturing process. Industry experts put
emphasis on the requirement for routinely design-
ing statistical procedures to monitor the presence of
defect clustering.
The entire industry is increasingly turning toward
statistical methods for quality control. ABCtronics has
also adopted a similar approach. Analysis of produc-
tion data has revealed that the probability of produc-
ing a defective chip is 0.004. The company is cur-
rently considering whether to incorporate a new IC
growth technology, namely, “defect-free manufactur-
ing,” which can bring down the probability of pro-
ducing a defective chip to 0.002.
ABCtronics currently applies the Lot Acceptance
Testing Method (LATM) for quality check. An auto-
mated machine is employed to take a sample of 25 IC
chips one by one randomly from a lot of 500 without
replacement. It means the chip already drawn from
the lot is not returned to the lot when the next chip
is selected, and the lot size goes down. If the sample
has less than two defectives then the lot is accepted;
otherwise it gets rejected. The in-house testing team
has conducted their analysis and found that every lot
of 500 IC chips contains two defectives on average.
The QRT has proposed a new design for quality
control to the board for approval. Scrapping the exist-
ing quality control policy, LATM, they want to bring
in a new type of testing called Individual Chip Testing
Method (ICTM), designed based on “defect-free man-
ufacturing.” According to the proposal from QRT, a
sample of 25 IC chips is to be taken one by one from
a lot of 500, but this time they will allow replace-
ment. It signifies the chip already drawn from the lot
is returned to the lot when the next chip is selected,
and the lot size remains the same for each selection. If
a defective chip is found, the rework will immediately
be done on that chip. The rework is usually done by
performing a functional test on the internal circuitry
of that IC chip (Tsai and Ho 2000). The proposal was
aimed at establishing ABCtronics as a reliable brand
in the IC chips market and also for retaining a good
relationship with its biggest customer, XYZsoft.
Adhikari, Biswas, and Bisi: Case: ABCtronics
INFORMS Transactions on Education 17(1), pp. 26–33, © 2016 INFORMS 29
Figure 3 ABCtronics Organizational Structure
Board of directors
Chairman
Vice-chairman
Audit
committee
Internal
audit team
Chief executive officer
Chief financial officer Chief operating officer Legal advisory team
Research and
development
Sales and
marketing team
Operations
(plant president)
Quality and
reliability team
Customer
interface team Marketing
Materials
management Manufacturing
Shareholders
XYZsoft, one of the major clients of ABCtronics,
uses IC chips on their personal computers (PCs). In
a component of each of these PCs, three IC chips are
connected in series. It is known that the life (mea-
sured in years) of any IC chip follows an exponen-
tial distribution. XYZsoft uses chips with an iden-
tical specification in series. The ABCtronics fabrica-
tion team is now contemplating the option of rework
on returned and defective IC chips from XYZsoft to
investigate any problem with series connectivity of
the IC chips. QRT is proposing immediate rework on
IC chips with ICTM.
6. Review Meeting with the
Manufacturing Unit and QRT
As Phil entered the board room, most of the execu-
tives of ABCtronics had already arrived. Mark, the
head of QRT, and Robert, the head of SMT, were
there. They were having a last look at their respec-
tive files before the start of the meeting. ABCtron-
ics was a hierarchy-centric organization, typical of
any manufacturing-based firm (see Figure 3). He saw
that Stuart, the president of the fabrication plant, was
scribbling down something in his file. Stuart would
give the first presentation of the review meeting. They
were waiting for Jim. Phil found a corner seat in the
room for himself.
As Phil was going to settle, Jim entered the room.
Like a meticulous taskmaster, he set the agenda for
the meeting first: “Gentlemen, last evening, I received
yet another complaint from XYZsoft. It is the third
time in last six months. I have assured them that I will
personally look into the matter and ensure all possible
rectification measures. So, let’s get started. We have a
number of things such as improvement of operational
efficiency, rework issue, customer feedback, and sales
growth potential to discuss today.”
Stuart immediately started. After detailing out the
overall production of IC chips, he said, “We are cur-
rently producing 500 IC chips per lot. However, our
plant is also facing the problem of downtime. This
issue is particularly critical with our ion implanter.” It
immediately reminded Phil about one of the reports
that he read earlier. Downtime of this equipment is a
matter of concern for semiconductor companies. Min-
imization of the downtime would lead to improve-
ment of operational efficiency (Globenewswire 2007).
At this point, Jim interrupted and asked, “In terms
of downtime, what are we looking at, Stuart? How
big is the problem?” Stuart said, “We ran some anal-
ysis and based on our estimate, average downtime
of the machine is 6 hours (see Table 1). We need to
curtail it down to 5.” “How good is that chance?”
Jim enquired. Stuart glanced at Mark, the head of
QRT, and said, “Mark and I agree that a replacement
Adhikari, Biswas, and Bisi: Case: ABCtronics
30 INFORMS Transactions on Education 17(1), pp. 26–33, © 2016 INFORMS
Table 1 Data on Downtime and Chemical Impurity
Excerpt from the fabrication plant report on downtime of ion implantation
Data on monthly downtime of ion implantation (for last one year)
Month Downtime
July, 2012 3 hours 36 minutes
August, 2012 4 hours 48 minutes
September, 2012 4 hours 36 minutes
October, 2012 6 hours 40 minutes
November, 2012 5 hours 24 minutes
December, 2012 5 hours 40 minutes
January, 2013 7 hours 20 minutes
February, 2013 9 hours 24 minutes
March, 2013 8 hours 40 minutes
April, 2013 5 hours 06 minutes
May, 2013 6 hours 20 minutes
June, 2013 4 hours 40 minutes
is needed. I have a minor disagreement with Mark
regarding the possibility of achieving 5 hours of the
downtime.”
Mark was ready with his reply, “QRT ran some
tests and found that downtime of ion implanter has
a gamma distribution pattern and based on that we
have calculated the chance of reducing the downtime
to 5 hours. But Stuart says the ion implanter impacts
the overall production. As per his opinion, downtime
of ion implanter and subsequent activities follows a
uniform distribution instead. Here lies the difference
in opinions.” Jim looked at Stuart and asked, “Are
you sure that we are left with no other option but
replacing this machine?” Stuart nodded silently. Jim
pondered over the matter for some time and said,
“Well, let me think over this. Now, I will look at the
status of the chemical impurity problem we had last
month.”
IC chip fabrication process of ABCtronics involves
the chemical vapor deposition method. In this tech-
nique, several chemicals such as ammonia, hydrochlo-
ric acid, sulfuric acid, etc., used in various steps
of manufacturing, often contain impurities. To avoid
contamination, the percentage of impurities per lot in
a chemical should not go beyond a specific limit; oth-
erwise, it results in producing defective IC chips. The
manufacturing company itself sets the upper thresh-
old of chemical impurity percentage based on the
desired operating level of the production process.
Stuart went on to explain, “From the analysis of his-
torical data, we have concluded that the percentage of
impurities per lot in a chemical approximately follows
a beta distribution (see Figure 4). Based on this, we
have decided that if the percentage of impurities per
lot in any chemical is more than 30%, it is not used in
the fabrication process.” Jim did not look convinced.
“Is that good enough?” Mark came to Stuart’s aid this
time. He said, “QRT has checked that the policy is
Figure 4 Report on Chemical Impurity Found in Raw Materials
(Probability Density Function of the Percentage of Impurity
in Chemical X)
0
0.5
1.0
1.5
2.0
2.5
0 10 20 30 40 50 60 70 80 90 100
Pr
ob
ab
ili
ty
d
en
sit
y
fu
nc
tio
n
Percentage of impurity
Note. The percentage of impurity in chemical X follows beta distribution with
shape parameters = 4 and = 2.
working fine.” Jim remained skeptical. He waved at
Phil. As Phil approached him, Jim handed him over
a stack of papers and said, “Keep these with you, we
will work on this issue, later.”
At this point Mark commented, “Regarding the
proposed quality check technique ICTM, we need to
make a decision. Are we going to adopt this method
or not?” Now, Stuart commented, “But don’t you see,
our current system is flawed, and client complaints
will not stop coming?” Robert supported Stuart and
opined that it would affect the product delivery sys-
tem. At the table, everybody seemed clueless. Jim
intervened and said, “QRT gave me a report on ICTM
last week. I have asked Phil to look into the analysis.
Next Monday we will take the final decision on the
matter of ICTM.”
Phil scribbled on his scratch-pad, “ICTM Report—
urgent.” The truth was that he was yet to figure out
the flaw in the current testing procedure. The meeting
at hand was entering into the second phase.
7. Review Meeting with SMT and CIT
“Where are we with our client Customer PQRsys-
tems?” Jim directed his question toward Robert.
“They are looking for specifications such that the
product can work for six years smoothly. I have asked
Mark to give you a copy of our internal RT report3
(see Figure 5). After looking at it, we have replied to
their query. I think we have a good chance of secur-
ing this order.” Mark commented, “I think we should
be able to meet their expectations. I have checked the
data myself. Our chips will easily last more than six
years.” After hearing this, Jim said, “Mark, send me
a copy of RT. I would also like to have a look at the
details.” Mark nodded.
3 RT report refers to Rigorous Testing (RT) report. This process
checks for the lifespan of any product.
Adhikari, Biswas, and Bisi: Case: ABCtronics
INFORMS Transactions on Education 17(1), pp. 26–33, © 2016 INFORMS 31
Figure 5 Related Portion of Rigorous Testing Report (Probability
Density Function of Time Before Failure (in Years) of
IC Chips)
Excerpt from the Rigorous Testing Report on Life Expectancy of IC
Chips
The following graph represents the Early Life Failure Test (the chip
is tested to check whether it can survive for 40 years or not) result
of the LM98XX chip that was carried out. The result is presented
in the figure below.
0.6
1.1
1.6
2.1
2.6
3.1
3.6
4.1
4.6
5.1
5.6
0 5 10 15 20 25 30 35 40
Pr
ob
ab
ili
ty
d
en
si
ty
f
un
ct
io
n
Time before IC chip failure (in years)
Salient data points from the Early Life Failure Test are presented
below.
Time before failure Cumulative distribution
(in years) function of failure time
5 22.55
10 40.00
15 53.53
20 64.01
25 72.13
30 78.41
Notes. The time before failure of IC chips is exponentially distributed. The
figure depicts the probability density function of IC chips failure time,
whereas the cumulative distribution function of the same is presented in the
table.
“OK. Now, the next issue is XYZsoft. Why has
the number of complaints increased? What has hap-
pened, Robert?” Jim said as he was finishing his cup
of coffee. Stuart immediately quipped, “They have
again started experimenting with their quality con-
trol.” Robert smiled at him and said, “A few months
ago, XYZsoft started a module-wise testing of their
product. Circuit module M (CM) has a path where
three chips from ABCtronics get connected in a series.
Before the new testing process, XYZsoft reported that
in a typical lot comprising 20 CMs they are finding
three defective items. In most of those cases, they
observed that the problem was with our chips. Now,
they have put a stricter policy in place. They have
now started to calculate the number of nondefectives
before they encounter a particular number of defec-
tives, and they started the count of 3. Till this point
of time nothing happened.” Robert paused.
Phil could sense that the new policy did not have
much impact compared to the previous one. He pon-
dered in his mind, how would that be possible?
Robert started after taking a sip of his coffee, “The
problem began as their testing team proposed that
they should send back the whole lot for rework and
recheck as soon as one defective item is found. All hell
has broken loose since then. We are now flooded with
requests for rechecks from XYZsoft.” At this point,
Mark commented, “I am telling you, Jim, we can eas-
ily tackle this problem if we implement ICTM at our
end.” Jim looked at Mark but said nothing. The entire
episode puzzled him. How can a change in the qual-
ity control policy of XYZsoft have serious implications
on the business of ABCtronics? Jim replied, “We need
to tackle this problem quickly. XYZsoft is our biggest
client. We simply cannot afford to lose their business.”
Phil also found this development of events to be fas-
cinating. He had quietly jotted down whatever Robert
has said and looked at his pad. “What am I missing?”
he wondered on his own.
Robert continued, “I had a talk with Stuart regard-
ing this matter. He told me that [the] Susceptible
High Voltage Problem (SHVP)4 could contribute to
this kind of issue. His team is looking into this aspect
on the priority basis.” After hearing this, Jim looked
at Stuart and said, “I thought we dealt with this prob-
lem a couple of years ago.” Stuart replied, “We are
checking to be sure of the fact that it is not due to
SHVP. Tests would be complete within next week.”
Jim replied, “Do you have any preliminary report on
this test?”
Stuart said, “As per historical data, our IC chips
produce [a] minimum 2.7 V output on an average, as
HIGH signal. The variance of the HIGH signal out-
put voltage remains 1.8 V. We have received a number
of complaints from XYZsoft that the IC chips are not
producing the expected voltage. Then, I ordered to
take a random sample of 100 IC chips across the lots
and test them for SHVP. Initial reports suggest the
average voltage produced by the IC chips is around
2.3 V.” At this point, Mark commented, “Is it possi-
ble that we are overestimating the output of the IC
chips?” Stuart replied, “It may be the case. We would
not know for sure unless the detailed report comes.”
Phil could sense that the entire XYZsoft episode had
caused a fluttering feeling across the power corridor
of ABCtronics. They need a fast fix for the problem
and as of now none was in sight. A stifled silence
prevailed over the board room.
4 Susceptible High Voltage Problem (SHVP): Ideally, every IC
should produce HIGH signal output at 3.5 V, LOW signal output at
0.1 V, whereas 0.4 V–2.4 V is the undefined region of signal where
signal remains neither HIGH nor LOW. In reality, it is often found
that the HIGH signal output is around 2.5 V–3.0 V for a good IC.
A problem starts if the HIGH signal output of an IC’s lies between
0 V and 2.4 V. Then the signal is LOW or undefined, whereas it
should be HIGH. This problem is defined as SHVP (Tokheim 2004).
Adhikari, Biswas, and Bisi: Case: ABCtronics
32 INFORMS Transactions on Education 17(1), pp. 26–33, © 2016 INFORMS
Table 2 Customer Score Sheet
Sl. no. Customer name Customer score Range
1 Customer A 79 Good
2 Customer B 56 Satisfactory
3 Customer C 33 Needs improvement
4 Customer D 79 Good
5 Customer E 66 Good
6 Customer F 49 Satisfactory
7 Customer G 47 Satisfactory
8 Customer H 34 Needs improvement
9 Customer I 88 Very good
10 Customer J 77 Good
11 Customer K 67 Good
12 Customer L 51 Satisfactory
13 Customer M 53 Satisfactory
14 Customer N 74 Good
15 Customer O 85 Very good
16 Customer P 56 Satisfactory
17 Customer Q 39 Needs improvement
18 Customer R 51 Satisfactory
19 Customer S 26 Needs improvement
20 Customer T 43 Satisfactory
21 Customer U 77 Good
22 Customer V 97 Very good
23 Customer W 73 Good
24 Customer X 57 Satisfactory
25 Customer Y 66 Good
26 Customer Z 45 Satisfactory
27 Customer AA 28 Needs improvement
28 Customer AB 33 Needs improvement
29 Customer AC 56 Satisfactory
30 Customer AD 68 Good
31 Customer AE 32 Needs improvement
32 Customer AF 93 Very good
33 Customer AG 60 Good
34 Customer AH 29 Needs Improvement
35 Customer AI 41 Satisfactory
36 Customer AJ 42 Satisfactory
37 Customer AK 72 Good
38 Customer AL 48 Satisfactory
39 Customer AM 59 Satisfactory
40 Customer AN 47 Satisfactory
Note. Customer Score Range is defined as follows—Below 40: Needs
improvement, 40–59: Satisfactory, 60–79: Good, and 80–100: Very good.
Breaking the silence of the room, Jim spoke, “This
brings us to the last issue of the discussion today.
Where are we, regarding the analysis of customer
feedback?” Robert picked up a thick file and said, “It’s
all here. We ran the survey through 40 randomly cho-
sen customers of the 74XX chip family. Four of them
have rated us Very Good, eight have rated us Needs
Improvement, and 28 have given us either Good or Sat-
isfactory (see Table 2). So, I think we are good. Most
of them are happy with our products.” Jim noted
something in his diary. Then he said to Robert, “That
would be your hunch. That cannot be your analy-
sis.” Then Jim remembered something from an earlier
meeting. He asked Robert, “In the last meeting, there
was some discussion on redesigning the survey alto-
Table 3 Historical Sales Figure of ABCtronics
ABCtronics’ sales Overall market demand Price per Economic
Year volume (in millions) (in millions) chip (in $) condition∗
2004 2.39 297 0.832 0
2005 3.82 332 0.844 1
2006 3.33 195 0.854 0
2007 2.49 182 1.155 1
2008 1.56 93 1.303 0
2009 0.97 98 1.265 0
2010 1.32 198 1.368 1
2011 1.42 188 1.208 0
2012 1.48 285 1.234 1
2013 1.85 264 1.282 1
Note. ∗Economic condition: 1 signifies favorable market condition and 0 sig-
nifies otherwise.
gether. Why?” Robert looked visibly uncomfortable
with this question. He cleared his throat and replied,
“We have a mean customer rating of about 56. But the
overall spread of the score is very high. If we want
to conduct the survey to be sure of this average score
with 90% confidence, even with a margin error of 4
the required sample size may exceed our total cus-
tomer base.” Jim said, “What is the total number of
customers for 74XX?” Robert said, “Including the new
clients, the total is 70.” Jim said, “Sample more cus-
tomers if needed. And tell me how good would be
our estimate of the customer score?”
Robert presented the report on the sales figure (see
Table 3). ABCtronics was using a simple linear regres-
sion model for predicting the sales figure. However,
the new interns, who recently joined SMT, indicated
a few problems with the existing method. They pro-
posed a new method to predict sales. They argued
that their multiple linear regression model had better
explanatory power and was devoid of multicollinear-
ity problems. Looking at the presentation, Phil figured
out that he was also asked to predict the sales figure
for various demand scenarios. But Phil adopted a dif-
ferent approach. While looking at the screen, he did
some quick mental calculation, and the results were
not matching with the one presented. He nervously
glanced at Jim. Jim was listening to Robert’s analysis
with rapt attention.
As the meeting was coming to an end, Phil could
sense that ABCtronics needed to deal with a number
of issues, and they needed to do it quickly. The com-
pany was entering into the second quarter of the year.
ABCtronics immediately had to take corrective mea-
sures; otherwise it might be too late. As Stuart, Mark,
and Robert were leaving the board room, Phil waited
quietly for Jim. He knew that he was asked to attend
today’s meeting for a reason.
Adhikari, Biswas, and Bisi: Case: ABCtronics
INFORMS Transactions on Education 17(1), pp. 26–33, © 2016 INFORMS 33
Table 4 Market Demand Estimate
Sales volume 4Y Y 5 (in millions)
Total market demand
for PCs (in millions) Y Y > 3 105 ≤ Y Y ≤ 3 Y Y < 105 Subtotal
XX > 200 0.10 0.20 0.10 0.40
100 ≤ XX ≤ 200 0.10 0.10 0.20 0.40
XX < 100 0.00 0.10 0.10 0.20
Sub-total 0.20 0.40 0.40 1
Joint probability matrix of the sales volume and total market demand
Total market demand of PCs 4XX5 (in millions) Average sales volume 4Y Y 5 (in millions)
XX > 200 2.385
100 ≤ XX ≤ 200 2.140
XX < 100 1.265
Average sales volume in different demand scenarios
Note. Based on the sales figure of ABCtronics in Table 3, Phil has calculated the joint probability matrix for the sales
volume and total market demand, as well as the average sales volume in different demand scenarios.
8. The Road Ahead
As everybody left the room, Jim turned to Phil and
asked, “What do you think of today’s meeting?” Phil
kept quiet. Many aspects of the meeting left him con-
fused. Jim went on, “I need an honest opinion before
going ahead. Look at all the reports and analyze. Tell
me what do you think? You have two days to pre-
pare. Let us meet on Monday.” As Jim uttered those
words, only one thought came to Phil’s mind, “There
goes my weekend plan!”
“Have you prepared an analysis of the sales fig-
ures I asked for?” Phil said yes (see Table 4). “Do
you think we can sell more than 3 million chips this
year?” Phil said nothing. He has done the calcula-
tion; the possibility stands below 50% level. Jim fig-
ured out the answer from Phil’s silence. He asked,
“What about 2.5?” Phil replied, “I think we will fall
short of that number, the industry overall is experi-
encing a medium level demand. But the difference
is not very large. If we get an order from Customer
PQRsystems, probably we can make it.” Jim sighed
for a moment and said, “Let us see where we arrive
independently with our analysis. On Monday, after
our meeting, I shall meet with Stuart and Mark. We
have gone through a prolonged rough patch. The time
has come to take some course correction; otherwise
competition will knock us to the ground.”
9. Note
This case has been prepared to form the basis for class
discussion rather than to illustrate either effective or
ineffective handling of a business situation.
Supplemental Material
Supplemental material to this paper is available at http://dx
.doi.org/10.1287/ited.2016.0158cs.
Acknowledgments
We sincerely thank the editor-in-chief, the associate edi-
tor, and two anonymous referees for their insightful com-
ments and helpful suggestions. Their efforts have signifi-
cantly improved the case.
References
CdrInfo Report (2014) Samsung, TSMC, and Micron Top List of IC
Capacity Leaders. Accessed November 2, 2015, http://www
.cdrinfo.com/.
Economist, The (2009) The semiconductor industry: Under new man-
agement. Accessed November 2, 2015, http://www.economist
.com/.
Globenewswire (2007) ATMI Announces Revolutionary Auto clean
Technology Offering Greater Process Efficiency by Reducing
Ion Implant Equipment Downtime. Accessed November 2,
2015, http://globenewswire.com/.
McKinsey Report (2011) Creating value in the semiconductor indus-
try. Accessed November 2, 2015, http://www.mckinsey.com/.
Moore GE (1965) Cramming more components onto integrated cir-
cuits. Electronics (April 19), 114-117.
Semiconductor Industry Association (2013) Global semiconductor
sales jump by largest margin in over three years. Accessed
November 2, 2015, http://www.semiconductors.org/.
Tokheim RL (2004) Digital Electronics: Principles and Applications, 6th
ed. (McGraw-Hill, New York).
Tsai MC, Ho HC (2000) U.S. Patent No. 6,159,838. Washington, DC:
U.S. Patent and Trademark Office, Alexandria, VA. (Accessed
November 2, 2015).
Wahlstrom P (2014) Mobile chips are driving strong demand for
TSMC’s manufacturing services. Accessed November 2, 2015,
http://analysisreport.morningstar.com/.
1
Reading Material for Students
Arnab Adhikari
Indian Institute of Management Calcutta, Joka, Kolkata 700104, India, arnaba10@email.iimcal.ac.in
Indranil Biswas
Indian Institute of Management Lucknow, Prabandh Nagar, Lucknow 226013, India, indranil@iiml.ac.in
Arnab Bisi
Johns Hopkins Carey Business School, 100 International Drive, Baltimore, Maryland 21202, abisi1@jhu.edu
Probability Distributions
Binomial distribution. The Binomial distribution describes the probability of exactly successes
out of trials; the probability associated with a success in a single trial is given by and that with
a failure is given by 1 − (also designated by ). The expression of the probability mass function
(pmf) of this distribution is as follows
(; , ) = (
)(1 − )−,
where the variable and the parameter are integers, satisfying the conditions 0 ≤ ≤ and
> 0. The parameter is a real quantity and ∈ [0,1]. The expected value and the variance of a
random variable X having binomial distribution can be expressed as follows:
E X N p and (1 )Var X N p p .
Hypergeometric distribution. The hypergeometric distribution describes the experiment where out
of total elements, possesses a certain attribute [and the remaining ( – ) does not]; if we
then choose elements at random without replacement, (; , , ) gives the probability that
exactly of the selected n elements have come from the group of elements that possesses the
attribute. Let the number of elements with that certain attribute be denoted by X. The probability
mass function (pmf) of X with hypergeometric distribution is given by
(; , , ) =
(
)(−
−
)
(
)
where is discrete and its range is given by: ∈ [max(0, − + ) , min (, )]. The
parameters , and are all integers and satisfy the following conditions: 1 ≤ ≤ , ≥ 1
2
and ≥ 1. Let probability of success be represented by
M
p
N
. Then, the expected value and
the variance of X under hypergeometric distribution can be expressed as follows:
npXE and
)1(
)()1(
N
nNpnp
XVar .
In real life, when a marketing group is trying to understand their customer base by testing a set
of known customers for over-representation of various demographic subgroups, they use
hypergeometric test designed based on hypergeometric distribution.
Negative Binomial distribution. The negative binomial distribution (also known as Pascal
distribution) gives the probability of waiting for exactly trials until ℎ success has occurred.
Let the number of trials before ℎ success be denoted by X. Here and (= 1 − ) designates
the probability of a success and a failure in a single trial, respectively. The probability mass
function (pmf) of this distribution is given by
(; , ) = (−1
−1
)(1 − )−,
where the variable and parameter are integers and satisfies the following condition: ≥ >
0. Now, the expected value and the variance of a random variable X under negative binomial
distribution can be expressed as follows:
p
pk
XE
)1(
and
2
)1(
p
pk
XVar
.
The negative binomial distribution has applications in the insurance industry, where for
example the rate at which people have accidents is affected by a random variable like the weather
condition.
Geometric distribution. The geometric distribution is a special case of the negative binomial
distribution discussed above with = 1. It expresses the probability of waiting for exactly x trials
before the occurrence of the first successful event. Let the number of trials before the first success
be denoted by X. Then, the probability mass function (pmf) of X with this distribution is given by
(; ) = (1 − )−1,
where p denotes the probability of success in each trial. The expected value and the variance of a
random variable X under geometric distribution can be expressed as follows:
3
p
p
XE
)1(
and
2
)1(
p
p
XVar
.
In real life, if a NGO wants to know the number of male births before one female birth
regarding the study of sex ratio in human population then it can use this kind of distribution.
Poisson distribution. The Poisson distribution gives the probability of finding occurrence of
exactly events in a given length of time when the events are independent in nature and happens
at a constant rate, given by . The probability mass function (pmf) of this distribution is given by
!
);(
x
e
xf
x
,
where the variable is a positive integer and the parameter is a real positive quantity. Now, the
expected value and the variance of a random variable X under Poisson distribution can be
expressed as follows:
XE and XVar .
When the value of N is very large and p is very small in the binomial distribution described
before, then it can be approximated by a Poisson distribution with expected value = Np. Poisson
distribution is applied to determine the probability of rare events like birth defects, genetic
mutations, car accidents, etc.
Uniform distribution. If a continuous random variable X follows the uniform distribution, then its
probability density function (pdf) is given by the expression
(; , ) =
1
−
for ≤ ≤ .
The expected value and the variance of a random variable X under uniform distribution can be
expressed as follows
2
ab
XE
and
12
2
ab
XVar
.
In oil exploration, the position of the oil-water contact in a potential prospect is often
considered to be uniformly distributed.
Exponential distribution. If a continuous random variable X follows the exponential distribution,
then its pdf can be expressed as follows:
4
(; ) =
1
−
,
where represents the scale parameter. The expected value and the variance of a random variable
X under exponential distribution are given by:
XE and 2XVar .
In real life, the radioactive or particle decays is considered to follow exponential distribution.
Normal distribution. The normal distribution (also called the Gauss distribution) is one of the most
important distributions in statistics. The pdf of normal distribution is given by the following
expression:
(; , 2) =
1
√2
−
1
2
(
−
)
2
,
where is the mean or expected value and 2 is the variance of the distribution. For = 0 and
= 1, the distribution is called the standard normal distribution. It has widespread applications in
natural and social sciences, financial models, etc.
Beta distribution. The beta distribution has been applied to model the behavior of random
variables limited to intervals of finite length in a wide variety of disciplines. The pdf of beta
distribution is given by:
(; , ) =
1
(,)
−1(1 − )−1,
where the shape parameters and are positive real numbers, and the variable satisfies the
condition 0 ≤ ≤ 1. (, ) designates the beta function and is given by the following expression
(, ) =
Γ()Γ()
Γ(+)
.
For ∈ ℝ+, the gamma function Γ() is defined by the integral
Γ() = ∫ −1−
∞
0
.
When = = 1 , the beta distribution assumes the form of the uniform distribution between
0 and 1; when = = 2 the distribution takes parabolic shape; when = 2 and = 1 or vise
versa the distribution takes triangular shaped distribution. The expected value and the variance of
a random variable X under beta distribution can be expressed as follows:
XE and
2
)1(
XVar .
5
Beta distribution is usually applied to determine the time allocation in project management/
control systems, heterogeneity in the probability of HIV transmission, etc.
Gamma distribution. It is a two-parameter family of continuous probability distributions.
Exponential distribution is a special case of the gamma distribution. The pdf of gamma distribution
can be represented by the following functional form:
(; , ) =
−1
−
Γ()
,
where the shape parameter and the scale parameter are positive real numbers ( ∈ ℝ+ and ∈
ℝ+) and the variable is also a positive real number ( ∈ ℝ+). The expected value and the
variance of a random variable X under gamma distribution are given by:
kXE and 2kXVar .
Sampling Distribution and Confidence Interval. If we take repeated samples from the same
population, samples means x would vary from sample to sample and form a sampling
distribution of sample means. It explains the random behavior of a sample mean. The variability
of x from can be obtained by determining the variance of x . The variance of the sample mean
with a sample of size n is given by:
n
x
2
2
.
Next, the confidence interval contains the true population parameter. A confidence interval
comprise point estimate, i.e., the best estimate of the population parameter from the sample statistic
and the margin of error or maximum sampling error (the maximum accepted difference between
the true population parameter and a sample estimate of that parameter). The confidence interval
where lies can be determined by the following expression:
n
zx
n
zx
2/2/
.
The confidence level is denoted by 100 1 % . The margin of error denoted by E is given
by the following formula:
n
zE
2/
.
6
From the formula given above, the required minimum sample size can be easily obtained and
it is given by:
2
)2/(
E
zn
.
Hypothesis Testing. Hypothesis testing is a technique to check with the help of a sample data
whether a claim or hypothesis about a population parameter is true or not. In hypothesis testing,
the stated conjecture defined as the null hypothesis can be disproved, but it cannot be proved.
However, by disproving the null hypothesis, one can prove that the contrary is true. The contrary
of the null hypothesis is termed as the alternative hypothesis. The test statistic represents the value
determined using the sample data. A test statistic for testing a hypothesis on population mean is
given by the following formula:
n
x
z
0 ,
where
0
denotes the hypothesized value of the population mean. Following are the null (
0
H )
and alternative (
A
H ) hypotheses for three standard tests on population mean:
The “Two-Tailed” Test.
The “One-Tailed” Test to the Right
0 0
0
0
0 / 2 / 2
:
:
reject if or .
A
H
H
x
z
n
H z z z z
0 0
0
0
0
:
:
reject if .
A
H
H
x
z
n
H z z
7
The “One-Tailed” Test to the Left
0 0
0
0
0
:
:
reject if .
A
H
H
x
z
n
H z z
Regression Models
Simple linear regression
Here we present a simple linear regression model to determine the relationship between the
dependent variable Y and the independent variable X, captured by the following equation:
E( | ) = + . Then the regression model can be designated as: = + + , where
= − E( | ) is a random variable or an error term with E() = 0 and ( ) = 2. If
̂ and ̂ denote the best estimates of the parameters α and β , respectively, then the estimated linear
regression equation of Y on X is:
̂ = ̂ + ̂.
Multiple linear regression
The effect of independent variables 1 , 2 and 3 on the dependent variable Y can be captured
by the following equation:
E( | 1, 2, 3) = + 11 + 22 + 33,
where = − E( | 1, 2, 3) is a random variable or an error term with E() =
0 and ( ) = 2. If ̂ , 1̂, 2̂, and 3̂ denote the best estimates of the parameters
α , β
1
, β
2
, and β
3
, respectively, then the estimated multiple linear regression equation of Y on
1 , 2 and 3 is given by:
̂ = + 1̂1 + 2̂2 + 3̂3.
Multicollinearity check
Often regression model is affected by linear relationship between independent variables termed as
‘multicollinearity’. Variance Inflation Factor (VIF) is one of the conventional techniques
employed to check whether any multicollinearity exists or not. VIF between two independent
variables X1 and X2 can be determined by the following expression:
8
VIF1,2 =
1
1−1,2
2 ,
where 1,2
2 denotes the co-efficient of determination between 1 and 2 . If the value of VIF is
greater than 5, then it indicates multicollinearity and the overall regression model gets affected by
it.
Sources
Anderson, D., Sweeney, D., Williams, T., Camm, J., Cochran, J. 2011. Statistics for
Business & Economics, 11th ed. Cengage Learning, Mason.
Berenson, M., Levine, D., Krehbiel, T. C. 2011. Basic business statistics: Concepts and
applications. Pearson Education, New Jersey.
Groebner, D. F., Shannon, P. W., Fry, P. C., Smith, K. D. 2013. Business statistics: a
decision making approach, 9th ed. Pearson Education, New Jersey.
Hildebrand, D. K. and O. Lyman. 1998. Statistical Thinking for Managers, 4th ed. Duxbury
Press, California.
Levin, R. I. and D. S. Rubin. 1997. Statistics for Management, 7th ed. Prentice Hall
International, New Jersey.
http://wps.aw.com/wps/media/objects/15/15512/formulas.pdf
http://www.nzqa.govt.nz/assets/qualifications-and-standards/qualifications/ncea/NCEA-
subject-resources/Mathematics/L3-Stats-Formulae-2013.pdf
学霸联盟
Transactions on Education
Vol. 17, No. 1, September 2016, pp. 26–33
ISSN 1532-0545 (online)
http://dx.doi.org/10.1287/ited.2016.0158cs
© 2016 INFORMS
Case
ABCtronics: Manufacturing, Quality Control, and
Client Interfaces
Arnab Adhikari
Indian Institute of Management Calcutta, Joka, Kolkata 700104, India, arnaba10@email.iimcal.ac.in
Indranil Biswas
Indian Institute of Management Lucknow, Prabandh Nagar, Lucknow 226013, India, indranil@iiml.ac.in
Arnab Bisi
Johns Hopkins Carey Business School, Baltimore, Maryland 21202, abisi1@jhu.edu
Keywords : probability distributions; sampling distribution; confidence interval; hypothesis testing; linear
regression; case study
History : Received: June 2014; accepted: December 2015.
1. Introduction
“Today is going to be long,” the thought came to Phil
McDermott, as he casually checked the time on his
watch while negotiating a busy crowd in the high-
speed rail station. After arriving at the station, he
started for his office. As an intern at ABCtronics, he
never anticipated he would be asked to attend today’s
meeting. But Jim was adamant; last evening he had
said, “Look at this as a good learning opportunity.”
After that, Phil was left with little or no option.
The steel and glass-structured gigantic office build-
ing was already in sight. The location houses the man-
ufacturing facility of eight-inch fabrication of ABC-
tronics, along with other departments, such as the
quality and reliability team (QRT), the sales and mar-
keting team (SMT), and the customer interface team
(CIT). Phil directly reported to Jim Morris, the chief
operating officer. Today, all important vertical heads
of the plant were meeting for the quarterly review.
Quarterly reviews are routine processes in every man-
ufacturing company. Today was different. Complaints
from a major client site had increased manifold, and
in spite of boom time in the chip industry, ABCtron-
ics had not done well over the last couple of quarters.
“Tempers are going to run high today,” Phil thought
as he threw his finished coffee cup in the trash can
nearby and entered the office. But Phil wondered,
as an intern, how he could add value to such a
discussion?
2. Company Background
ABCtronics, a semiconductor manufacturing com-
pany, was established in 1997. It started its opera-
tions on a small scale. Over time, it had become a
medium scale enterprise. The company offers a vari-
ety of wafer product lines, such as a mixed-signal
integrated circuit, analog, and high-voltage circuit
boards. ABCtronics is dependent on one major client
(XYZsoft) for a good portion of its business. On the
other hand, the semiconductor manufacturing indus-
try is affected by a highly cyclical demand pattern.1
During upturns, semiconductor manufacturers have
to ensure they have sufficient production capacity
to meet high customer demand. During downturns,
companies must contend with excess capacity because
of weaker demand and high fixed costs associated
with manufacturing facilities. Market analysts already
predicted that the integrated chip market would face
1 According to noted market analyst Wahlstrom (2014), “Given the
cyclical nature of the chip industry, foundries tend to add too much pro-
duction capacity as they attempt to meet burgeoning demand during the
good times, yet are left with excess capacity and are on the hook for the
high fixed costs associated with their equipment during the bad times.”
26
Adhikari, Biswas, and Bisi: Case: ABCtronics
INFORMS Transactions on Education 17(1), pp. 26–33, © 2016 INFORMS 27
shrinkage in 2008–2009, and growth would resume
from 2010 onward (The Economist 2009). Despite the
increasing market size from 2010, sales revenue of
ABCtronics is not up to the mark compared to its ini-
tial year’s sales.
3. Industry Background
With a global sales figure approaching $24.70 billion,
the semiconductor industry has experienced continu-
ous sales revenue increases over the last three years
(Semiconductor Industry Association 2013). Samsung
has the highest installed wafer capacity with a pro-
duction capacity of approximately 1.9 million wafers
per month that represents 12.6% of the world’s total
capacity (CdrInfo Report 2014). Other industry lead-
ers are Taiwan Semiconductor Manufacturing Com-
pany (10%), Micron (9.3%), Toshiba/SanDisk (8%),
and SK Hynix (7%).
The industry faces an economic challenge for two
reasons: (i) cyclical nature of demand and (ii) the high
cost associated with research and development (R&D)
(McKinsey Report 2011). Ever increasing costs related
to upgrading the existing fabrication plants compli-
cates the scenario further. Demand cycles, though bad
for the entire industry, have proved to be a blessing
in disguise for underperformers. Semiconductor chip
manufacturers have invested heavily in R&D to meet
the expectations of Moore’s Law.2 As a result, com-
plexities of the chip design and costs have naturally
gone up.
A high cost of R&D has also led to substantial cap-
ital requirements for building state-of-the-art facili-
ties for wafer fabrication. A McKinsey report indi-
cates that “R&D spending amounted to approximately
17% of industry revenue for semiconductor companies (up
from 14% a decade earlier) versus 3% for automakers,”
(McKinsey Report 2011). As a result, the industry
also focuses on quality assurance processes. In the
semiconductor manufacturing lines, the uncertainty
regarding the health of processes and wafers often
leads to “major scrap events” as well as higher cost.
In case of mixed-signal IC chips, it can be as high as
50% of the manufacturing cost. One of the main ways
of tackling the quality problem is by monitoring some
key parameters for deviations. These kinds of controls
stem from statistical process control techniques.
2 Moore’s Law: The number of transistors in a dense integrated
circuit doubles approximately every two years. There is no fun-
damental obstacle to achieving device yields of 100%. At present,
packaging costs so far exceed the cost of the semiconductor struc-
ture itself that there is no incentive to improve yields, but they can
be raised as high as is economically justified (Moore 1965).
Figure 1 Wafer Shaping Process
Back seal
poly-silicon
Quality check
and packaging
Crystal growth
Ingot surface grinding
or notching
Wafer slicing by
multiple wire saw
Edge rounding
Lapping/Grinding
Donor anneal
Double/Single
side polish
Final polish
(CMP)
Edge polish
Epitaxy
Note. CMP: Chemical Mechanical Polishing.
4. Semiconductor Fabrication Process
The fabrication of integrated circuit (IC) chips is a
highly complex process that involves hundreds of
separate steps. The overall process lasts for several
weeks. At ABCtronics, hundreds of IC chips are fab-
ricated simultaneously on a six by eight-inch disc of
silicon, termed as a “wafer” (see Figure 1). The wafers
are processed in groups called “lots.” The circuit ele-
ments such as transistors, resistors, and capacitors are
manufactured in layers on the wafer, with alternate
deposition of material and exposure to light through a
mask; finally they are subjected to an etching process
that removes the unexposed material. The exposure
to light is referred as photo-lithography (see Figure 2).
This process itself contributes to the variability in the
quality of the IC chips. The features created in this
way are currently as small as 0.16 m (1 microme-
ter (m) = 1 × 10−6 meter), and therefore, fabrication
is required to be done in a virtually sterile environ-
ment, often referred to as a “clean room.” ABCtronics
Adhikari, Biswas, and Bisi: Case: ABCtronics
28 INFORMS Transactions on Education 17(1), pp. 26–33, © 2016 INFORMS
Figure 2 Photo-Lithography
Plasma etching
Metal deposition
Ion implantation
Oxidation
Photoresist
coating
Stepper exposure Optical mask
Photoresist
development
Acid etch
Spin, rinse,
and dry
Process steps
Photoresist
removal
enforces a strict quality control policy. Samples of
wafers from each lot are subjected to quality checks at
various steps during the process to assess the impact
of particular defects, the thickness of different lay-
ers, and the performance of test structures created in
the areas between the chips. At the end of the line,
each chip on every wafer is subjected to functional-
ity tests (to reduce probing time, testing of each chip
is stopped after the first failed functionality test; this
serves as an equivalent of a rejection rule).
The quantity and the complexity of the process and
associated testing have forced the quality improve-
ment efforts to focus primarily on summary statistics
such as the number of salable IC chips per lot. This
kind of figure has the advantage of suggesting sim-
ple and unambiguous screening rules (e.g., mark a
lot if less than a particular proportion of the IC chips
are usable). However, such one-dimensional analy-
sis fails to address more pressing issues such as the
possible causes of the defects, and crucial informa-
tion for process improvement. Large manufacturing
houses of IC chips are therefore always looking for
avenues to improve upon the quality checks and cus-
tomized design of such control measures. Similarly,
ABCtronics is also concerned about maintaining high
quality. From an internal investigation, the fabrication
process is currently facing two major problems: high
downtime and chemical impurities.
5. Quality Control Tests
The cost of quality assurance in manufacturing IC
chips is very high. Moreover, as the complexity of IC
chip design increases, the probability of faulty pro-
duction also goes up. Since it is practically impossi-
ble to attain 100% yield on any IC chip manufactur-
ing, quality checks are incorporated at several stages
of the manufacturing process. Industry experts put
emphasis on the requirement for routinely design-
ing statistical procedures to monitor the presence of
defect clustering.
The entire industry is increasingly turning toward
statistical methods for quality control. ABCtronics has
also adopted a similar approach. Analysis of produc-
tion data has revealed that the probability of produc-
ing a defective chip is 0.004. The company is cur-
rently considering whether to incorporate a new IC
growth technology, namely, “defect-free manufactur-
ing,” which can bring down the probability of pro-
ducing a defective chip to 0.002.
ABCtronics currently applies the Lot Acceptance
Testing Method (LATM) for quality check. An auto-
mated machine is employed to take a sample of 25 IC
chips one by one randomly from a lot of 500 without
replacement. It means the chip already drawn from
the lot is not returned to the lot when the next chip
is selected, and the lot size goes down. If the sample
has less than two defectives then the lot is accepted;
otherwise it gets rejected. The in-house testing team
has conducted their analysis and found that every lot
of 500 IC chips contains two defectives on average.
The QRT has proposed a new design for quality
control to the board for approval. Scrapping the exist-
ing quality control policy, LATM, they want to bring
in a new type of testing called Individual Chip Testing
Method (ICTM), designed based on “defect-free man-
ufacturing.” According to the proposal from QRT, a
sample of 25 IC chips is to be taken one by one from
a lot of 500, but this time they will allow replace-
ment. It signifies the chip already drawn from the lot
is returned to the lot when the next chip is selected,
and the lot size remains the same for each selection. If
a defective chip is found, the rework will immediately
be done on that chip. The rework is usually done by
performing a functional test on the internal circuitry
of that IC chip (Tsai and Ho 2000). The proposal was
aimed at establishing ABCtronics as a reliable brand
in the IC chips market and also for retaining a good
relationship with its biggest customer, XYZsoft.
Adhikari, Biswas, and Bisi: Case: ABCtronics
INFORMS Transactions on Education 17(1), pp. 26–33, © 2016 INFORMS 29
Figure 3 ABCtronics Organizational Structure
Board of directors
Chairman
Vice-chairman
Audit
committee
Internal
audit team
Chief executive officer
Chief financial officer Chief operating officer Legal advisory team
Research and
development
Sales and
marketing team
Operations
(plant president)
Quality and
reliability team
Customer
interface team Marketing
Materials
management Manufacturing
Shareholders
XYZsoft, one of the major clients of ABCtronics,
uses IC chips on their personal computers (PCs). In
a component of each of these PCs, three IC chips are
connected in series. It is known that the life (mea-
sured in years) of any IC chip follows an exponen-
tial distribution. XYZsoft uses chips with an iden-
tical specification in series. The ABCtronics fabrica-
tion team is now contemplating the option of rework
on returned and defective IC chips from XYZsoft to
investigate any problem with series connectivity of
the IC chips. QRT is proposing immediate rework on
IC chips with ICTM.
6. Review Meeting with the
Manufacturing Unit and QRT
As Phil entered the board room, most of the execu-
tives of ABCtronics had already arrived. Mark, the
head of QRT, and Robert, the head of SMT, were
there. They were having a last look at their respec-
tive files before the start of the meeting. ABCtron-
ics was a hierarchy-centric organization, typical of
any manufacturing-based firm (see Figure 3). He saw
that Stuart, the president of the fabrication plant, was
scribbling down something in his file. Stuart would
give the first presentation of the review meeting. They
were waiting for Jim. Phil found a corner seat in the
room for himself.
As Phil was going to settle, Jim entered the room.
Like a meticulous taskmaster, he set the agenda for
the meeting first: “Gentlemen, last evening, I received
yet another complaint from XYZsoft. It is the third
time in last six months. I have assured them that I will
personally look into the matter and ensure all possible
rectification measures. So, let’s get started. We have a
number of things such as improvement of operational
efficiency, rework issue, customer feedback, and sales
growth potential to discuss today.”
Stuart immediately started. After detailing out the
overall production of IC chips, he said, “We are cur-
rently producing 500 IC chips per lot. However, our
plant is also facing the problem of downtime. This
issue is particularly critical with our ion implanter.” It
immediately reminded Phil about one of the reports
that he read earlier. Downtime of this equipment is a
matter of concern for semiconductor companies. Min-
imization of the downtime would lead to improve-
ment of operational efficiency (Globenewswire 2007).
At this point, Jim interrupted and asked, “In terms
of downtime, what are we looking at, Stuart? How
big is the problem?” Stuart said, “We ran some anal-
ysis and based on our estimate, average downtime
of the machine is 6 hours (see Table 1). We need to
curtail it down to 5.” “How good is that chance?”
Jim enquired. Stuart glanced at Mark, the head of
QRT, and said, “Mark and I agree that a replacement
Adhikari, Biswas, and Bisi: Case: ABCtronics
30 INFORMS Transactions on Education 17(1), pp. 26–33, © 2016 INFORMS
Table 1 Data on Downtime and Chemical Impurity
Excerpt from the fabrication plant report on downtime of ion implantation
Data on monthly downtime of ion implantation (for last one year)
Month Downtime
July, 2012 3 hours 36 minutes
August, 2012 4 hours 48 minutes
September, 2012 4 hours 36 minutes
October, 2012 6 hours 40 minutes
November, 2012 5 hours 24 minutes
December, 2012 5 hours 40 minutes
January, 2013 7 hours 20 minutes
February, 2013 9 hours 24 minutes
March, 2013 8 hours 40 minutes
April, 2013 5 hours 06 minutes
May, 2013 6 hours 20 minutes
June, 2013 4 hours 40 minutes
is needed. I have a minor disagreement with Mark
regarding the possibility of achieving 5 hours of the
downtime.”
Mark was ready with his reply, “QRT ran some
tests and found that downtime of ion implanter has
a gamma distribution pattern and based on that we
have calculated the chance of reducing the downtime
to 5 hours. But Stuart says the ion implanter impacts
the overall production. As per his opinion, downtime
of ion implanter and subsequent activities follows a
uniform distribution instead. Here lies the difference
in opinions.” Jim looked at Stuart and asked, “Are
you sure that we are left with no other option but
replacing this machine?” Stuart nodded silently. Jim
pondered over the matter for some time and said,
“Well, let me think over this. Now, I will look at the
status of the chemical impurity problem we had last
month.”
IC chip fabrication process of ABCtronics involves
the chemical vapor deposition method. In this tech-
nique, several chemicals such as ammonia, hydrochlo-
ric acid, sulfuric acid, etc., used in various steps
of manufacturing, often contain impurities. To avoid
contamination, the percentage of impurities per lot in
a chemical should not go beyond a specific limit; oth-
erwise, it results in producing defective IC chips. The
manufacturing company itself sets the upper thresh-
old of chemical impurity percentage based on the
desired operating level of the production process.
Stuart went on to explain, “From the analysis of his-
torical data, we have concluded that the percentage of
impurities per lot in a chemical approximately follows
a beta distribution (see Figure 4). Based on this, we
have decided that if the percentage of impurities per
lot in any chemical is more than 30%, it is not used in
the fabrication process.” Jim did not look convinced.
“Is that good enough?” Mark came to Stuart’s aid this
time. He said, “QRT has checked that the policy is
Figure 4 Report on Chemical Impurity Found in Raw Materials
(Probability Density Function of the Percentage of Impurity
in Chemical X)
0
0.5
1.0
1.5
2.0
2.5
0 10 20 30 40 50 60 70 80 90 100
Pr
ob
ab
ili
ty
d
en
sit
y
fu
nc
tio
n
Percentage of impurity
Note. The percentage of impurity in chemical X follows beta distribution with
shape parameters = 4 and = 2.
working fine.” Jim remained skeptical. He waved at
Phil. As Phil approached him, Jim handed him over
a stack of papers and said, “Keep these with you, we
will work on this issue, later.”
At this point Mark commented, “Regarding the
proposed quality check technique ICTM, we need to
make a decision. Are we going to adopt this method
or not?” Now, Stuart commented, “But don’t you see,
our current system is flawed, and client complaints
will not stop coming?” Robert supported Stuart and
opined that it would affect the product delivery sys-
tem. At the table, everybody seemed clueless. Jim
intervened and said, “QRT gave me a report on ICTM
last week. I have asked Phil to look into the analysis.
Next Monday we will take the final decision on the
matter of ICTM.”
Phil scribbled on his scratch-pad, “ICTM Report—
urgent.” The truth was that he was yet to figure out
the flaw in the current testing procedure. The meeting
at hand was entering into the second phase.
7. Review Meeting with SMT and CIT
“Where are we with our client Customer PQRsys-
tems?” Jim directed his question toward Robert.
“They are looking for specifications such that the
product can work for six years smoothly. I have asked
Mark to give you a copy of our internal RT report3
(see Figure 5). After looking at it, we have replied to
their query. I think we have a good chance of secur-
ing this order.” Mark commented, “I think we should
be able to meet their expectations. I have checked the
data myself. Our chips will easily last more than six
years.” After hearing this, Jim said, “Mark, send me
a copy of RT. I would also like to have a look at the
details.” Mark nodded.
3 RT report refers to Rigorous Testing (RT) report. This process
checks for the lifespan of any product.
Adhikari, Biswas, and Bisi: Case: ABCtronics
INFORMS Transactions on Education 17(1), pp. 26–33, © 2016 INFORMS 31
Figure 5 Related Portion of Rigorous Testing Report (Probability
Density Function of Time Before Failure (in Years) of
IC Chips)
Excerpt from the Rigorous Testing Report on Life Expectancy of IC
Chips
The following graph represents the Early Life Failure Test (the chip
is tested to check whether it can survive for 40 years or not) result
of the LM98XX chip that was carried out. The result is presented
in the figure below.
0.6
1.1
1.6
2.1
2.6
3.1
3.6
4.1
4.6
5.1
5.6
0 5 10 15 20 25 30 35 40
Pr
ob
ab
ili
ty
d
en
si
ty
f
un
ct
io
n
Time before IC chip failure (in years)
Salient data points from the Early Life Failure Test are presented
below.
Time before failure Cumulative distribution
(in years) function of failure time
5 22.55
10 40.00
15 53.53
20 64.01
25 72.13
30 78.41
Notes. The time before failure of IC chips is exponentially distributed. The
figure depicts the probability density function of IC chips failure time,
whereas the cumulative distribution function of the same is presented in the
table.
“OK. Now, the next issue is XYZsoft. Why has
the number of complaints increased? What has hap-
pened, Robert?” Jim said as he was finishing his cup
of coffee. Stuart immediately quipped, “They have
again started experimenting with their quality con-
trol.” Robert smiled at him and said, “A few months
ago, XYZsoft started a module-wise testing of their
product. Circuit module M (CM) has a path where
three chips from ABCtronics get connected in a series.
Before the new testing process, XYZsoft reported that
in a typical lot comprising 20 CMs they are finding
three defective items. In most of those cases, they
observed that the problem was with our chips. Now,
they have put a stricter policy in place. They have
now started to calculate the number of nondefectives
before they encounter a particular number of defec-
tives, and they started the count of 3. Till this point
of time nothing happened.” Robert paused.
Phil could sense that the new policy did not have
much impact compared to the previous one. He pon-
dered in his mind, how would that be possible?
Robert started after taking a sip of his coffee, “The
problem began as their testing team proposed that
they should send back the whole lot for rework and
recheck as soon as one defective item is found. All hell
has broken loose since then. We are now flooded with
requests for rechecks from XYZsoft.” At this point,
Mark commented, “I am telling you, Jim, we can eas-
ily tackle this problem if we implement ICTM at our
end.” Jim looked at Mark but said nothing. The entire
episode puzzled him. How can a change in the qual-
ity control policy of XYZsoft have serious implications
on the business of ABCtronics? Jim replied, “We need
to tackle this problem quickly. XYZsoft is our biggest
client. We simply cannot afford to lose their business.”
Phil also found this development of events to be fas-
cinating. He had quietly jotted down whatever Robert
has said and looked at his pad. “What am I missing?”
he wondered on his own.
Robert continued, “I had a talk with Stuart regard-
ing this matter. He told me that [the] Susceptible
High Voltage Problem (SHVP)4 could contribute to
this kind of issue. His team is looking into this aspect
on the priority basis.” After hearing this, Jim looked
at Stuart and said, “I thought we dealt with this prob-
lem a couple of years ago.” Stuart replied, “We are
checking to be sure of the fact that it is not due to
SHVP. Tests would be complete within next week.”
Jim replied, “Do you have any preliminary report on
this test?”
Stuart said, “As per historical data, our IC chips
produce [a] minimum 2.7 V output on an average, as
HIGH signal. The variance of the HIGH signal out-
put voltage remains 1.8 V. We have received a number
of complaints from XYZsoft that the IC chips are not
producing the expected voltage. Then, I ordered to
take a random sample of 100 IC chips across the lots
and test them for SHVP. Initial reports suggest the
average voltage produced by the IC chips is around
2.3 V.” At this point, Mark commented, “Is it possi-
ble that we are overestimating the output of the IC
chips?” Stuart replied, “It may be the case. We would
not know for sure unless the detailed report comes.”
Phil could sense that the entire XYZsoft episode had
caused a fluttering feeling across the power corridor
of ABCtronics. They need a fast fix for the problem
and as of now none was in sight. A stifled silence
prevailed over the board room.
4 Susceptible High Voltage Problem (SHVP): Ideally, every IC
should produce HIGH signal output at 3.5 V, LOW signal output at
0.1 V, whereas 0.4 V–2.4 V is the undefined region of signal where
signal remains neither HIGH nor LOW. In reality, it is often found
that the HIGH signal output is around 2.5 V–3.0 V for a good IC.
A problem starts if the HIGH signal output of an IC’s lies between
0 V and 2.4 V. Then the signal is LOW or undefined, whereas it
should be HIGH. This problem is defined as SHVP (Tokheim 2004).
Adhikari, Biswas, and Bisi: Case: ABCtronics
32 INFORMS Transactions on Education 17(1), pp. 26–33, © 2016 INFORMS
Table 2 Customer Score Sheet
Sl. no. Customer name Customer score Range
1 Customer A 79 Good
2 Customer B 56 Satisfactory
3 Customer C 33 Needs improvement
4 Customer D 79 Good
5 Customer E 66 Good
6 Customer F 49 Satisfactory
7 Customer G 47 Satisfactory
8 Customer H 34 Needs improvement
9 Customer I 88 Very good
10 Customer J 77 Good
11 Customer K 67 Good
12 Customer L 51 Satisfactory
13 Customer M 53 Satisfactory
14 Customer N 74 Good
15 Customer O 85 Very good
16 Customer P 56 Satisfactory
17 Customer Q 39 Needs improvement
18 Customer R 51 Satisfactory
19 Customer S 26 Needs improvement
20 Customer T 43 Satisfactory
21 Customer U 77 Good
22 Customer V 97 Very good
23 Customer W 73 Good
24 Customer X 57 Satisfactory
25 Customer Y 66 Good
26 Customer Z 45 Satisfactory
27 Customer AA 28 Needs improvement
28 Customer AB 33 Needs improvement
29 Customer AC 56 Satisfactory
30 Customer AD 68 Good
31 Customer AE 32 Needs improvement
32 Customer AF 93 Very good
33 Customer AG 60 Good
34 Customer AH 29 Needs Improvement
35 Customer AI 41 Satisfactory
36 Customer AJ 42 Satisfactory
37 Customer AK 72 Good
38 Customer AL 48 Satisfactory
39 Customer AM 59 Satisfactory
40 Customer AN 47 Satisfactory
Note. Customer Score Range is defined as follows—Below 40: Needs
improvement, 40–59: Satisfactory, 60–79: Good, and 80–100: Very good.
Breaking the silence of the room, Jim spoke, “This
brings us to the last issue of the discussion today.
Where are we, regarding the analysis of customer
feedback?” Robert picked up a thick file and said, “It’s
all here. We ran the survey through 40 randomly cho-
sen customers of the 74XX chip family. Four of them
have rated us Very Good, eight have rated us Needs
Improvement, and 28 have given us either Good or Sat-
isfactory (see Table 2). So, I think we are good. Most
of them are happy with our products.” Jim noted
something in his diary. Then he said to Robert, “That
would be your hunch. That cannot be your analy-
sis.” Then Jim remembered something from an earlier
meeting. He asked Robert, “In the last meeting, there
was some discussion on redesigning the survey alto-
Table 3 Historical Sales Figure of ABCtronics
ABCtronics’ sales Overall market demand Price per Economic
Year volume (in millions) (in millions) chip (in $) condition∗
2004 2.39 297 0.832 0
2005 3.82 332 0.844 1
2006 3.33 195 0.854 0
2007 2.49 182 1.155 1
2008 1.56 93 1.303 0
2009 0.97 98 1.265 0
2010 1.32 198 1.368 1
2011 1.42 188 1.208 0
2012 1.48 285 1.234 1
2013 1.85 264 1.282 1
Note. ∗Economic condition: 1 signifies favorable market condition and 0 sig-
nifies otherwise.
gether. Why?” Robert looked visibly uncomfortable
with this question. He cleared his throat and replied,
“We have a mean customer rating of about 56. But the
overall spread of the score is very high. If we want
to conduct the survey to be sure of this average score
with 90% confidence, even with a margin error of 4
the required sample size may exceed our total cus-
tomer base.” Jim said, “What is the total number of
customers for 74XX?” Robert said, “Including the new
clients, the total is 70.” Jim said, “Sample more cus-
tomers if needed. And tell me how good would be
our estimate of the customer score?”
Robert presented the report on the sales figure (see
Table 3). ABCtronics was using a simple linear regres-
sion model for predicting the sales figure. However,
the new interns, who recently joined SMT, indicated
a few problems with the existing method. They pro-
posed a new method to predict sales. They argued
that their multiple linear regression model had better
explanatory power and was devoid of multicollinear-
ity problems. Looking at the presentation, Phil figured
out that he was also asked to predict the sales figure
for various demand scenarios. But Phil adopted a dif-
ferent approach. While looking at the screen, he did
some quick mental calculation, and the results were
not matching with the one presented. He nervously
glanced at Jim. Jim was listening to Robert’s analysis
with rapt attention.
As the meeting was coming to an end, Phil could
sense that ABCtronics needed to deal with a number
of issues, and they needed to do it quickly. The com-
pany was entering into the second quarter of the year.
ABCtronics immediately had to take corrective mea-
sures; otherwise it might be too late. As Stuart, Mark,
and Robert were leaving the board room, Phil waited
quietly for Jim. He knew that he was asked to attend
today’s meeting for a reason.
Adhikari, Biswas, and Bisi: Case: ABCtronics
INFORMS Transactions on Education 17(1), pp. 26–33, © 2016 INFORMS 33
Table 4 Market Demand Estimate
Sales volume 4Y Y 5 (in millions)
Total market demand
for PCs (in millions) Y Y > 3 105 ≤ Y Y ≤ 3 Y Y < 105 Subtotal
XX > 200 0.10 0.20 0.10 0.40
100 ≤ XX ≤ 200 0.10 0.10 0.20 0.40
XX < 100 0.00 0.10 0.10 0.20
Sub-total 0.20 0.40 0.40 1
Joint probability matrix of the sales volume and total market demand
Total market demand of PCs 4XX5 (in millions) Average sales volume 4Y Y 5 (in millions)
XX > 200 2.385
100 ≤ XX ≤ 200 2.140
XX < 100 1.265
Average sales volume in different demand scenarios
Note. Based on the sales figure of ABCtronics in Table 3, Phil has calculated the joint probability matrix for the sales
volume and total market demand, as well as the average sales volume in different demand scenarios.
8. The Road Ahead
As everybody left the room, Jim turned to Phil and
asked, “What do you think of today’s meeting?” Phil
kept quiet. Many aspects of the meeting left him con-
fused. Jim went on, “I need an honest opinion before
going ahead. Look at all the reports and analyze. Tell
me what do you think? You have two days to pre-
pare. Let us meet on Monday.” As Jim uttered those
words, only one thought came to Phil’s mind, “There
goes my weekend plan!”
“Have you prepared an analysis of the sales fig-
ures I asked for?” Phil said yes (see Table 4). “Do
you think we can sell more than 3 million chips this
year?” Phil said nothing. He has done the calcula-
tion; the possibility stands below 50% level. Jim fig-
ured out the answer from Phil’s silence. He asked,
“What about 2.5?” Phil replied, “I think we will fall
short of that number, the industry overall is experi-
encing a medium level demand. But the difference
is not very large. If we get an order from Customer
PQRsystems, probably we can make it.” Jim sighed
for a moment and said, “Let us see where we arrive
independently with our analysis. On Monday, after
our meeting, I shall meet with Stuart and Mark. We
have gone through a prolonged rough patch. The time
has come to take some course correction; otherwise
competition will knock us to the ground.”
9. Note
This case has been prepared to form the basis for class
discussion rather than to illustrate either effective or
ineffective handling of a business situation.
Supplemental Material
Supplemental material to this paper is available at http://dx
.doi.org/10.1287/ited.2016.0158cs.
Acknowledgments
We sincerely thank the editor-in-chief, the associate edi-
tor, and two anonymous referees for their insightful com-
ments and helpful suggestions. Their efforts have signifi-
cantly improved the case.
References
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http://analysisreport.morningstar.com/.
1
Reading Material for Students
Arnab Adhikari
Indian Institute of Management Calcutta, Joka, Kolkata 700104, India, arnaba10@email.iimcal.ac.in
Indranil Biswas
Indian Institute of Management Lucknow, Prabandh Nagar, Lucknow 226013, India, indranil@iiml.ac.in
Arnab Bisi
Johns Hopkins Carey Business School, 100 International Drive, Baltimore, Maryland 21202, abisi1@jhu.edu
Probability Distributions
Binomial distribution. The Binomial distribution describes the probability of exactly successes
out of trials; the probability associated with a success in a single trial is given by and that with
a failure is given by 1 − (also designated by ). The expression of the probability mass function
(pmf) of this distribution is as follows
(; , ) = (
)(1 − )−,
where the variable and the parameter are integers, satisfying the conditions 0 ≤ ≤ and
> 0. The parameter is a real quantity and ∈ [0,1]. The expected value and the variance of a
random variable X having binomial distribution can be expressed as follows:
E X N p and (1 )Var X N p p .
Hypergeometric distribution. The hypergeometric distribution describes the experiment where out
of total elements, possesses a certain attribute [and the remaining ( – ) does not]; if we
then choose elements at random without replacement, (; , , ) gives the probability that
exactly of the selected n elements have come from the group of elements that possesses the
attribute. Let the number of elements with that certain attribute be denoted by X. The probability
mass function (pmf) of X with hypergeometric distribution is given by
(; , , ) =
(
)(−
−
)
(
)
where is discrete and its range is given by: ∈ [max(0, − + ) , min (, )]. The
parameters , and are all integers and satisfy the following conditions: 1 ≤ ≤ , ≥ 1
2
and ≥ 1. Let probability of success be represented by
M
p
N
. Then, the expected value and
the variance of X under hypergeometric distribution can be expressed as follows:
npXE and
)1(
)()1(
N
nNpnp
XVar .
In real life, when a marketing group is trying to understand their customer base by testing a set
of known customers for over-representation of various demographic subgroups, they use
hypergeometric test designed based on hypergeometric distribution.
Negative Binomial distribution. The negative binomial distribution (also known as Pascal
distribution) gives the probability of waiting for exactly trials until ℎ success has occurred.
Let the number of trials before ℎ success be denoted by X. Here and (= 1 − ) designates
the probability of a success and a failure in a single trial, respectively. The probability mass
function (pmf) of this distribution is given by
(; , ) = (−1
−1
)(1 − )−,
where the variable and parameter are integers and satisfies the following condition: ≥ >
0. Now, the expected value and the variance of a random variable X under negative binomial
distribution can be expressed as follows:
p
pk
XE
)1(
and
2
)1(
p
pk
XVar
.
The negative binomial distribution has applications in the insurance industry, where for
example the rate at which people have accidents is affected by a random variable like the weather
condition.
Geometric distribution. The geometric distribution is a special case of the negative binomial
distribution discussed above with = 1. It expresses the probability of waiting for exactly x trials
before the occurrence of the first successful event. Let the number of trials before the first success
be denoted by X. Then, the probability mass function (pmf) of X with this distribution is given by
(; ) = (1 − )−1,
where p denotes the probability of success in each trial. The expected value and the variance of a
random variable X under geometric distribution can be expressed as follows:
3
p
p
XE
)1(
and
2
)1(
p
p
XVar
.
In real life, if a NGO wants to know the number of male births before one female birth
regarding the study of sex ratio in human population then it can use this kind of distribution.
Poisson distribution. The Poisson distribution gives the probability of finding occurrence of
exactly events in a given length of time when the events are independent in nature and happens
at a constant rate, given by . The probability mass function (pmf) of this distribution is given by
!
);(
x
e
xf
x
,
where the variable is a positive integer and the parameter is a real positive quantity. Now, the
expected value and the variance of a random variable X under Poisson distribution can be
expressed as follows:
XE and XVar .
When the value of N is very large and p is very small in the binomial distribution described
before, then it can be approximated by a Poisson distribution with expected value = Np. Poisson
distribution is applied to determine the probability of rare events like birth defects, genetic
mutations, car accidents, etc.
Uniform distribution. If a continuous random variable X follows the uniform distribution, then its
probability density function (pdf) is given by the expression
(; , ) =
1
−
for ≤ ≤ .
The expected value and the variance of a random variable X under uniform distribution can be
expressed as follows
2
ab
XE
and
12
2
ab
XVar
.
In oil exploration, the position of the oil-water contact in a potential prospect is often
considered to be uniformly distributed.
Exponential distribution. If a continuous random variable X follows the exponential distribution,
then its pdf can be expressed as follows:
4
(; ) =
1
−
,
where represents the scale parameter. The expected value and the variance of a random variable
X under exponential distribution are given by:
XE and 2XVar .
In real life, the radioactive or particle decays is considered to follow exponential distribution.
Normal distribution. The normal distribution (also called the Gauss distribution) is one of the most
important distributions in statistics. The pdf of normal distribution is given by the following
expression:
(; , 2) =
1
√2
−
1
2
(
−
)
2
,
where is the mean or expected value and 2 is the variance of the distribution. For = 0 and
= 1, the distribution is called the standard normal distribution. It has widespread applications in
natural and social sciences, financial models, etc.
Beta distribution. The beta distribution has been applied to model the behavior of random
variables limited to intervals of finite length in a wide variety of disciplines. The pdf of beta
distribution is given by:
(; , ) =
1
(,)
−1(1 − )−1,
where the shape parameters and are positive real numbers, and the variable satisfies the
condition 0 ≤ ≤ 1. (, ) designates the beta function and is given by the following expression
(, ) =
Γ()Γ()
Γ(+)
.
For ∈ ℝ+, the gamma function Γ() is defined by the integral
Γ() = ∫ −1−
∞
0
.
When = = 1 , the beta distribution assumes the form of the uniform distribution between
0 and 1; when = = 2 the distribution takes parabolic shape; when = 2 and = 1 or vise
versa the distribution takes triangular shaped distribution. The expected value and the variance of
a random variable X under beta distribution can be expressed as follows:
XE and
2
)1(
XVar .
5
Beta distribution is usually applied to determine the time allocation in project management/
control systems, heterogeneity in the probability of HIV transmission, etc.
Gamma distribution. It is a two-parameter family of continuous probability distributions.
Exponential distribution is a special case of the gamma distribution. The pdf of gamma distribution
can be represented by the following functional form:
(; , ) =
−1
−
Γ()
,
where the shape parameter and the scale parameter are positive real numbers ( ∈ ℝ+ and ∈
ℝ+) and the variable is also a positive real number ( ∈ ℝ+). The expected value and the
variance of a random variable X under gamma distribution are given by:
kXE and 2kXVar .
Sampling Distribution and Confidence Interval. If we take repeated samples from the same
population, samples means x would vary from sample to sample and form a sampling
distribution of sample means. It explains the random behavior of a sample mean. The variability
of x from can be obtained by determining the variance of x . The variance of the sample mean
with a sample of size n is given by:
n
x
2
2
.
Next, the confidence interval contains the true population parameter. A confidence interval
comprise point estimate, i.e., the best estimate of the population parameter from the sample statistic
and the margin of error or maximum sampling error (the maximum accepted difference between
the true population parameter and a sample estimate of that parameter). The confidence interval
where lies can be determined by the following expression:
n
zx
n
zx
2/2/
.
The confidence level is denoted by 100 1 % . The margin of error denoted by E is given
by the following formula:
n
zE
2/
.
6
From the formula given above, the required minimum sample size can be easily obtained and
it is given by:
2
)2/(
E
zn
.
Hypothesis Testing. Hypothesis testing is a technique to check with the help of a sample data
whether a claim or hypothesis about a population parameter is true or not. In hypothesis testing,
the stated conjecture defined as the null hypothesis can be disproved, but it cannot be proved.
However, by disproving the null hypothesis, one can prove that the contrary is true. The contrary
of the null hypothesis is termed as the alternative hypothesis. The test statistic represents the value
determined using the sample data. A test statistic for testing a hypothesis on population mean is
given by the following formula:
n
x
z
0 ,
where
0
denotes the hypothesized value of the population mean. Following are the null (
0
H )
and alternative (
A
H ) hypotheses for three standard tests on population mean:
The “Two-Tailed” Test.
The “One-Tailed” Test to the Right
0 0
0
0
0 / 2 / 2
:
:
reject if or .
A
H
H
x
z
n
H z z z z
0 0
0
0
0
:
:
reject if .
A
H
H
x
z
n
H z z
7
The “One-Tailed” Test to the Left
0 0
0
0
0
:
:
reject if .
A
H
H
x
z
n
H z z
Regression Models
Simple linear regression
Here we present a simple linear regression model to determine the relationship between the
dependent variable Y and the independent variable X, captured by the following equation:
E( | ) = + . Then the regression model can be designated as: = + + , where
= − E( | ) is a random variable or an error term with E() = 0 and ( ) = 2. If
̂ and ̂ denote the best estimates of the parameters α and β , respectively, then the estimated linear
regression equation of Y on X is:
̂ = ̂ + ̂.
Multiple linear regression
The effect of independent variables 1 , 2 and 3 on the dependent variable Y can be captured
by the following equation:
E( | 1, 2, 3) = + 11 + 22 + 33,
where = − E( | 1, 2, 3) is a random variable or an error term with E() =
0 and ( ) = 2. If ̂ , 1̂, 2̂, and 3̂ denote the best estimates of the parameters
α , β
1
, β
2
, and β
3
, respectively, then the estimated multiple linear regression equation of Y on
1 , 2 and 3 is given by:
̂ = + 1̂1 + 2̂2 + 3̂3.
Multicollinearity check
Often regression model is affected by linear relationship between independent variables termed as
‘multicollinearity’. Variance Inflation Factor (VIF) is one of the conventional techniques
employed to check whether any multicollinearity exists or not. VIF between two independent
variables X1 and X2 can be determined by the following expression:
8
VIF1,2 =
1
1−1,2
2 ,
where 1,2
2 denotes the co-efficient of determination between 1 and 2 . If the value of VIF is
greater than 5, then it indicates multicollinearity and the overall regression model gets affected by
it.
Sources
Anderson, D., Sweeney, D., Williams, T., Camm, J., Cochran, J. 2011. Statistics for
Business & Economics, 11th ed. Cengage Learning, Mason.
Berenson, M., Levine, D., Krehbiel, T. C. 2011. Basic business statistics: Concepts and
applications. Pearson Education, New Jersey.
Groebner, D. F., Shannon, P. W., Fry, P. C., Smith, K. D. 2013. Business statistics: a
decision making approach, 9th ed. Pearson Education, New Jersey.
Hildebrand, D. K. and O. Lyman. 1998. Statistical Thinking for Managers, 4th ed. Duxbury
Press, California.
Levin, R. I. and D. S. Rubin. 1997. Statistics for Management, 7th ed. Prentice Hall
International, New Jersey.
http://wps.aw.com/wps/media/objects/15/15512/formulas.pdf
http://www.nzqa.govt.nz/assets/qualifications-and-standards/qualifications/ncea/NCEA-
subject-resources/Mathematics/L3-Stats-Formulae-2013.pdf
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