Download BBP712S_Cengage Chapter 5

Chapter 5
Process Analysis
1
Analysis

Analysis is the examination of processes,
facts, and data to gain an understanding of
why problems occur and where
opportunities for improvement exist.
– Statistical inference is the process of drawing
conclusions about unknown characteristics of a
population from which data were taken.
– Predictive statistics focus on cause-and-effect
relationships and predictions of future
performance from historical data.
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Basic Probability Concepts

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An experiment is a process that results in
some outcome.
The outcome of an experiment is a result
that we observe
The collection of all possible outcomes of an
experiment is called the sample space.
Probability is the likelihood that an outcome
occurs.
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Probability Properties

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Label the n outcomes in a sample space as
O1, O2, … On, where Oi represents the ith
outcome in the sample space.
The probability associated with any outcome
must be between 0 and 1
– 0 ≤ P(Oi) ≤ 1 for each outcome Oi

The sum of the probabilities over all
possible outcomes must be 1.0
– P(O1) + P(O2) + … + P(On) = 1
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Events
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An event is a collection of one or more
outcomes from a sample space
If A is any event, the complement of A,
denoted as Ac, consists of all outcomes in
the sample space not in A.
Two events are mutually exclusive if they
have no outcomes in common.
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Calculating Probabilities

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Rule 1: The probability of any event is
the sum of the probabilities of the
outcomes that compose that event.
Rule 2: The probability of the
complement of any event A is P(Ac) =
1 – P(A).
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Calculating Probabilities

Rule 3: If events A and B are mutually
exclusive, then P(A or B) = P(A) +
P(B)

Rule 4: If two events A and B are not
mutually exclusive, then P(A or B) =
P(A) + P(B) – P(A and B)
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Conditional Probability

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Conditional probability is the probability of
occurrence of one event A, given that
another event B is known to be true or have
already occurred.
Multiplication rule of probability:
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Tree Diagram
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Calculation of Joint
Probabilities
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Independent Events

Two events A and B are independent if
P(A | B) = P(A).
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Random Variables

A random variable, X, is a numerical
description of the outcome of an experiment.
Formally, a random variable is a function that
assigns a numerical value to every possible
outcome in a sample space.
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Probability Distributions
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A probability distribution, f(x), is a
characterization of the possible values that a
random variable may assume along with the
probability of assuming these values.
The cumulative distribution function, F(x),
specifies the probability that the random variable
X will assume a value less than or equal to a
specified value, x, denoted as P(X ≤ x).
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Important Probability
Distributions

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Discrete
– Binomial
– Poisson
Continuous
– Normal
– Exponential
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Binomial Distribution

The binomial distribution describes the
probability of obtaining exactly x
“successes” in a sequence of n
identical experiments, called trials.
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Computing the Binomial
Distribution using Excel
BINOM.DIST(number_s, trials, probability_s,
cumulative)
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Poisson Distribution
 = expected value or average number of
occurrences
x = 0, 1, 2, 3, …
e = 2.71828…
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Computing the Poisson
Distribution Using Excel
POISSON.DIST(x, mean, cumulative)
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Probability Density Function

A curve that characterizes outcomes of a
continuous random variable is called a
probability density function, and is described
by a mathematical function f(x).
– Probabilities are only defined over intervals.
– The cumulative distribution function, F(x),
represents the probability P(X ≤ x).
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Normal Distribution

Familiar bell-shaped curve.
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Standard Normal Distribution

If a normal random variable has a mean μ =
0 and a standard deviation σ = 1, it is called
a standard normal distribution, represented
by z.
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Calculating Normal Probabilities

If x is any value from a normal distribution
with mean μ and standard deviation σ, we
may easily convert it to an equivalent value
from a standard normal distribution using:
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Calculating Normal Probabilities
Using Excel
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Excel function NORM.DIST(x, mean, standard
deviation, true) calculates the cumulative
probability F(x) for a specified mean and
standard deviation.
The Excel function NORM.S.DIST(z)
calculates the cumulative probability for the
standard normal distribution.
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NORM.INV Function

The Excel function
NORM.INV(probability, mean,
standard_dev) can be used when we
know the cumulative probability
(probability) but don’t know the value
of x.
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Exponential Distribution
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The exponential distribution models the time
between randomly occurring events, such as the
time to or between failures of mechanical or
electrical components.
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Calculating the Exponential
Distribution Using Excel
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The Excel function
EXPON.DIST(x, lambda, true)
can be used to compute
cumulative exponential
probabilities.
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Sampling Distributions
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A sampling distribution is the distribution of
a statistic for all possible samples of a fixed
size.
Sampling distribution of the mean
– Expected value of the sample mean is the
population mean
– Standard deviation of the sample mean (called
the standard error of the mean) is the
population standard deviation divided by the
square root of the sample size
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Central Limit Theorem
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Illustrating the Central Limit
Theorem
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Confidence Intervals

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A confidence interval (CI) is an interval
estimate of a population parameter that also
specifies the likelihood that the interval
contains the true population parameter. This
probability is called the level of confidence,
denoted by 1 − α, and is usually expressed
as a percentage.
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Common Confidence Intervals
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Confidence Interval Template

The Student Companion Site provides an
Excel workbook, Confidence Intervals.xlsx,
with worksheet templates for formulas
(5.17) through (5.19).
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Hypothesis Testing

Hypothesis testing involves drawing inferences
about two contrasting propositions
(hypotheses) relating to the value of a
population parameter, one of which is
assumed to be true in the absence of
contradictory data (called the null hypothesis),
and the other which must be true if the null
hypothesis is rejected (called the alternative
hypothesis).
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Hypothesis Testing Process

Steps
1. Formulate the hypotheses to test.
2. Select a level of significance.
3. Determine a decision rule on which to base a
conclusion.
4. Collect data and calculate a test statistic.
5. Apply the decision rule to the test statistic and
draw a conclusion.
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Excel Procedures
35
Regression Analysis

Regression analysis is a tool for building
statistical models that characterize
relationships between a dependent variable
and one or more independent variables, all
of which are numerical.
– A regression model that involves a single
independent variable is called simple regression.
A regression model that involves several
independent variables is called multiple
regression.
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Correlation
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Correlation is a measure of a linear
relationship between two variables, X
and Y, and is measured by the
(population) correlation coefficient.
Correlation coefficients will range from
−1 to +1.
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Analysis of Variance

Analysis of Variance, or ANOVA, is a
hypothesis-testing methodology for drawing
conclusions about equality of means of
multiple populations.
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One Way Analysis of Variance
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In its simplest form—one-way ANOVA—we
are interested in comparing means of
observed responses of several different
levels of a single factor.
ANOVA tests the hypothesis that the means
of all populations are equal against the
alternative hypothesis that at least one
mean differs from the others.
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Multi-Vari Studies
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A multi-vari study investigates three
types of process variation: positional
(variation within the same item or
sample), cyclical (variation between
parts or samples), and temporal (over
time, such as between different
production shifts).
40
Design of Experiments

A designed experiment is a test or series
of tests that enables the experimenter to
compare two or more methods to
determine which is better, or determine
levels of controllable factors to optimize
the yield of a process or minimize the
variability of a response variable.
41
Factorial Experiments

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Factorial experiment – one that considers all
combinations of levels of each factor
The simplest type of factorial experiment is one with
two factors at two levels
Each combination of different levels of the factor is
called a treatment.
42
Main Effects

A main effect measures the difference that a
factor has on the response.
– What is the effect of increasing the temperature
regardless of the value of the reaction time?
– What is the effect of increasing the reaction time
regardless of the temperature?
Main effect =
(Average response at high level) - (Average response at low level)
(5.22)
43
Interactions

An interaction is the effect of changing one
factor has on the level of other factors.
– For example, increasing factor 1 when factor 2 is at the low
level might result in an increase in the response variable;
however, increasing factor 1 when factor 2 is at the high
level might result in a decrease in the response variable.
Interaction effect = (Average response with both factors at the same
level) – (Average response with both factors at opposite levels)
(5.23)
44
Excel Templates for Factorial
Experiments

The Student
Companion Site
has Excel
templates for
factorial
experiments.
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Note to Instructors
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46
The following slides provide an
experiential exercise found on the
Web for applying DOE.
Students usually have a lot of fun with
this.
46
Paper Helicopter Design
1.
2.
3.
4.
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Cut along all the solid lines
on the diagram to the
right.
Fold flap A forward and
flap B to the back.
Fold flaps C and D both
forward along the dotted
lines.
Fold along the line E
upward to give a weight at
the bottom.
47
Making the Helicopter
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Helicopter Project Description
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Response goal: Maximize the length of time before
hitting the ground from a fixed height.
Identify 3 potential sources that might influence
the response, for example, AB length, CD length,
and width
Develop an experimental design; collect data, and
analyze results. Each group should submit a report
of their findings next class. Show all results;
calculate the main effects and interactions, and
draw conclusions about what is the best design.
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Root Cause Analysis

Root cause – “that condition (or interrelated
set of conditions) having allowed or caused
a defect to occur, which once corrected
properly, permanently prevents recurrence
of the defect in the same, or subsequent,
product or service generated by the
process.”
50
Five Why Technique

Redefine a problem statement as a
chain of causes and effects to identify
the source of the symptoms by asking
why, ideally five times.
51
Cause-and-Effect Diagrams

Cause-and-effect diagram – a simple
graphical method for presenting a chain of
causes and effects and for sorting out
causes and organizing relationships between
variables.
52
Project Review – Analyze
(1 of 2)
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Team members have received any necessary “justin-time” training
Team members understood how to use analysis
tools appropriately and effectively
The data collected in the Measure Phase have been
fully understood and studied
Appropriate statistical tools have been used to
conduct the analyses of data
Variation is thoroughly understood
Root causes and hypotheses that explain problems
have been identified
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Project Review – Analyze
(2 of 2)
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Data provide confirmation of key conclusions
and validation of root causes
Process maps are accurate and representative
of actual or desired process flow (in the case of
a re-design activity)
The process has been studied to identify
bottlenecks, sources of error, and non-value
added activities
Preliminary improvement or re-design goals
have been set
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