Transcript Chapter 1 Material

A look ahead
The Role of Statistics
in Engineering
ENM 500
Chapter 1
The adventure begins…
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1-1 The Engineering Method and
Statistical Thinking
Figure 1.1 The engineering method
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The Engineering or Scientific Method
• Figure 1-1 Describes the Scientific or Engineering
Method.
• Several steps rely on statistical methods
– Conduct experiments – how are efficient experiments
designed?
– Identify the important factors – how do we account for
variability when we measure these factors?
– Confirm the solution – how do we accept or reject a
solution/hypothesis based on measurements?
• Variability complicates the task.
• Statistical methods help us understand and deal with
variability.
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1-1 The Engineering Method and
Statistical Thinking
• Statistical techniques are useful for describing and
understanding variability.
• By variability, we mean successive observations of a
system or phenomenon do not produce exactly the same
result.
• Statistics gives us a framework for describing this
variability and for learning about potential sources of
variability.
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Why is variability important to us?
• We want to predict results and control results with
accuracy. Variability makes predictions and control
more difficult and less accurate.
• If a particular part was required to be 1” + 0.010” and
the actual standard deviation was 0.010”, almost onethird of the parts would be out of tolerance, even if
their mean was exactly 1.000”!
• Would you rather work in a room that had a constant
temperature of 70o or one where the temperature
alternated between 50o and 90o every 30 minutes?
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Why Do We Study Probability & Statistics?
• Statistics – deals with the collection, presentation, analysis, and use
of data to make decisions, solve problems, and design products &
processes.
– use statistics to draw inferences. Examples: quality, performance, or
durability of a product, weather forecasts, utilization or loading of system.
• Probability – allows us to use information & data to make intelligent
statements & forecasts about future events.
– Probability helps quantify the risks associated with statistical
inferences
• Prob & Stat are foundations for other coursework, e.g. reliability
and quality courses, robust design, simulation, design of
experiments, decision analysis, forecasting, time-series analysis, and
operations research.
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What do we want to know about our data?
A measure of central tendency: Average or mean -
x1  x2  ....  xn 1 n
x
  xi
n
n i1
A measure of variability: Sample variance –
n
1
2
s2 
(
x

x
)
 i
n  1 i1
Sample Standard Deviation s  s2
We build models to
explain this variability
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An Example
Sample 1
Sample 2
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x1  20
x 2  20
s1  2.16
X
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X  3s2
X  3s1
20
10
X  3s1
X  3s2
s2  3.62
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Sample vs. Population Measures –
Statistical Inference
• The sample mean ( x ) estimates the population
mean ( )
• The sample variance ( s 2 ) estimates the population
variance (  2)
SAMPLE
POPULATION
MEAN:
1 n
x   xi
n i1
n
1
2
2
(
x

x
)
VARIANCE: s 
 i
n  1 i1
1 N
   xi
N i 1
1 N
   ( xi  x)2
N i1
2
The population can sometimes be conceptual and
essentially have infinite size.
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Sample vs. Population Measures
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We use sample measures (x, s ) to draw conclusions
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about the population measures (  ,  ).
• The sample will be a (random) subset of the
population
• The population may not yet exist, so the sample may
be from a small set of prototypes (analytic)
– There is an issue of stability – do the prototypes accurately
reflect the prospective population?
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Sample Data – May be obtained from:
• Observational Study – sample is drawn randomly
from current process or system
• Designed experiment – deliberate changes are
made to the controllable variables of a process or
system. The system output is observed & inferences
made about the effects of controlling the input.
• Retrospective Study – Historical observations.
Were you fortunate enough that the needed variables
were actually collected accurately!?
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Concept of Models
• Common engineering/physical models:
– F = ma
– I = E/R
– d = vt
• Mechanistic models: used when we understand the
physical mechanism relating these variables.
• Empirical models: use our engineering & scientific
knowledge of the phenomena, but are not built on
first-principle understanding of the underlying
mechanism. They are data driven.
Let the data do the
talking, right?
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1-3 Mechanistic and Empirical Models
A mechanistic model is built from our underlying
knowledge of the basic physical mechanism that relates
several variables.
Example: Ohm’s Law
Current = voltage/resistance
I = E/R
I = E/R + 
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1-3 Mechanistic and Empirical Models
An empirical model is built from our engineering and
scientific knowledge of the phenomenon, but is not
directly developed from our theoretical or firstprinciples understanding of the underlying mechanism.
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1-3 Mechanistic and Empirical Models
Example
Suppose we are interested in the average molecular weight
(Mn) of a polymer. Now we know that Mn is related to the
viscosity of the material (V), and it also depends on the
amount of catalyst (C) and the temperature (T ) in the
polymerization reactor when the material is manufactured.
The relationship between Mn and these variables is
Mn = f(V,C,T)
say, where the form of the function f is unknown.
where the b’s are unknown parameters.
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1-3 Mechanistic and Empirical Models
In general, this type of empirical model is called a
regression model.
The estimated regression line is given by
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Figure 1-15 Three-dimensional plot of the wire and pull strength
data.
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Figure 1-16 Plot of the predicted values of pull strength from the
empirical model.
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Designing Engineering Experiments
• Experiments are often used to confirm theory or to
evaluate various design options
– Often, several factors may be important
– Each factor may have more than one level of concern
• Full factorial design – considers all factors at all
levels of interest
– For K factors, each having two levels, a total of 2K
experiments are required
– For K = 4, N = 16
– For K = 8, N = 256
• Fractional factorial design – only a subset of factor
combinations are actually tested
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Design of Experiments (DOE)
• Assume you want to investigate the impact of three
factors on the pull-off force of a connector:
– Wall thickness (3/32” and 1/8”)
– Cure times (1 hour and 24 hours)
– Cure temperature (70o F and 100o F)
• We can now conduct an experiment to assess the
impact of each of these variables (separately &
interacting), each variable being assessed at two
different levels
• Since other sources of variability may be present,
we would do multiple experiments (replicate) at each
design point.
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Full Factorial Design
Figure S1-1 The factorial experiment for the connector wall thickness problem.
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Importance of Factor Interactions
Figure S1-2 The two-factor interaction between cure time and cure temperature.
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The Key Distinction
• The key difference between observational studies
and experimental designs is this:
– In a proper experiment you can eliminate confounding
factors and isolate effects of interest.
– In an observational study you take existing data. This may
make it impossible to distinguish the effects of two factors
that appear to explain observations equally well.
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Time Series
• The correct analysis and interpretation of data
collected over time is very important in assessing &
controlling the performance of a system or process.
–
–
–
–
When is performance normal & when is it out of control?
What factors are driving a system out of control?
What corrections should be applied to regain control?
When has a change occurred – a fundamental shift in the
process behavior?
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1-2.5 Observing Processes Over Time
Figure 1-11 Adjustments applied to random disturbances over control
the process and increase the deviations from the target.
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1-2.5 Observing Processes Over Time
Figure 1-12 Process mean shift is detected at observation number 57,
and one adjustment (a decrease of two units) reduces the deviations
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from target.
1-2.6 Observing Processes Over Time
Figure 1-13 A control chart for the chemical process concentration
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data.
1-4 Probability and Probability Models
• Probability models help quantify the
risks involved in statistical inference, that
is, risks involved in decisions made every
day.
• Probability provides the framework for
the study and application of statistics.
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Let’s Toss a Coin
• There are 1000 coins one of which contains two heads;
the others are fair. A coin is selected at random and
tossed 10 times. If heads appear on all ten tosses, what
is the probability that the coin selected is the two-headed
coin?
P(two-headed is selected) = .001
P(toss 10 heads in a row – fair coin) = (1/2)10 = 1/1024  .001
Therefore P(two-headed coin selected given 10 heads observed)  .5
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