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Unformatted text preview: Introduction to
Engineering Statistics We must draw conclusions from
partial information.
What
What time should I leave to make it on time to
class?
How
How many hours do we guarantee a light bulb
will operate?
Will
Will the pen caps fit the pen barrels?
Is
Is the grocery bag strong enough to hold a 20
pound bag of potatoes?
Where
Where should we target a filing machine so that
no more than 1% of “16 oz” bottles of Pepsi
have less than 16 oz? We Draw Conclusions about Populations
Populations
Examples:
Examples:
– All possible driving times from home to class.
– Lives in hours of a certain type of light bulb
manufactured by a certain company.
– Outside diameters of all pen barrels produced
and inside diameters of all pen caps
produced.
– Tensile strength of paper used in the
manufacture of grocery bags.
– Fill on all bottles of Pepsi coming off a filling
machine. 1 Population – a Definition
A population is the set of all possible
population
observations of interest.
A population has two parts to it:
population
– The individual being measured.
– The measurement being taken. Example
Example
– Life in hours of every light bulb manufactured.
– Inside diameter in mm of all pen caps
produced. Our Conclusions are Based on Samples
Examples:
Examples:
– Random sample of 10 possible driving times
from home to class.
– Life in hours of 50 randomly chosen light
bulbs manufactured by a certain company.
– Outside diameters of 25 randomly chosen pen
barrels and 25 randomly chosen pen caps.
– Tensile strength of 20 random samples of
paper used in the manufacture of grocery
bags.
– Fill in 100 randomly chosen bottles of Pepsi. Sample – a Definition
A sample is a subset of the population of
sample
interest.
It
It also has two parts:
– The individual being measured.
– The measurement being taken. Example
Example
– Life in hours of 50 randomly chosen light
bulbs.
bulbs.
– Inside diameter in mm of 25 randomly chosen
pen caps. 2 Why Use Sampling?
Time
Time
Money
Money
Accuracy
Accuracy
Destructive
Destructive testing Simple Random Sample
A sample of size n is a simple random
sample
sample if every possible sample of size n
taken from the population has the same
chance of being selected.
Choosing
Choosing randomly means we can use the
laws of probability to draw our
conclusions.
We
We can use the laws of probability to
know the chance we are wrong. Other Sampling Schemes
Stratified
Stratified random sample
– Divide population into several nonoverlapping
nonpopulations and then take a simple random
sample from each subpopulations.
– Stratification is an engineering decision based
on sources of variation. Systematic
Systematic random sample
– We sample every mth item, starting the
process with a randomly selected item for the
first m. 3 An engineer applies basic scientific
principles to real world problems.
Develop
Develop models to explain real world
phenomena.
Collect
Collect data to test the models.
The
The collected data is a sample (partial
information).
– Does the collected data support the model?
– What is the chance we are wrong? Statistical Thinking & Engineering
All
All work occurs in a system of
interconnected processes.
Variation
Variation exists in all processes.
Understanding
Understanding and reducing variation are
the keys to success. It takes engineering and statistics
to improve a process.
Process
Process improvement generally comes from
simplifying a process and removing unwanted
sources of variation.
“Six
“Six Sigma” – Define the problem.
– Measure important characteristics with validated
measurement systems. – Analyze potential sources of variation.
– Improve process through experimentation and data
analysis. – Control the process to maintain improvements. 4 Deterministic Models
Deterministic
Deterministic models make no attempt to
explain variability.
Ideal
Ideal Gas Law: P= nRT
V Ohm’s
Ohm’s Law: V = IR Probabilistic or Statistical Models
If
If we add variation to a deterministic model we
get a probabilistic or statistical model. P= nRT
+ε
V V = IR + ε Where ε is random error or the variation
we inherently expect in nature. A Probabilistic Model Describes a
Population
If
If 20 students measure the voltage for a
simple circuit with known resistance and
current, we will get an array of voltages,
all close to the voltage predicted by Ohm’s
Law.
These
These 20 voltages are a sample of the
population represented by V = IR + ε 5 Consider the Pen Barrel Outside Diameter
Let
Let yi be the observed outside diameter for the
ith pen barrel inspected.
Let represent
Let represent the true average or mean
outside barrel diameter.
Deterministic
Deterministic Model: y = µ
i Probabilistic
Probabilistic Model: yi = µ + ε i Where εi represents the variation from the mean
associated with the ith pen barrel. Strength (S) of a Polymer Filament depends
on amount of catalyst (C), polymerization
temperature (T), and polymerization
pressure (P).
Deterministic
Deterministic Model: S = β 0 + β1C + β 2T + β 3 P
Probabilistic
Probabilistic Model: yi = β 0 + β1C + β 2T + β 3 P + ε i Data Collection – A Retrospective
Study
Analyze
Analyze previously collected data.
Example:
Example:
– Use previously collected tensile strength data
for paper samples. Draw
Draw backs:
– Conditions may change over time.
– Data often missing or hard to coordinate with
other data. 6 Data Collection – Observational
Study
Observe
Observe a population through sampling.
Example:
Example:
– Pen caps are not fitting onto pen barrels.
– We are bottling Pepsi and want to be sure
that virtually all of our product will pass FDA
approval which requires a minimum of 16 oz
in each bottle Observational Study
Population (select) Sample (describes) (calculate) µ ,σ 2 x, s2 Parameter (estimate) Statistic Statistics and Parameters
A statistic is calculated from a random sample. It
statistic
describes the sample and is used to estimate a
parameter. Statistics change from sample to
sample. x, s2 A parameter describes a population. It is fixed
parameter
and usually unknown.
2 µ ,σ 7 Data Collection  Designed
Experiments
Manipulate
Manipulate factors according to a well defined
strategy in order to obtain a desired result.
Factors
Factors are the variables we manipulate in
order to produce a desired result.
– Catalyst, temperature and pressure are factors for
controlling the strength of a polymer filament. The
The values a factor is set to are called levels.
levels
Treatment
Treatment is a given combination of the levels
of each factor. Examples
Examples where You might use an
Experimental Design
Developing
Developing a process to produce polymer
filaments of a certain strength.
– Process Development Changing
Changing a process to produce stronger
polymer filaments.
– Process Improvement Experimental & Observational
Units
The Experimental
The Experimental Unit is the smallest
unit to which we apply a treatment
combination.
Experimental
Experimental error is a measure of the
variability among the experimental units. 8 Experimental Unit
A given treatment (C,T,P) is given to a
given
batch of polymer. The batch of polymer is
the experimental unit.
experimental
Variability
Variability from batch to batch of polymer
produced at the same C,T,P is the
experimental
experimental error. Experimental & Observational
Units
The Observational
The Observational Unit is the unit upon
which we make the measurement.
Observational
Observational error is a measure of the
variability among the observational units. Observational Units
We
We measure the strength of a filament
made from the polymer. The filament is
the observational unit.
observational
Variability
Variability from filament to filament from
within the same batch is the
observational
observational error. 9 Experimental & Observation Units
Another Example
A manufacturer of clay pots is interested in their
manufacturer
strength. Firing temperature determines
strength. He fires 10 pots at a time in the oven.
Experimental
Experimental Unit: 10 pots in an oven
Experimental
Experimental Error: variability that results from
trying to reproduce the firing.
Observational
Observational Unit: a pot
Observational
Observational error: variability among the 10
pots in a single firing. Basic Principles of Experimental
Design
Replication means
Replication means that each treatment
combination is applied to multiple experimental
units. Replication allows us to estimate
experimental error.
Randomization
Randomization means that the experimental
runs are performed in random order. This
minimizes the effects of any systematic changes
that occur during the experiment.
Local
Local Control of Error seeks to control
anything other than the factors that might affect
the response. This reduces random error among
experimental units. A study was carried out on the coating
thickness of a panel produced by a paint
operation.
operation. Viscosity is thought to impact the
coating
coating thickness. Only two levels of
viscosity are used (low and high). The
experimenters produced in random order 6
batches of paint, 3 at each level of
viscosity. 5 panels are painted at a time
with each batch of paint. The goal of the
experiment was to determine if higher
viscosity leads to thicker coating. 10 Identify
Response
Response of Interest
Factor(s)
Factor(s)
Factor
Factor Levels
Treatments
Treatments
Replications
Replications
Randomization
Randomization
Local
Local control of error
Experimental
Experimental unit and error
Observational
Observational unit and error Experimental Designs
A Completely Randomized Design randomly
Completely
allocates all the treatment combinations to the
experimental units.
A Randomized Complete Block Design
Randomized
randomly allocates treatment combinations to
the experimental units within the blocks. Blocks
are determined using known sources of variation
such as vendors.
Paired
Paired Designs are a variation of Blocked
Designs in which two treatment combinations
are applied to one experimental unit. Generally,
we are comparing two processes or methods. Examples
Examples of Engineering Experiments
Screening
Screening Factors to determine which actually
affect the response. We would consider
C,T,P,X,Y,Z’s effect on polymer filament
strength.
Predict
Predict the behavior of a response over a
specified range of factors. Use responses to test
a model. If data supports the model, we can use
the model to predict the average responses.
Optimization
Optimization is an attempt to find the best
combination of factor levels to produce a desired
response.
Robust
Robust Design attempts to make products and
processes robust to known sources of variation.
We are trying to achieve a target over a wide
range of operating conditions. 11 ...
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 Spring '10
 Wheatley
 Statistics

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