etm 620 - 09u 1 statistical inference “statistical thinking will one day be as necessary for...
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ETM 620 - 09U1
Statistical inference
“Statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write.” (H.G. Wells, 1946)
“There are three kinds of lies: white lies, which are justifiable; common lies, which have no justification; and statistics.” (Benjamin Disraeli)
“Statistics is no substitute for good judgment.” (unknown)
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Statistical inferenceSuppose –
A mechanical engineer is considering the use of a new composite material in the design of a vehicle suspension system and needs to know how the material will react under a variety of conditions (heat, cold, vibration, etc.)
An electrical engineer has designed a radar navigation system to be used in high performance aircraft and needs to be able to validate performance in flight.
An industrial engineer needs to validate the effect of a new roofing product on installation speed.
A motorist must decide whether to drive through a long stretch of flooded road after being assured that the average depth is only 6 inches.
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Statistical inference
What do all of these situations have in common?
How can we address the uncertainty involved in decision making?
a priori
a posteriori
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ProbabilityA mathematical means of determining how likely
an event is to occur.Classical (a priori): Given N equally likely outcomes,
the probability of an event A is given by,_______________
where n is the number of different ways A can occur.
Empirical (a posteriori): If an experiment is repeated M times and the event A occurs mA times, then the probability of event A is defined as,
____________________
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Descriptive statisticsNumerical values that help to characterize the
nature of data for the experimenter.Example: The absolute error in the readings
from a radar navigation system was measured with the following results:
17, 31, 22, 39, 28, 147, and 52
the sample mean, ̅x = _________________________
the sample median, x = _____________
the sample mode = ________________
~
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Descriptive StatisticsMeasure of variability
Our example:17, 31, 22, 39, 28, 147, and 52
sample range:
sample variance:
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Variability of the data
sample variance,
sample standard deviation,
n
i
i
n
xxs
1
22
1
2ss
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Other descriptorsDiscrete vs Continuous
discrete:
continuous:
Categorical and identifyingcategorical:
unit identifying:
Distribution of the data“What does it look like?”
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Graphical methodsDot diagram and scatter plot
useful for understanding relationships between factor settings and output
example (pp. 174-175)
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Using graphical methods …
Which factor(s) (or independent variable(s)) appears to have an effect on the output (or dependent variable), and what does that relationship look like?
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Graphical methods (cont.)Stem and leaf plot
example (radar data): 17, 31, 22, 39, 28, 147, and 52
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Another exampleBottle-bursting strength data (pg. 176)
Stem Leaf Frequency
17 6 1
18 7 2
19 7 3
20 0 5 8 6
21 0 4 5 9
22 0 1 3 8 13
23 1 1 4 5 5 5 19
24 2 2 3 5 6 8 8 26
25 0 0 0 1 3 4 4 7 8 8 8 37
26 0 0 0 0 1 2 3 3 4 4 5 5 5 5 5 5 7 7 8 9 9 (21)
27 0 1 1 2 4 4 4 4 5 6 6 7 8 8 42
28 0 0 0 0 1 1 3 3 6 7 28
29 0 3 4 6 8 9 9 18
30 0 1 7 8 11
31 7 8 7
32 1 8 5
33 4 7 3
34 6 1
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Graphical methods (cont.)
Frequency Distribution (histogram)equal-size class intervals – “bins”‘rules of thumb’ for interval size
7-15 intervals per data set√ nmore complicated rules
Identify midpointDetermine frequency of occurrence in each
binCalculate relative frequencyPlot frequency vs midpoint
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Relative frequency histogramExample: stride lengths (in inches) of 25 male
students were determined, with the following results:
What can we learn about the distribution of stride lengths for this sample?
Stride Length
28.60 26.50 30.00 27.10 27.80
26.10 29.70 27.30 28.50 29.30
28.60 28.60 26.80 27.00 27.30
26.60 29.50 27.00 27.30 28.00
29.00 27.30 25.70 28.80 31.40
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Constructing a histogram
Determining relative frequencies
Class Interval Class Midpt.
Frequency, F
Relative frequency
25.7 - 26.9 26.3 5 0.2
27.0 - 28.2 27.6 9 0.36
28.3 - 29.5 28.9 8 0.32
29.6 - 30.8 30.2 2 0.08
30.8 - 32.0 31.4 1 0.04
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Relative frequency graph
Stride Length Relative Frequency
Histogram
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
26.3 27.6 28.9 30.2 31.4
Stride Length (inches)
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What can you see?
Unimodal, Bimodal, or Multi-modal distribution
Recognizable distribution?
Skewness
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Another example …Bottle-bursting strength data (pg. 176)
Bursting strength (psi)
Frequency
330300270240210180
20
15
10
5
0
Histogram of bottle bursting strength
0
5
10
15
20
25
30
35
176 193 210 227 244 261 278 295 312 329 More
Bursting strength (psi)F
req
uen
cy
(from Minitab)
(from Excel)
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Other useful graphical methodsBox plot (aka, box and whisker plot)
bottle bursting data and another example (viscosity measurement, pg. 181)
Bu
rsti
ng
str
en
gth
(p
si)
360
320
280
240
200
Boxplot of Bursting strength (psi)
Da
ta
Mixture 3Mixture 2Mixture 1
27
26
25
24
23
22
21
20
Boxplot of Mixture 1, Mixture 2, Mixture 3
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Other useful graphical methods (cont.)Pareto diagram
frequency count for categorical data arranged in descending order of frequency of occurrence
useful for identifying “high value” targetssources of defectslevel of effort required in maintenance activitiesetc.
Time plotplot of observed values vs a time scale (hour of
day, day, month, etc.)useful for identifying patterns
effect of time of day on electricity usageseasonal effectsetc.
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Your turn** …
Look at problem 8-8 on page 194do parts a & bdraw conclusions
** - time permitting (Note: this also makes a good study problem)
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