# etm620lesson1 by jizhen1947

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• pg 1
```									    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 inference
 Suppose –
 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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Probability
 A 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 statistics
 Numerical values that help to characterize the nature of data for
the experimenter.
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 Statistics
 Measure 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,

s 
2
n     x i  x 
2


i 1     n 1

 sample standard deviation,

   s  s2 

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Other descriptors
 Discrete vs Continuous
 discrete:

 continuous:

 Categorical and identifying
 categorical:

 unit identifying:

 Distribution of the data
 “What does it look like?”
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Graphical methods
 Dot diagram and scatter plot
 useful for understanding relationships between factor settings
and output
 example (pp. 174-175)
70

60

50
0    10        20   30         40      50   60        70   80

Pull strength
Pull Strength                                          40

30

20

10

0
0          5         10           15             20        25                        0   5       10        15       20
Wire Length                                                           Wire length

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Using graphical methods …

70                                                      70

60                                                      60

50                                                      50
Pull strength

Pull strength
40                                                      40

30                                                      30

20                                                      20

10                                                      10

0                                                      0
0     5       10        15   20                         0   100   200      300       400   500   600
Wire length                                                   Die height

 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 example
 Bottle-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

12      34    6                                                                                           1
ETM 620 - 09U
Graphical methods (cont.)
 Frequency Distribution (histogram)
 equal-size class intervals – “bins”
 „rules of thumb‟ for interval size
 7-15 intervals per data set
 √n
 more complicated rules
 Identify midpoint
 Determine frequency of occurrence in each bin
 Calculate relative frequency
 Plot frequency vs midpoint

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Relative frequency histogram
 Example: stride lengths (in inches) of 25 male students were
determined, with the following results:

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

 What can we learn about the distribution of stride lengths for
this sample?

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Constructing a histogram
 Determining relative frequencies

Class                      Frequency,    Relative
Interval     Class Midpt.       F        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

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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)

Histogram of bottle bursting strength                     35
20
30

25
15

Frequency
20
Frequency

15
10

10

5                                                                    5

0
0
176   193   210   227    244   261   278   295    312   329 More
180   210       240         270         300   330
Bursting strength (psi)
Bursting strength (psi)

(from Minitab)                                                                 (from Excel)

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Other useful graphical methods
 Box plot (aka, box and whisker plot)
 bottle bursting data and another example (viscosity
measurement, pg. 181)
Boxplot of Bursting strength (psi)                 Boxplot of Mixture 1, Mixture 2, Mixture 3
27
360

26

320
25
Bursting strength (psi)

24

Data
280
23

22
240

21

200                                               20
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” targets
 sources of defects
 level of effort required in maintenance activities
 etc.

 Time plot
 plot of observed values vs a time scale (hour of day, day, month, etc.)
 useful for identifying patterns
 effect of time of day on electricity usage
 seasonal effects
 etc.

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