ETM 620 - 09U 1 Statistical inference “Statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write.” (H.G.

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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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