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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)



1                                                                   ETM 620 - 09U
    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.
2                                                                     ETM 620 - 09U
    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




3                                                               ETM 620 - 09U
    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,
                    ____________________

       We‟ll talk more about this next time …
4                                                                         ETM 620 - 09U
    Descriptive statistics
     Numerical 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 = ________________
5                                                                    ETM 620 - 09U
    Descriptive Statistics
     Measure of variability
       Our example:
           17, 31, 22, 39, 28, 147, and 52



       sample range:



       sample variance:




6                                            ETM 620 - 09U
    Variability of the data
     sample variance,


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

     sample standard deviation,


         s  s2 




7                                       ETM 620 - 09U
    Other descriptors
     Discrete vs Continuous
       discrete:


       continuous:


     Categorical and identifying
       categorical:


       unit identifying:


     Distribution of the data
       “What does it look like?”
8                                   ETM 620 - 09U
        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




9                                                                                                              ETM 620 - 09U
                          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?
10                                                                                                                   ETM 620 - 09U
     Graphical methods (cont.)
      Stem and leaf plot
        example (radar data): 17, 31, 22, 39, 28, 147, and 52




11                                                               ETM 620 - 09U
     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



13                                                       ETM 620 - 09U
     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?


14                                                                  ETM 620 - 09U
     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




15                                                           ETM 620 - 09U
     Relative frequency graph




16                              ETM 620 - 09U
     What can you see?
      Unimodal, Bimodal, or Multi-modal distribution




      Recognizable distribution?




      Skewness




17                                                      ETM 620 - 09U
                  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)



18                                                                                                                                      ETM 620 - 09U
     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




19                                                                                                                                 ETM 620 - 09U
 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.

20                                                                       ETM 620 - 09U
     Your turn** …
      Look at problem 8-8 on page 194
        do parts a & b
        draw conclusions




** - time permitting (Note: this also makes a good study problem)
21                                                         ETM 620 - 09U

				
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