Docstoc

Introduction to Nonparametric Statistical Methods

Document Sample
Introduction to Nonparametric Statistical Methods Powered By Docstoc
					STAT 2010, Business Stat         2006                     Jaimie Kwon



           STAT 2010, Elements of Statistics for
                    Business and Economics
                            Lecture Notes

                            Prof. Jaimie Kwon
                              Statistics Dept
                            Cal State East Bay


Disclaimer
These lecture notes are for internal use of Prof. Jaimie Kwon, but are
provided as a potentially helpful material for students taking the course.
A few things to note:


 The lecture in class always supersedes what’s in the notes
 These notes are provided “as-is” i.e. the accuracy and relevance of
   the contents are not guaranteed
 The contents are fluid due to constant update during the lecture
 The contents may contain announcements etc. that are not relevant
   to the current quarter
 Students are free to report typos or make suggestions on the notes
   via emailing or in person to improve the material, but they need to
   understand the above nature of the notes
 Do not distribute these notes outside the class




                                  -1-
STAT 2010, Business Stat          2006                      Jaimie Kwon




Best Practice for note-taking in class
 I do not recommend students relying on this lecture notes in place of
   actual notes he/she writes down
 Bring a notepad and write down materials that I go over in the class,
   using this lecture notes as the independent reference; you don’t
   miss a thing by not having a printout of this lecture note in (and
   outside) the class
 If you still want to print these notes, it’d be better to print them 4
   pages on a single page (using “pages per sheet” feature in MS
   Word), preferably double sided (to save trees)




                                   -2-
STAT 2010, Business Stat       2006                     Jaimie Kwon



 Some canonical examples:
   Benefit of low-fat diet (Jan 2006)
   # of supporters of Bush/Gore in Florida exit poll (Florida, 2000)
   Is driving an SUV more dangerous than driving a passenger car?
   To cash in now and retire or keep working, for GM workers (Mar
     2006)?
   When do I have to leave home to be at school on time (this
     morning)?
   Has consumer confidence in the US increased or decreased from
     last to this month (March 2006)?
   Where do I put this $1,000? Google stock? Coca-Cola stock? A
     mutual fund? Certificate of deposit (CD)? What are expected
     returns and risks? (pay day)
   The number of mothers opting for cesarean birth is on the rise.
     On the other hand, cesarean babies have higher risk of breathing
     problem (March 30, 2006)
   Arnold is back (almost). The Californian governor’s approval
     rating is 47% now, a 7% increase in a single month. (March 30,
     2006)
   What’s the daily number of reports related to statistics? Interval
     variable? Categorical?
 What’s common in above examples: decision under uncertainty




                                -3-
STAT 2010, Business Stat        2006                    Jaimie Kwon




1 What is statistics?
 Statistics: a way to extract information from data
 Descriptive statistics: methods of organizing, summarizing, and
   presenting data in such a way that useful information is produced
    Graphical methods
    Numerical summary of data
 Inferential statistics: a body of methods used to draw conclusions or
   inferences about characteristics of population based on sample data
 Key paradigm of statistics
 Population: the group of all items of interest
 Parameter: a descriptive measure of a population
 Sample: a set of data drawn from the population
 Statistic: a descriptive measure of a sample
 Statistical inference: the process of making and estimate, prediction
   or decision about a population based on sample data
 Exercises 1.3, 4

2 Graphical and tabular descriptive statistics
2.1 Types of data
 Variable: some characteristic of a population or sample
 The values of the variable are the possible observations of the
   variable. (Integers b/w 0-100, real numbers, M/F, A-F)
 Data are the observed values of a variable (plural for datum)
 Types of data/variable




                                 -4-
STAT 2010, Business Stat         2006                      Jaimie Kwon



   Interval data/variable are real numbers, a.k.a. quantitative or
     numerical
   Nominal data/variable have categorical values without orders,
     a.k.a. qualitative or categorical
   Ordinal data/variable are similar to nominal but their values can
     be ordered
   (“Categorical variable” is the generic name for nominal and
     ordinal variables)
 Hierarchy? (Course grade: score to letter grade to pass/fail)
 Exercises 2.1-2.3

2.2 Techniques for nominal data
 Frequency distribution: a table of the categories and their counts
 Relative frequency distribution : shows the proportion (not count) of
  each category
 A bar chart is used to display frequencies
 A pie chart shows relative frequencies
 Exercises 2.11

2.3 Graphical techniques for interval data
 How to visualize the data? Histogram
 E.g. Items with defects (Xr02-35)
x=c(4, 9, 13, 7, 5, 8, 12, 15, 5, 7, 3, 8, 15, 17, 19, 6, 4, 10, 8, 22,
16, 9, 5, 3, 9, 19, 14, 13, 18, 7); hist(x)

 Example (recycle below): mean time spent on the internet; 0, 7, 12,
  5, 33, 14, 8, 0, 9, 22 (hrs /month)
x=c(0, 7, 12, 5, 33, 14, 8, 0, 9, 22); hist(x, nclass=4)




                                 -5-
STAT 2010, Business Stat         2006                   Jaimie Kwon



 We’ve all seen histograms. Here’s how you draw one:
   Build class intervals, equally wide, non-overlapping intervals that
     cover the complete range of observations.
   Create a frequency distribution, by counting the # of observations
     that fall into each class interval
   Draw the histogram, rectangles whose bases are class intervals
     and heights are frequencies
   How many class intervals?
      More class intervals for {more, less} data points.
      Table 2.6 for the rule of thumbs;
      Sturges’ formula: “1+3.3 log(n)”
      My favorite: eyeballing
   How wide is each interval? Round (range/# of classes) to
     something convenient.
 Reading histograms…
   Symmetry and Skewness (positively/negatively)
   How many peaks? unimodal, bimodal
   Bell shape (symmetric, unimodal; important)
 Which variables are likely to have
   A positively skewed distribution?
   A negatively skewed distribution?
   Symmetric distribution?
   Symmetric, bell shaped distribution?
   Bimodal distribution?




                                  -6-
STAT 2010, Business Stat        2006                    Jaimie Kwon



 Stem-and-leaf display
 Ogive
 Ex. 2.33, 35(a)(c)

2.4 Describing the relationship between two variables
 Bivariate methods are used to study the relationship between two
  variables (Cf. Univariate methods)
 Dependent variable (Y) vs. independent variable (X)
 Four possible combinations: {categorical, integer} {X, Y} variable
 Two categorical variables:
 E.g. Gender and choice of doctorate, 1998 (Ex. 2.56, Xr02-56)
   Example: Blue collar/white collar/professional vs NYTimes/USA
     today/SF Chronicles; ad targeting
   A contingency table lists the frequency of each combination of
     the values of two categorical variables
   To study the differences in the row variable among the column
     variable; compute the column totals and divide each frequency
     by it to obtain column relative frequencies
 Two interval variables:
 E.g. Size vs. price of home (100 ft2 vs K dollars) which are
  dependent and independent variable? Use of X and Y. (e.g. Xm02-
  09)
   Draw scatter diagram using X and Y
 Interpreting scatter diagrams:




                                   -7-
STAT 2010, Business Stat        2006                      Jaimie Kwon



   Linear relationship: most of the points fall close to a straight line
     through points (cf. least squares method)
   Two main characteristics of linear relationship:
      Strength (strong, medium, weak, none)
      Direction (positively linear, negatively linear)
   Nonlinear relationship
 Ex. 2.55 (Xr02-55), 56 (Xr02-56)

2.5 Time series data
 Bankrate, Hbrhomes graph (<> cross-sectional data)
 Ex 2.73 (Xr02-73)

3 Art and science of graphical presentations
 graphical excellence
 graphical deception
 presenting statistics: writing reports and oral presentations




                                 -8-
STAT 2010, Business Stat              2006                             Jaimie Kwon




4 Numerical descriptive techniques
4.1 Measures of central location
 Label observations in a sample as x1 , x2 ..., xn
 We typically use n for the sample size, N for population size
    Population quantities are usually not computable, especially
      when N=
 Example (recycle below): mean time spent on the internet; 0, 7, 12,
   5, 33, 14, 8, 0, 9, 22 (hrs /month)
x=c(0, 7, 12, 5, 33, 14, 8, 0, 9, 22);mean(x);hist(x)

 Three measures of central location
    Arithmetic mean:
                           n                                N

                           xi                              x     i
      sample mean x      i 1
                                   ; population mean:     i 1

                               n                              N
    Median: the observation that falls in the middle of the sorted data
    Mode: value that occurs with the greatest frequency
 Which to use?
    Mode is usually a poor measure.
    Compared to mean, median is less sensitive to extreme
      observations and in many cases more interpretable
 Geometric mean: useful for finance, when averaging growth rate
   over years
    Let Ri be the rate of return in period i. The geometric mean Rg of
      the returns R1,…,Rn is (1+Rg)n = (1+R1)…(1+Rn); Solving for Rg,




                                       -9-
STAT 2010, Business Stat                    2006                             Jaimie Kwon



      we have 1  Rg  n (1  R1 )...(  Rn ) ; example with R1=100% and R2=-
                                     1

      50%. ($1,000 -> $2,000 -> $1,000 again)
 Ex 4.3, 4.10 (geometric mean)

4.2 Measures of variability
 Measure of spread or variability of the data
 Example: 8, 4, 9, 11, 13 (# of hours the students spent studying stat
  last week)
   Range = largest value observed - smallest value observed (too
      simple)
                                                      n

                                                      x         x
                                                                    2
                                                             i
   Variance: sample variance s 2  s x2            i 1
                                                                        , population
                                                            n 1
                              N

                               x        
                                            2
                                     i
      variance  2   x2    i 1

                                     N
       Why n-1? We will see in Chapter 10.1;
       Compute “deviations” first and squaring, summing, dividing.
       Why squaring? (absolute value is also possible; MAD)
       The unit? (square of the original unit)
                                                      n  
                                                              2

                                                       xi  
                                         1  n 2  i1  
     Shortcut for sample variance: s 
                                     2
                                                 xi  n 
                                        n  1  i1
                                                               
                                              
                                                               
                                                                
   Standard deviation (SD): sample standard deviation s  s 2 ,
      population standard deviation    2
       Same unit as the original data; easy to interpret
   s2=2 =0 if and only if ___



                                            - 10 -
STAT 2010, Business Stat           2006                    Jaimie Kwon



 Empirical Rule: Given a set of n measurements that is
   approximately normal (bell-shaped), it follows that the interval with
   endpoints
    xs      contains ~ 68% of the measurements
    x  2s   contains ~ 95% of the measurements
    x  3s   contains almost all of the measurements
 E.g. Analysis of the monthly returns on an investment shows the
   distribution is approximately bell shaped and mean=10% and
   sd=4%. What can you say about the distribution of the return?
hist(rnorm(240, 10, 4), col=’red’)

    How often is the return between 6 to 14%?
    How often is the return larger than 14%?
                                   s    
 Coefficient of variation (CV):     or
                                   x    
 Ex 4.23, 24((b) and (c) only; also compute standard deviations as
   well), 27, 28

4.3 Percentiles and box plots
 Percentiles are everywhere (test scores…)
 The p’th percentile: the value for which p percent of observations
   are less than that value and (100-p)% are greater than that value
 Quartiles are 25th, 50th, 75th percentiles (divide the data into
   quarters),
   each called first/lower quartile, median, and third/upper quartile
   each labeled Q1, Q2, Q3
   (cf. quintiles and deciles)



                                   - 11 -
STAT 2010, Business Stat          2006                      Jaimie Kwon



 Location of a p’th percentile in the sorted numbers is approximately
                    p
   L p  (n  1)
                   100
 Recycle the internet data example:
    Simple, rounding approach
    Detailed approach
 Relationship between the skewness and distribution of quartiles
    If Q2 is closer to Q1 than Q3, then ____ skewed
    If Q2 is closer to Q3 than Q1, then ____ skewed
 Inter-quartile range (IQR) : Q3-Q1; spread of the middle 50% of the
   observations
 (horizontal) Box plots:
    Q1, Q2, Q3 for the box boundaries;
    Left and right ‘whiskers’ extend outward from the box boundaries
      to the outermost values that are within 1.5 * IQR from the box
      boundaries
    Points outside the whiskers are ‘outliers’ (>1.5*IQR outward from
      Q1 or Q3); interesting or incorrect points
 Multiple box plots: Great tool for comparing distribution of multiple
   groups
 Ex 4.37, 4.43, 4.48 (do only “describe your findings” part; the
   boxplot is provided in the handout; feel free to try Minitab to draw
   the boxplot per in class instruction but it’s not required)




                                  - 12 -
STAT 2010, Business Stat                2006                                             Jaimie Kwon



4.4 Measures of linear relationship
 Numerical measure for direction and strength of the linear
    relationship
 Example: (which are X and which are Y?)
     baseball wins vs. home/road attendance (Baseball attendance);
     GMAT score vs. MBA GPA (xm04-16)
 Covariance between variables X and Y:
                                              N

                                               ( x   )( y  
                                                      i         x   i       y   )
     Population covariance  xy             i 1
                                                                                    ,
                                                                N
                                        n

                                       ( x  x )( y  y )
                                               i            i
     Sample covariance: sxy          i 1
                                                                        ,
                                                     n 1
                                                                 n      n
                                                                             
                                                1    n              xi  yi 
                                                                 i 1
       Shortcut for sample covariance: s xy         xi yi  i 1         
                                               n  1  i 1            n     
                                                     
                                                                            
                                                                             
     Manual calculation:
I            xi        yi         xi  x              yi  y                xi  x  yi  y    xi yi

1            2         13
…            6         20
N            7         27
Total
Average


         Xi=2,6,7; yi=13, 20, 27;
           How about yi=27, 20, 13?
           How about yi=20, 27, 13?



                                        - 13 -
STAT 2010, Business Stat                2006                        Jaimie Kwon



   Look at the sign (direction) and magnitude (strength) –
   How do we judge magnitude of covariance?
 Coefficient of correlation
                                          xy                           s
   Population correlation                   ; sample correlation r  xy
                                         x y                         sx s y
   Correlation is between -1 and 1
   Java Applet for correlation coefficient
 Least squares method: an objective way of producing a straight line
  through data points in scatter diagram
   It produces a straight line such that the sum of squared
      deviations between the points and the line is minimized
   Equation for a line:
          y  b0  b1 x ,
          ˆ

      where
      b 0 : intercept

      b1 : slope

          y : the (predicted) value of y determined by the line
          ˆ
                                                                            n
   Use calculus to find coefficients b0, b1 which minimizes              (y
                                                                           i 1
                                                                                  i    yi ) 2
                                                                                        ˆ

 Least squares line coefficients are given by
          sxy
   b1     2
                and   b0  y  b1 x .
          sx
 Ex 4.55, 56, 58 (xr04-58; computer use is OK but show your work)




                                        - 14 -
STAT 2010, Business Stat        2006                    Jaimie Kwon



4.5 Comparing graphical and numerical techniques
 Comparing returns on two investment; centers=expected return;
  spreads=risks (low-risk vs high-risk)
 Business stat marks vs. math stat marks: unimodal, bimodal, …
 Relationship b/w price and size of houses

4.6 General guidelines for exploring data
 Look at the shape of the distribution; find Center; spread; peaks;
  skewness (bell curve?)
 Shapes guide on which numerical techniques to use
 Optional (won't be graded): Ex 4.84, 4.86 (you have to use the
  computer, preferrably Minitab, for these two problems)




                                - 15 -
STAT 2010, Business Stat        2006                     Jaimie Kwon




5 Data collection and sampling
5.1 Methods of collecting data
 Direct observation (observational data): aspirin vs. heart attack
  example; limitations; inexpensive
   Surveys: Gallup Poll example; market research; response rate
   Personal interview
   Telephone interview
   Self-administered survey
   Questionnaire design
 Experiment (experimental data): same example
 Ex 5.1

5.2 Sampling
 The chief motif for a sample rather than population: cost
 Use sample quantities as ‘estimates’ for the corresponding
  population quantities
   E.g. Nielson ratings (what is watched by 1000 television viewers);
     quality control
 “Target population” (the population about which we want to draw
  inferences) vs. “sampled population” (the actual population from
  which the sample has been taken)
   E.g. The Literary Digest : predicted Alfred Landon’s 3 to 2 victory
     over the incumbent Franklin D. Roosevelt based on 10 million
     sample ballots
      That are sampled from phone directory


                                - 16 -
STAT 2010, Business Stat       2006                     Jaimie Kwon



      Of which “only” 2.3 million were returned (‘self-selected
        samples’)
 Ex. 5.6, 5.7

5.3 Sampling plans
 A “simple random sample” is a sample selected in such a way that
  every possible sample with the same # of observations is equally
  likely to be chosen
   Simple and good (do it “randomly”!!)
   How to do it?? (random sample; jar; …)
 A “stratified random sample” is obtained by separating the
  population into mutually exclusive sets, or strata, and then drawing
  simple random samples from each stratum
   To extract more information
   Criteria for separating a population into strata include: gender,
     age, occupation,…
   Sampling procedure and analysis can be complicated: plan
     ahead and consult stat pros!
 A “cluster sample” is a simple random sample of groups or clusters
  of elements
   Reduce geometric distances the surveyor must cover to gather
     data (reduce cost)
   Increases sampling error
 Sample size and accuracy: The larger the sample size is, the more
  accurate the sample estimates becomes



                               - 17 -
STAT 2010, Business Stat        2006                     Jaimie Kwon



   Details in Chapters 10 and 12
 Ex 5.11, 14-16

5.4 Sampling and nonsampling errors
 Sampling error: differences between the sample and the population
  that exist only because of the observations that happened to be
  selected for the sample
   E.g. the mean annual income of North American blue-collar
     workers
   Estimate the mean income  of the population by the mean x of
     the sample. The value of x will deviate from  simply by chance
   This deviation can be large simply due to bad luck
   The only way to reduce the expected size of this error is to take a
     larger sample
   Given a fixed sample size, we state the probability that the
     sampling error is less than certain amount (Ch. 10)
 Nonsampling error: more serious; taking a larger sample won’t help
  here; due to mistakes made in the acquisition of data or due to the
  sample observations being selected improperly
   Error in data acquisition
   “Non-response error”: error or bias introduced when responses
     are not obtained from some members of the sample
   Selection bias
 Ex 5.17, 5.18




                                - 18 -
STAT 2010, Business Stat           2006                      Jaimie Kwon




6 Probability
 Probability is critical in statistical inference since it provides the link
   between the population and the sample

6.1 Assigning probability to events
 A “random experiment” is a process that leads to one of several
   possible outcomes
    E.g. coin flipping; grade on a stat test; time to assemble
      computer; party preference
 A “sample space’ of a random experiment is a set of all possible
   outcomes of the experiment (exhaustive and mutually exclusive)
    S  {O1 , O2 ,..., Ok }
 Requirements of probabilities: given a sample space S, the
   probabilities assigned to outcome must satisfy two requirements:
    The probability of any outcome must be between 0 and 1, i.e.
      0  POi   1

    The sum of the probabilities of all the outcomes in the sample
      space must be 1, i.e.    PO   1
                               i
                                    i



 Three approaches to assigning probabilities
    The classical approach
    The relative frequency approach
    The subjective approach
 An “event” is a set of outcomes in a sample space
    A “simple event” is an individual outcome




                                   - 19 -
STAT 2010, Business Stat        2006                     Jaimie Kwon



 The “probability of an event” is the sum of probabilities of the simple
   events that constitute the event
 Most useful way to interpret probability is the relative frequency
   approach for a hypothetical, infinite number of experiments
 Ex. 6.1-3 (in class), 8

6.2 Joint, marginal, and conditional probability
 Want to consider ‘combinations’ of events
 Example: relationship between whether a mutual fund outperforms
   market and whether the manager of the fund has an MBA from a
   top-20 program


 Consider a population of 1,000 mutual funds


                  Mutual fund         Mutual fund       Totals
                  outperforms         does not
                  market              outperform
                                      market
The manager       110                 290
has MBA
The manager       60                  540
does not have
MBA
Totals                                                  1,000




                                - 20 -
STAT 2010, Business Stat          2006                      Jaimie Kwon



 The “intersection of events A and B,” denoted “A and B,” is the
   event that occurs when both A and B occurs.
 The probability of the intersection is called the “joint probability”
 P(A randomly selected mutual fund outperforms and its manager
   has an MBA degree) =
 What is the joint probability if we sample a mutual fund from the
   above population?


                   Mutual fund         Mutual fund        Totals
                   outperforms         does not
                   market              outperform
                                       market
The manager        .11                 .29
has MBA
The manager        .06                 .54
does not have
MBA
Totals
 “Marginal probabilities” are computed by adding across rows or
   down columns
 P(A randomly selected mutual fund manager has MBA degree) = ?
       i.e., When a mutual fund is randomly selected, the probability
         that its manager has an MBA is ___
       i.e., ___ all mutual fund managers have an MBA



                                  - 21 -
STAT 2010, Business Stat        2006                     Jaimie Kwon



 Try, P(A randomly selected mutual fund outperforms the market) = ?
 “Given that a fund is fund is managed by an MBA, what’s the
  probability that it outperforms the market?”
   Given A, what’s the probability of B?
 The “Conditional probability of B given A”, written P(B|A), is the
  probability of event B given the occurrence of another related event
  A.
   Formally, it can be computed as P(B|A)=P(A and B)/P(A)
 Two events A and B are “independent” if P(A|B)=P(A) or
  P(B|A)=P(B)
   i.e., the probability of one event is not affected by the occurrence
       of the other event
 Checking dependence: For the table like above, we can check all
  four combinations but showing it for only one of them [P(B)  P(B|A)
  for some A and B] is enough. On the other hand, showing
  independence would be more work
 The “union” of events A and B is the event that occurs when either A
  or B or both occur. It is denoted as “A or B”
   E.g. determine P(A1 or B1)
   Approach #1 : sum the components
   #2 : 1- P(the other component)
 Ex 6.86




                                - 22 -
STAT 2010, Business Stat          2006                 Jaimie Kwon



6.3 Probability rules and trees
 Want to calculate the probability of more complex events from the
  probability of simpler events
 Complement rule: the “complement” of event A is … and is denoted
  by AC. The rule says P(AC)=1-P(A); e.g.
 Multiplication rule: P(A and B) = P(A|B)P(B) or, P(B|A)P(A)
   Proof:
   If independent,… it reduces to:
      The joint probability of any two independent events A and B is
        P(A and B)=P(A)P(B)
 Ex 6.5: 7 males and 3 females. P(two randomly selected students
  are both female)?
 Ex 6.5: 7 males and 3 females. P(two randomly selected students
  by two professors to answer questions are both female)?
 Addition rule: P(A or B)=P(A)+P(B)-P(A and B)
   [revisit the above example]
 When two events are mutually exclusive (two events cannot occur
  together), the joint prob is 0, thus the above reduces to…
 P(paper A)=?, P(paper B)=?, P(both papers)=?. Then P(either
  paper)=?
 Probability trees
   First choice, second choice, joint probability
   {F,M}, {F,M}|F and {F,M}|M, {FF, FM , MF, MM} (for the two
     cases above)
 Ex. 6.47, 51-55, 67, 68


                                  - 23 -
STAT 2010, Business Stat     2006    Jaimie Kwon



6.4 Bayes’ Law
 Skip

6.5 Identifying the correct method
 Read




                            - 24 -
STAT 2010, Business Stat          2006                       Jaimie Kwon




7 Random variables and discrete probability
  distributions
Motivation: Want to tell if a coin is fair. Throw it 100 times. Reject the
null hypothesis that the coin is fair if # of heads is too large or small.
But where do we draw the line? 90? 70? How extreme is the observed
value? Need to know probability distribution of the number of heads
from a balanced coin.

7.1 Random variables and probability distributions
 E.g. # of heads in flipping of two coins; total of two dice
 Random variable : a function or rule that assigns a number to each
   outcome of an experiment
 Two types of random variable :
    Discrete random variable: takes on a countable number of
      values; e.g.
    Continuous random variable: takes on uncountable number of
      values.
 Probability distribution: a table, formula, or graph, that describes the
   values of a random variable and the probability associated with
   these values.
 X vs. x: X: name of a random variable; x: value of the random
   variable
 P(X=x) or P(x)
 Requirements for a discrete probability distribution function
   (distribution of a discrete random variable):



                                  - 25 -
STAT 2010, Business Stat           2006                     Jaimie Kwon



       0  P( x)  1 for all x
        P( x)  1
         x

 Example. Consider a game where the player draws a card from a
    deck of cards and wins $100 for spade ace, $5 for any heart and $0
    for anything else. If we let X be the winning (in $), specify P(x).


x                       P(x)




 Example. Consider investing money to a start-up company. After a
    year, it either fails, has moderate success, or has a big success with
    probabilities 0.8, 0.15 and __, respectively. In each case, the
    investment return is given by $0, $1,000 and $10,000.
     What’s the quantity ot consider as a random variable X?
     What’s P(X>0)? What’s P(X=0)?


x                       P(x)




                                  - 26 -
STAT 2010, Business Stat                      2006                      Jaimie Kwon



 Population mean:   E ( X )   xP( x) (“the expected value of X”)
                                              x

   Population variance:   V ( X )   ( x   ) 2 P( x)
                                   2

                                                  x

 Shortcut calculation for population variance: V ( X )   x 2 P( x)   2
                                                                    x

 Population SD :    2
 Note that we’re using the same terms as in Chapter 2. It’s not a
   coincidence. Consider a population consisting of N individuals and
   assume that for a variable X, the population relative frequency of the
   value x, (# of individuals that are x)/N, is given by P(x). Then the
                             N

                            x         i
   population mean        i 1
                                           as a descriptive measure for the
                              N
   population is same as   E ( X )   xP( x) , the expected value of X.
                                                  x

   Same can be said for the population variance and standard
   deviation.
 Laws of expected value and variance: for a random variable X and a
   constant c,
    E(c)=c
    E(X+c) = E(X)+c
    E(cX)=cE(X)
    V(c)=0
    V(X+c)=V(X)
    V(cX)=c2V(X)
 Example. The monthly sales at a computer store has a mean of
   $25,000 and SD of $4,000. Also,



                                             - 27 -
STAT 2010, Business Stat         2006                     Jaimie Kwon




   Profits = 30% of the sales – fixed costs of $6000.


   Find the mean and SD of monthly profits
   [conventional method vs. empirical rule method]


 Ex. 7.1(d), 2(d), 7, 19(a)(d), 39 (in answering 7.39, use the fact that
   answers to 7.38 is E(X) =4.00 and V(X) = 2.40)

7.2 Bivariate distributions
 Bivariate distribution provides probabilities of combinations of two
   random variables (Cf. univariate distribution)
 Joint probability are written P(x,y): again, table or formula


 X and Y are # of houses sold by two agents, Xavier and Yvonne per
   day; P(x,y) = P(X=x,Y=y) are given below:


                                           x
                                     0               1
                    y    0     .11             .29
                         1     .06             .54




 Requirements:
    0P(x,y) 1 for all x,y



                                 - 28 -
STAT 2010, Business Stat                  2006                      Jaimie Kwon



    xy P(x,y) = 1
 Marginal probabilities P(x)= all_y P(x,y), P(y) = all_x P(x,y)
 E(X)=X
 V(X)=2X
 X
 E(Y)=Y
 V(Y)=2Y
 Y
 Covariance: Cov( X , Y )   x   x y   y P( x, y)
                                 x   y

    Shortcut calculation: Cov( X , Y )   xyP( x, y)   x  y
                                                    x   y

                                     Cov( X , Y )
 Coefficient correlation  
                                          XY
 Two discrete random variables X and Y are independent if two
   events {X=x} and {Y=y} are independent for any x and y
    In other words, if P(x,y)=P(x)P(y) for any x and y
    More informally, if X and Y don’t affect each other
 Laws of expected value and variance of the sum of two variables
    X+Y, P(x+y)
    X+Y
    E(X+Y) = E(X) + E(Y)
    V(X+Y) = V(X) + V(Y) + 2COV(X,Y)
    If X and Y are indep, COV(X,Y)=0 and =0
 Total # of houses sold by Xavier and Yvonne
 Ex. 7.43-46, 55, 56



                                          - 29 -
STAT 2010, Business Stat       2006     Jaimie Kwon




 Quiz #1 scores (out of 36)
 Mean = 32.09
 Median = 32
 SD = 2.6




                               - 30 -
STAT 2010, Business Stat           2006                   Jaimie Kwon



7.3 Binomial distribution
 Binomial random variable is the number of successes in n
   independent trials with a constant success probability p. We write
   X~bin(n,p) to describe that a random variable X follows such a
   binomial distribution.
 Such experiment is called a binomial experiment:
    Consists of a fixed # of trials (n)
    Two possible outcomes (‘success’ and ‘failure’)
    The success probability is p.
    The trials are indep.
 Each trial is called a ‘Bernoulli process’
 E.g. Flipping coin; draw cards (not binomial); political survey (not
   quite but come close)
 E.g. a clueless student takes an exam consists of 5 multiple choice
   (1 out of 4) questions.
    Delineate n and p
    What’s the probability that he gets no answers correct? P(X=0);
      two answers correct? P(X=2)=?
    What’s the chance that P(fail the quiz) = P(X2)=?
    For a class full of similar studnets, What’s the mean score? SD?
    hist(rbinom(20, 5, 1/4))
 Mathematically, we can show that if X~bin(n,p),
                         n
    P(x) = P(X=x) =   p x 1  p n x for x=0,1,…,n
                      x
                         




                                   - 31 -
STAT 2010, Business Stat        2006                      Jaimie Kwon



             n    n!
            x  x!(n  x)! , which reads “n choose x,” is the number of
   Here,   
            
    different ways of choosing x objects from n objects.
 P(Xx) : cumulative probability
 Binomial table: Table 1 in appendix B provides values of cumulative
  probability for selected n and p. (x, P(X<=x))
 P(X3) =?
   Can compute by (1-P(X2)
   In general, P(Xx) = 1-P(X x-1)
 P(X=3)=?
   P(X=x) = P(Xx) – P(X x-1)
 General formula for mean and var of a binomial random variable :
     np
    2  np(1  p)
 Ex. 7.81-83, 89 (computer), 90

7.4 Poisson Distribution
 Another useful discrete probability distribution. # of occurrences of
  events in an interval of time or specific region of space.
 Some examples
                  e   x
 Formula: P(x) =          where e=2.71828…
                     x!
 Skip




                                - 32 -
STAT 2010, Business Stat           2006                    Jaimie Kwon




8 Continuous probability distributions
8.1 Probability density functions
 Need a completely different approach to deal with a continuous
   random variables since
    There are infinitely, uncountably many possible values
    the probability of individual value is virtually zero, i.e. P(X=x) = 0
      for any x
 Example. duration of a commute
    Table of (intervals: relative frequency)
    E.g. 0-10 min: .3, 10-20 min: .5, 20-30 min: .2
 We can only determine the probability of a range of values only
 The probabilities sum up to 1
 If we divide relative frequency by interval width, we have a set of
   rectangles whose area equals the probability that the random
   variable will fall into each interval.
 Imagine very large # of small intervals. A function f(x) that
   approximates the curve is called a probability density function (pdf):
 Requirements for a pdf over a range a ≤ x ≤ b
    f(x)≥0 for all x
    the total area under the curve between a and b is 1
 Probability of an interval: the area under the curve
 Integral calculus helps… but we don’t want to do it.
 Uniform distribution
    Uniform pdf is given by f(x) = 1/(b-a) where a ≤x ≤b



                                   - 33 -
STAT 2010, Business Stat           2006                   Jaimie Kwon



    P(x1 < X < x2) = (x2-x1)*(1/(b-a))
 Ex. 8.1, 9,10

8.2 Normal distribution
 The most important distribution in probability and statistics
                         1          (x  )2 
 Normal pdf: p( x)            exp          where e=2.71828… and
                        2 2         2 2 
   =3.14159…
 We write: X ~ N(,2), or X follows a normal distribution with mean 
   and standard deviation 




 Example: For a certain professor, the duration of the morning
   commute follows a normal distribution with mean 30 and standard
   deviation 10, i.e. the commute duration X ~ N(30, 102). Then we
   want to answer questions like:
    What’s the probability that the trip will take more than 50 minute?




    What’s the probability that the trip will take between 20 and 50
      minutes?




    On 2.5% of days, the trip will take longer than ___ minutes.




                                   - 34 -
STAT 2010, Business Stat       2006                    Jaimie Kwon



 Example: For a certain population (say, a large school), the
  student’s SAT score is normal distributed with mean 500 and
  standard deviation 50, i.e. the SAT score X ~ N(500, 502). Then we
  want to answer questions like:
   What’s the probability that a randomly selected student scores
     more than 600?




   What’s the probability that a randomly selected student scores
     between 400 and 550?




   To be in top 5% in the population, how much does a student
     need to score?




   To be in bottom 5% in the population, how much does a student
     need to score?




   Symmetrical, bell shaped




                               - 35 -
STAT 2010, Business Stat        2006                       Jaimie Kwon



   Centered around the mean 
   The spread specified by the variance 2
 Try applets:
   Normal Distribution Parameters
   Normal Distribution Areas
 Calculating normal probabilities




   Compute the area in the interval under the curve.
   Use the probability table
   Need a separate table for different  and ? No - by
     standardizing the random variable
 If X~ N(,2), the transformed variable, denoted by Z, is called the
                                              X 
  “standard normal random variable”: Z              ~ N(0, 1)
                                               


 “probability statement about X  probability statement about Z”


 If X ~ N(30, 102), what is P(25 < X < 40)?




                      25  30 X  30 40  30 
  P(25 < X < 40) = P
                                           = P(-.5 < Z < 1)
                         10    10       10    



                                - 36 -
STAT 2010, Business Stat        2006                   Jaimie Kwon



 “Z=-.5 corresponds to a value of X that is one-half a standard
  deviation below the mean”
 The table gives P(0 < Z < z) for positive z.
   P(Z > 0) =


   P(Z < 0) =


   P(Z > 2) =




     1-P(0 < Z < 2) =


   P(Z < -3) =




     P(Z>3) =


   P(0 < Z < 1) =


   P(-.5 < Z < 0) =




                                - 37 -
STAT 2010, Business Stat       2006     Jaimie Kwon




     P(0 < Z < .5) =


   P(-.5 < Z < 1) =




     P(-.5 < Z < 0) + P(0 < Z < 1)
     = P(0 < Z < .5) + P(0 < Z < 1) =


   P(1 < Z < 2) =




     P(0 < Z < 2) – P(0 < Z < 1) =


   P(-2 < Z < -1) =




                               - 38 -
STAT 2010, Business Stat         2006                     Jaimie Kwon




      P(1 < Z < 2) =


 We sometimes need to compute ZA, the value z such that the area
   to the right under the standard normal curve is A, i.e., such that P(Z
   > ZA)=A
    Use the table backward
    Z0.025 =


    Z0.05 =


    ZA = 100(1-A)th percentiles of a standard normal random variable




 If X ~ N(, 2), find x such that P(X > x) = A
    For example, if X ~ N(600, 502), find x such that P(X > x) = 0.05


    Convert the problem to Z




                                - 39 -
STAT 2010, Business Stat         2006                   Jaimie Kwon



   Find z0.05




   Convert back to X space




   How about top 10 percent? How about top 1 percent?
 Ex.8.19-24, 31-32, 37-41, 58

8.3 Exponential distribution
8.4 Other continuous distributions
 Student-t distribution
   Very commonly used in statistical inference. (chapters 12, 13, 15,
     17, 18, 19)
   We use symbol T() to denote the random variable that follows
     the student-t distribution with  degrees of freedom.
   This we write as T() ~ t() (a la X ~ N(, 2))
   We sometime write T() as T, if  is clear from the context.
 Example: if a random variable T follows the student-t distribution
  with 10 degrees of freedom, then:
   What’s the probability that T will be greater than 1.812?




   What’s the value of t such that T is greater than t 5% of time?




                                - 40 -
STAT 2010, Business Stat         2006                     Jaimie Kwon




   What’s the value of t such that T is less than t 5% of time?




 The distribution looks very close to the standard normal; the larger v
  is the closer it is.
   E(T) = 0 and V(T) = /(-2) for >2




 Computing student-t probabilities
   Student-t probabilities can be computed using computer (TDIST
     in Excel)
   Finding student t values such that P(T  t A,v )  A (TINV in Excel)
      Table 4 of the book
      t.05,10 = 1.812
      t.05,25 = 1.708
      -t.05,25 = -1.708
 Chi-squared distribution
   X2 ~ 2(v)
   Looks like …. For different v
   Finding chi-squared values P( 2   A, )  A
                                         2




                                 - 41 -
STAT 2010, Business Stat           2006     Jaimie Kwon



      (use table 5)
      2.05,8 = 15.5073
      2.95,25 = 2.73264
 F distribution
   F~F(v1, v2)
   Finding P( F  FA, , )  A
                        1   2



 Ex. 8.83, 84




                                   - 42 -
STAT 2010, Business Stat      2006     Jaimie Kwon



 Midterm score (Out of 60)
    mean(x) = 48.4
 median(x) = 49





                                      >




                              - 43 -
STAT 2010, Business Stat       2006                    Jaimie Kwon




9 Sampling distributions
9.1 Sampling distribution of the mean
 Example (same as above): For a certain population (say, a large
  school), the student’s SAT score is normally distributed with mean
  500 and standard deviation 50, i.e. the SAT score X ~ N(500, 502).
 If we randomly sample 25 students from the school and have them
  take SAT, what can we say about the distribution of the sample
  mean SAT score? In particular,
   What’s the mean of X ?




   What’s the standard deviation of X ?




   What’s the distribution of X ?




   How does a conclusion changes if the original distribution of the
     inidividual score was not normal?




                              - 44 -
STAT 2010, Business Stat         2006                    Jaimie Kwon



      (exact; also, we don’t need many n)


   In particular, what’s P( X > 550)? What’s P( X < 450)? What’s
      P(450 < X < 550)




      Compare this with P(X > 550), P(X < 450) and P(450 < X < 550)
      [the effect of smaller standard deviation]
  




 Fair die example; 1 die; 2 dice; 5? 10? Sampling distribution of the
  mean of fair dice and CLT


 Let X be the outcome of a single throw of a fair die
   Distribution of X




                                 - 45 -
STAT 2010, Business Stat          2006                    Jaimie Kwon



         X and  X are computed to be 3.5 and 2.92
                   2




   The “sampling distribution of the mean” of two fair dice, X .




      Takes on what values?




         1.0, 1.5, 2.0, …., 5.5, 6.0


      The “sampling distribution” of a statistic is created by repeated
         sampling from one population.
         X and  X , computed to be 3.5 and 1.46 (half of the original)
                   2



      Consistent with what the theory tells us. See below.




   Sampling distribution of X for larger n=5, 10, 25.
     X  X




                                 - 46 -
STAT 2010, Business Stat           2006                   Jaimie Kwon



                2
         
          2
          X
                     (distribution becomes narrower when n increases) or
                n
                
         X 
                 n
      Sampling distribution of X becomes increasingly bell shaped.




                                   - 47 -
STAT 2010, Business Stat             2006                 Jaimie Kwon



 To summarize….
 The sampling distribution of the sample mean X :
    X   X , and
          2                             
   X 
    2
               or, equivalently,  X        .
          n                              n
  Also, the distribution is approximately normal regardless of the
  original population distribution, for a sufficiently large sample size
  (say, n  30). The larger the sample size is, the more closely the
  sampling distribution of X will resemble a normal distribution.
  If the original distribution of X is normal, then X is exactly normal.



 The result is called the Central Limit Theorem (CLT):
 The sampling distribution of the sample mean of a random sample
  drawn from any population is approximately normal for a sufficiently
  large sample size (say, n  30). The larger the sample size is, the
  more closely the sampling distribution of X will resemble a normal
  distribution.


 Implication for the inference?
   A claim has been made that the SAT score for a private school
      has the distribution X~N(550, 100^2). To check this claim, a
      sample of 25 people have been surveyed and the sample mean
      was found to be 500. What is the P(X-bar < 500) if the dean’s
      claim was true.



                                     - 48 -
STAT 2010, Business Stat               2006                          Jaimie Kwon



    P(X-bar < 500) = P(Z<(500-550)/(100/5)) =P(Z<-2.5) = …
    What’s the conclusion?? The precursor of hypothesis testing
    Z.025 = 1.96
    P(-1.96<Z<1.96)=.95
                   X 
      P(1.96          1.96 )  .95
                  / n
      P(   1.96 / n  X    1.96 / n )  .95
    In general, P(  z / 2 / n  X    z / 2 / n )  1  
        For the above example, P(760.8<X-bar < 839.2) = .95
        P(748.5 < X-bar < 851.5) = .99
        the precursor of interval estimation
 Ex. 9.5, 6, 7, 9, 10, 11, 15, 16

9.2 Sampling distribution of a proportion
 Among a very large population, 48% support a certain bill and 52%
   do not. If we randomly select 100 people, what can we say about
   the sampling distribution of the sample proportion of the people who
   support the bill? Among others,
    What’s the mean of the sample proportion?




    What’s the standard deviation of the sample proportion?




                                       - 49 -
STAT 2010, Business Stat        2006                    Jaimie Kwon



   What’s the distribution of the sample proportion?




   What’s the chance that the sample proportion is greater than
     50%?




 In binomial experiment, the estimator of the population proportion of
                                           X
  successes is the sample proportion p 
                                     ˆ       , the # of successes
                                           n
  divided by the sample size.




                                - 50 -
STAT 2010, Business Stat                 2006                       Jaimie Kwon



 Normal approximation to binomial experiment: Distribution of a
  sample proportion is given by:
   E ( p)  p
       ˆ
                  p(1  p)
  V ( p)   P 
      ˆ      2
             ˆ               or
                       n
           p (1  p )
  P 
   ˆ                  .
               n
                                    ˆ
                                   P p
  Also, the variable Z =                         is approximately standard normal,
                                  p(1  p) / n

  provided that n is large. (i.e. both np ≥ 5 and n (1  p) ≥ 5)
 Ex. 9.30, 34

9.3 Sampling distribution of the difference between two means
 For two separate population A and B (say, two large schools), the
  SAT score of individual student follows N(550, 502) and N(500, 502)
  distributions, respectively. In other words, if we let X1 and X2 to
  denote respective random variables, X1 ~ N(550, 502) and X2 ~
  N(500, 502). If we randomly select 25 students each from population
  A and B, what is the distribution of the difference between two
  sample means, X 1  X 2 ? In particular,
   What’s the mean of X 1  X 2 ?




   What’s the standard deviation of X 1  X 2 ?




                                         - 51 -
STAT 2010, Business Stat                          2006                        Jaimie Kwon



     What’s the distribution of X 1  X 2 ?




     What’s P( X 1  X 2 > 60)=?




     How do the above change if X1 and X2 don’t follow a normal
           distribution?





 For independent random samples of size n1 and n2 drawn from of
    two normal populations N(1, 12) and N(2, 22), respectively, the
    difference of sample means                    X1  X 2   has a normal distribution. Even
    when the two original distributions are not normal, the distribution of
    X1  X 2       is approximately normal if both n1 and n2 are large (say both
    n1  30 and n2  30). Also,
                                            12       2
                                                       2
     X X        1  2 and  X  X 
                                 2
                                                  
       1     2                    1   2
                                           n1         n2
 Ex. 9.45, 46




                                                  - 52 -
STAT 2010, Business Stat          2006                  Jaimie Kwon




10 Intro to estimation
 So far, we assumed known parameters and study the sampling
  distribution of various statistics.




 What if, we don’t know the value of parameters but have observed a
  single value of a statistic?
   We want to say something about the parameters.




 For a certain population (say, a large school), the student’s SAT
  score is normally distributed with mean 500 and standard deviation
  50.
 If we randomly sample 25 students from the school and have them
  take SAT, what can we say about the distribution of the sample
  mean SAT score? In particular,
 A certain school has the population mean score of 500 and standard
  deviation of 50. If we randomly sample 25 students from the school
  and have them take SAT, what can we say about the distribution of
  the sample mean SAT score? In particular, ….




 A certain school has the unknown population mean score  and
  standard deviation of 50. When we randomly sampled 25 students



                                  - 53 -
STAT 2010, Business Stat        2006                    Jaimie Kwon



    from the school and had them take SAT, we observed x = 520.
    What can we say about the distribution of the sample mean SAT
    score?




 A certain school has the unknown population mean score  and
    unknown standard deviation . When we randomly sampled 100
    students from the school and had them take SAT, we observed x =
    52 and s = 45. What can we say about the distribution of the sample
    mean SAT score?

 Two general procedures for inference: estimation and hypothesis
    testing

10.1 Concept of estimation
 A “point estimator” draws inferences about a population by
    estimating the value of an unknown parameter using a single value
    or point
 An “interval estimator” draws inferences about a population by
    estimating the value of an unknown parameter using an interval
     E.g. mean weakly income of sample of 25 students is $400. (can
       also use $380-$420)
 An “unbiased estimator” of a population parameter is an estimator
    whose expected value is equal to that parameter



                                - 54 -
STAT 2010, Business Stat              2006                   Jaimie Kwon



 An unbiased estimator is said to be “consistent” if the difference
   between the estimator and the parameter grows smaller as the
   sample size grows larger
      X is consistent estimator of 

10.2 Estimating the population mean when the population SD is
     known
 In general, confidence interval is of the form:


   (the estimate)  (a constant) (standard error of the estimate)




 E.g. (SAT score) We believe X ~ N(, 502). For a certain school, if x
   = 520 for n = 25, what is 95% CI for ? 90% CI? 99% CI?




 100(1-)% confidence interval estimator of the unknown population
                                                       
   mean  is  x  z / 2 , x  z / 2  , or x  z / 2
                                       
                          n           n                n
    “Lower confidence limit” and “upper confidence limit”
    The probability 1- is called the “confidence level”




                                      - 55 -
STAT 2010, Business Stat          2006                   Jaimie Kwon



 Table of (100(1 – )%, , /2, z/2) for 90%, 95%, 99% confidence
  levels




 Why? : P[100(1 – )% confidence interval containing ] = (1 – )
                       X 
   The variable Z         is standard normal or approximately
                       / n
     standard normal
   CI is random, but  is not.


 Interpreting the CI: It’s important to realize that we observe only one
  sample and only one value of x . Cannot be correct all the time. Aim
  to be correct 95% of time.
                                                                    
 The sampling error of 100(1 – )% confidence interval is z / 2       .
                                                                    n
   We want {larger, smaller} sampling error, or {wider, narrower} CI


   Larger  leads to {narrower, wider} interval


   Increasing the confidence level 100(1-)% leads to {narrower,
     wider} interval


   Increasing sample size n leads to {narrower, wider} interval




                                  - 56 -
STAT 2010, Business Stat      2006                    Jaimie Kwon



 Ex. 10.9, 10, 11, 15, 21

10.3 Selecting the sample size
 Sample size to estimate the mean within W at (1-) confidence level,
       z  
                  2

   n    /2 
        W 
 Ex. In estimating the population mean SAT score, I want to estimate
  it with W=10 for confidence level = .90 or alpha=1. How many
  samples do I need?
 Ex. 10.41, 42, 51




                              - 57 -
STAT 2010, Business Stat        2006                     Jaimie Kwon




11 Introduction to hypothesis testing
 Are there enough statistical evidences to enable us to conclude that
  a belief or hypothesis about a parameter is supported by the data.

11.1 Concepts of hypothesis testing
 E.g. Is a particular school A has the mean SAT score greater than
  the national average of 0 = 500? We assume X ~ N(, 502) and just
  observed x = 510 for n=100.


   The null hypothesis H0:
      “the private school has the same mean SAT score as the
        national average of 500 (usually specified as the status quo)”


   The alternative hypothesis H1:
      “the private school has the mean SAT score higher than 500”


   There are two possible decisions: accept H0 or reject H0.


                                           Decision
                                  Accept H0           Reject H0
            Truth H0 true
                   H1 true


      More common to say “cannot reject H0” than “Accept H0”




                                - 58 -
STAT 2010, Business Stat         2006                       Jaimie Kwon



   The decision is either correct or wrong. When the decision is
     wrong, we commit either:


      Type I error: wrongfully reject H0
         = P(Type I error)


      Type II error: wrongfully accept H0
         = P(Type II error)


   The type I error probability of a certain testing procedure is called
     the significance level of the test, or sometimes just level of the
     test and is written as .


   The test statistic: a statistic which we base our decision upon
      “The observed sample mean score”
      If the value of test stat is inconsistent with H0 (and more
        consistent with H1), we reject H0.
      “Sufficient evidence” = “evidence beyond a reasonable doubt”
      We use “sampling distribution” of the test statistic to decide
        how sufficient the evidence is


   The rejection region is a range of values such that if the test
     statistic falls into that range, we reject the H0 in favor of H1




                                 - 59 -
STAT 2010, Business Stat         2006                     Jaimie Kwon



   Testing begins by assuming H0 is true. We reject the H0 if the test
     statistic has the value that is inconsistent with H0 but is consistent
     with H1. But how inconsistent does it have to be for us to reject it?
     That’s up to us. How aggressive do we want to be in rejecting the
     null?


      Aggressive
         More likely to reject the correct H0
         More likely to commit type I error
         Test with a larger 


      Conservative
         Less likely to reject the correct H0
         Less likely to commit type I error
         Test with a smaller 


   A particular decision rule (test) is obtained by deciding on level ,
     the type I probability we are willing to accept. Typically,  = 0.05
     = 5% is used.


   The conclusion of the test is stated either as:
      “We reject H0 at 5% significance level”
      “We cannot reject H0 at 5% significance level”, etc.




                                 - 60 -
STAT 2010, Business Stat        2006                      Jaimie Kwon



   Somehow, we don’t say
      “accept H0 at 5% significance level” or
      “accept H1 at 5% significance level”.




   The P-value of a test is the probability of observing a test statistic
     at least as extreme as the one observed given that H0 is true.


   In the example, H0:  = 500. The alternative H1 could be of the
     form:
      H1:  > 500
      H1:  < 500
      H1:  ≠ 500


      Testing either the first two are called one-sided hypothesis
        testing and testing the third called two-sided hypothesis
        testing.



11.2 Testing the population mean when the population standard
     deviation is known
 Recall the example: Is a particular school A has the mean SAT
  score greater than the national average of 0 = 500? We assume X
  ~ N(, 502) and just observed x = 510 for n=100.




                                - 61 -
STAT 2010, Business Stat           2006                      Jaimie Kwon



 For testing H0:  = 0 vs. H1:  > 0,
   the test at level  rejects H0 if
        x  0
   z            > z
        / n
   The p-value = P(Z > z)


    That test has type I error probability = .


 If H0 is rejected, we say “the result is statistically significant at
   significance level ”


 Can we reject H0 at  = .10?




    At  = .05?




    At  = .01?




        (1.28, 1.64, 2.33 for 10%,5%,1%)


 P-value = P(Z > observed z)




                                   - 62 -
STAT 2010, Business Stat           2006                       Jaimie Kwon



    In the example, P-value = P( X > 510, given that H0 is true)




        = P(Z>2) = .0228
   
    Large P-value suggests H0 is more likely
    Small P-value suggests H1 is more likely


    The level  hypothesis testing is equivalent to “Reject H0 if p is
        { > or  } ”


    It is a better practice to report the P-value than just “accept” or
        “reject”


 The test at level  reject H0 if and only if P-value < .
   Equivalently, P-value is the smallest significance level at which a
   test can reject H0.


 For testing H0:  = 0 vs. H1:  < 0,
   the test at level  rejects H0 if
        x  0
   z            < z .
        / n
   The p-value = P(Z< z)


 E.g. For another school B, the test score X ~ N(, 502). Is school B
   significantly worse than the national average?



                                   - 63 -
STAT 2010, Business Stat           2006                 Jaimie Kwon



   We observed x = 492.5 for n = 100.


   Can we reject H0 at  = 0.05?




   What’s the P-value?




        P-value = P(Z<-1.5)= 0.0668




 For testing H0:  = 0 vs. H1:  ≠ 0,
   the test at level  rejects H0 if
            x  0
   | z |            > z/2 .
            / n
   (Or, equivalently, if z < z/2 or z > z/2)
   The p-value = 2 P(Z>|z|)


 E.g. For another school C, the test score X ~ N(, 502). Is school C
   significantly different from the national average? Observed x =
   492.5 for n=100.


   Can we reject H0 at  = 0.05?



                                   - 64 -
STAT 2010, Business Stat             2006                   Jaimie Kwon




   What’s the P-value?




    P-value =2 P(Z>1.5) = 0.1236


 Interpreting the test results


 Ex. 11.7-9, 13-15, 28

11.3 What about type II error probability?
 Recall that a test at level  rejects H0:  = 0 for H1:  > 0 rejects if
        x  0
   z          > z. Let’s suppose that indeed H1 is true, specifically, the
        / n
   true  = 1 > 0. Then


   P(Type II error when  = 1)
   = P(Not reject H0 when  = 1)
          x  0
   = P(           z when  = 1)
          / n
   = P( x  0  z / n when  = 1)
        x  1         1
   = P(         z  0     when  = 1)
        / n         / n
                  
   = P( Z  z  0 1 )
                 / n



                                     - 65 -
STAT 2010, Business Stat        2006                      Jaimie Kwon



  : increases as 1 (> 0) gets closer to 0 (i.e., as the problem
  becomes harder)


 Decreasing Type I error (smaller ) leads to larger type II error
   There’s no free lunch


 If n increases, Type II error decreases for the given 
   Since there is more information


 Power of the test = 1  P(Type II error) = P(correctly rejecting the
  null)
 OC (operating characteristic) curve
 Ex. 11.48, 49, 61 (??)

11.4 The road ahead
 You’re pretty much done for the quarter!


 Three steps
   Define the problem
   Identify the appropriate method
   Interpret the results
 Describe a population
 Compare two populations
 Compare two or more populations
 Analyze the relationship between two variables



                                - 66 -
STAT 2010, Business Stat           2006                    Jaimie Kwon




12 Inference about a population


12.1 Inferences about population mean  for a normal population,
      unknown
 Is a particular school has the mean SAT score greater than the
    national average of 0 = 500? We assume X ~ N(,2) with unknown
    . We just observed x = 510 and s = 45.0 for n = 25.


 100(1-)% confidence Interval for ,  unknown
                     s
    x  t / 2,n1
                      n


 For H0:  = 0 vs. H1:  > 0,
    test at significance level  rejects H0 if
     t > t,n-1
    The P-value is P(t  computed t)



 For H0:  = 0 vs. H1:  < 0,
    test at significance level  rejects H0 if
     t <  t,n-1
    The P-value is P(t  computed t)




                                   - 67 -
STAT 2010, Business Stat              2006               Jaimie Kwon



 For H0:  = 0 vs. H1:   0,
  test at significance level  rejects H0 if
    |t| > t/2,n-1
  (Or, equivalently, if t < t/2,n-1 or t > t/2,n-1)
  The P-value is 2P(t  |computed t|)




 The effect of non-normality on the inference based on t distribution
   What kind of non-normality? (skewed, heavy tailed)
   Effect on the power, level of test, etc.
 Consider “robust methods” of estimation and inference
 Checking required condition
   Normality by histogram; if n is large, OK.
 Ex. 12.1, 2d, 3d, 4d, 8a, 9, 13, 21

12.2 Inference about a population variance
 Skip

12.3 Inference about a population proportion
 E.g. from an exit poll of n = 765 voters, x = 407 people were
  observed to have voted for a bill.


  What’s p ?
         ˆ



  What’s the 95% Confidence interval?



                                     - 68 -
STAT 2010, Business Stat                      2006          Jaimie Kwon




   For H0: p = .5 vs. H1: p > .5, can we reject H0 at  = 5%?


        x
 p
  ˆ
        n
 If np  5 and n(1  p)  5 , the distribution of p can be approximated by
                                                   ˆ
                       p(1  p)  2 
   N (  p ,  )  N  p, 
              2
                                   
                                      
         ˆ    ˆ
              p
                           n
                                 



 100(1-)% confidence interval for the population proportion p is
                                               p (1  p )
                                               ˆ      ˆ
   given by       p  z / 2 p
                  ˆ          ˆˆ   where  ˆ 
                                         ˆ
                                                    n


 For H0: p = p0 vs. H1: p > p0, the test at level  rejects H0 if
        p  p0
        ˆ
   z             > z.
         p
          ˆ


   P-value = P(Z > z)



 For H0: p = p0 vs. H1: p < p0, the test at level  rejects H0 if
        p  p0
        ˆ
   z             < z.
         p
          ˆ


   P-value = P(Z < z)




                                              - 69 -
STAT 2010, Business Stat           2006                      Jaimie Kwon



 For H0: p = p0 vs. H1: p > p0, the test at level  rejects H0 if
         p  p0
         ˆ
   z             > z/2.
          p
           ˆ


   (Or, equivalently, if z < z/2 or z > z/2)
   P-value = 2 P(Z > |z|)


 Sample size for estimating p within W at confidence level alpha =
                            2
       z     p (1  p ) 
   n    /2             . Conservative estimate is given by the formula for
             n          
                        
   p = ½.
   ˆ

    Wish to estimate the above proportion within .03. What’s required
      n?
 Ex. 12.54, 58, 66




                                   - 70 -
STAT 2010, Business Stat         2006                     Jaimie Kwon




13 Inference about comparing two population
 Keywords: pooled variance estimator; equal-variances test statistic,
 How to tell if two variances are equal? Methods are there but
  informal method would be fine for now.
 If there is no strong evidence against equal variance, it’s usually
  “better” to assume the equal variance one. (why?)
 Checking required conditions
   Draw histogram to check normality; if sample size is large, we’re
      OK; this t-testis robust too; if not normal there are nonparametric
      methods
 E.g. comparing mean SAT score for school 1 and school 2
   X1~N(1, 12), X2~N(2, 22)
   For n1=25 and n2=25 samples, x1 =530; x2 =500 and s1=90 and
      s2=120
   Assuming equal variances; sp=106; denom=30; 95% CI for mu1-
      mu2? (-30, 90); df=48; P-value = .16
   Assuming unequal variances; nu=44.5 or 45 or 44; 95% CI for
      mu1-mu2= (-30, 90) (slightly larger than the previous one)

13.1 Inference about the difference between two means:
     independent samples
 Consider three cases
   Case 1. Both population distributions are normally distributed
      with  1   2
   Case 2. Both sample sizes n1 and n2 are large



                                - 71 -
STAT 2010, Business Stat                            2006        Jaimie Kwon



     Case 3. The sample sizes n1 or n2 are small and the population
        distributions are non-normal
 Concentrate on Case 1 for now
     Population distributions are normal with equal variances
 Statistics: y1 , y2 , s1 , s2
 Confidence interval for 1   2 , independent samples:

     y1  y2   t / 2 (n1  n2  2)s p   1 1
                                                    where
                                            n1 n2
            (n1  1) s12  (n2  1) s2
                                     2
    sp 
                  n1  n2  2

     Why? (sample distribution of y1  y2 )
     Reasonably stable for mound-shaped distributions and
        approximately equal SD
 A statistical test for 1   2 , independent samples:
     H0: 1  2  D0
     Ha: 1  2  D0
         (D0 is a specified value, often 0)
     T.S. : t 
                         y1  y2   D0
                              1 1
                         sp     
                              n1 n2
     R.R. : for a level , reject H0 if t  t (n1  n2  2)
 Check assumptions and draw conclusions
     Test whether the mean score of school 1 is higher than school 2.
        Use =.05.
     P-value of the test?
     95% confidence interval on the difference of means?



                                                    - 72 -
STAT 2010, Business Stat                   2006                                           Jaimie Kwon



   sp=?
 Three critical conditions
   Two random samples are independent
   Population distributions are normal or mound-shaped
   Two population variances are equal
 Approximate t-test for independent samples, unequal variance
   T.S. t ' 
                 ( y1  y2 )  D0
                                    and d.f. is  
                                                           s
                                                            2
                                                            1   / n1  s 2 / n2
                                                                         2
                                                                                         [ or
                     s12 s2
                        
                          2
                                                      s
                                                       2
                                                       1 / n1     
                                                                  2
                                                                s2 / n
                                                               2 2
                                                                                      
                                                                                      2


                     n1 n2                             n1  1    n2  1
                    (n1  1)( n2  1)                     s2 / n
      df                                      where c  21 12 (round down to the
             (1  c) 2 (n1  1)  c 2 (n2  1)          s1 s2
                                                            
                                                         n1 n2
     nearest integer)]
   Similar for confidence interval.
 Ex. 13.1a, 2a, 3a, 5b, 7

13.2 Observational and experimental (controlled study) data
 The latter is more expensive but can shed more right on causality
 E.g. Slytherine may not be a better school than Gryffindor; it may be
  that simply there are more good students going there; what kind of
  experimental study would be possible?

13.3 Inference about the difference between two means: matched
     pairs experiment
 What if each measurement in one sample is “matched” or “paired”
  with a particular measurement in the other sample?
   E.g. comparing repair estimates from two garages for each of 15
     cars damaged by accidents



                                           - 73 -
STAT 2010, Business Stat             2006                     Jaimie Kwon



    Two-sample t-test gives a nonsense result. Why?
 Also called ‘paired t-test’ (more common than ‘matched t-test’
 Ask: is there some natural relationship exist between each pair of
   observations?
 E.g. SAT score for before-and-after attending certain prep school;
   
 Not the two-sample t-test. But run the regular t-test on the
   differences
 Solution: use differences di  y1i  y2i and obtain its sample mean and
   SD, d , sd .
 Test hypotheses about d  1  2
 Paired t test
    H0: d  D0
    Ha: d  D0
        (D0 is a specified value, often 0)
                  d  D0
    T.S. : t 
                  sd / n
    R.R. : for a level , reject H0 if t  t (n  1)
    Check assumptions and draw conclusions
 Confidence interval for  d based on paired data: d  t / 2 (n  1)
                                                                         sd
                                                                          n
 Of course, assuming
    the distribution of the d i s is (close to) a normal distribution
    the differences are independent




                                     - 74 -
STAT 2010, Business Stat                           2006        Jaimie Kwon



13.4 Inference about the ratio of two variances
 Skip

13.5 Inference about the difference between two population
     proportions
                    y1          y
 Use  1 
       ˆ               and  2  2
                            ˆ
                    n1          n2
 Confidence intervals for 1-2 are given by  1   2  z / 2 ˆ ˆ where
                                               ˆ     ˆ           ˆ 1   2


                    1 (1   1 )  2 (1   2 )
    ˆ ˆ 
    ˆ                            
      1   2
                        n1                n2

 Statistical test for H0: 1-20 etc. is based on
                       1   2
                        ˆ    ˆ
   z                                       .
               1 (1   1 )  2 (1   2 )
                             
                  n1                 n2
 E.g. accident rate of vehicles with ABS and those without ABS




                                                   - 75 -
STAT 2010, Business Stat               2006                            Jaimie Kwon




14 Statistical inference: review of chapters 12 and 13



 Graphical methods and numerical measures for univariate data
    Variable type                          Methods
    Categorical         Frequency, relative frequency, p ;
                                                       ˆ

                        bar-chart, pie-chart
X
    Interval            x , median, s , percentiles;
                        histogram, boxplot, (stem-and-leaf, ogive)


 Graphical methods and numerical measures for bivariate data
                                                  Y
                             Categorical                         Interval
    Categorical contingency table , p1 , p 2 ; x1 , x2 , s1 , s2 ;
                                    ˆ ˆ

                    bar-chart                         side-by-side boxplots
X
    Interval        ?                                 r,   Cov(X,Y), y  b0  b1 x ;
                                                                     ˆ

                                                      scatter plot




 Univariate statistical inference techniques
    Variable type                       Methods
    Categorical         One-sample proportion
X
    Interval            One-sample z-test; one-sample t-test



                                       - 76 -
STAT 2010, Business Stat        2006                        Jaimie Kwon




 Bivariate statistical inference techniques
                                         Y
                       Categorical               Interval
    Categorical Two-sample proportion; Two-sample t-test;
X               Chi-squared analysis         ANOVA
    Interval    Logistic regression          Regression




                                - 77 -
STAT 2010, Business Stat   2006     Jaimie Kwon




15 Analysis of variance
 Skip

16 Chi-squared tests
 Skip




                           - 78 -
STAT 2010, Business Stat                       2006                        Jaimie Kwon




17 Simple linear regression and correlation
 Regression analysis is used to predict the value of one variable (Y)
   on the basis of other variables ( X 1 , X 2 ,... X k )
 E.g. midterm score vs. final score for a class

17.1 Model
 Simple linear regression model:
   y   0  1 x  

   where  is the “error variable”

17.2 Estimating the coefficients
 Least squares regression line is obtained by finding b0, b1 which
                      n
   minimizes         (y
                     i 1
                            i    yi ) 2 , where y , the (predicted) value of y, is
                                  ˆ              ˆ

   determined by the line
   y  b0  b1 x
   ˆ

 “Least squares line coefficients” are given by
          sxy
   b1     2
                and b0  y  b1 x .
          sx
 See the old notes for formula for s xy , s x , etc.
 E.g. Computing the regression line from basic statistics

17.3 Required conditions on the error variable
 Conditions:
    The probability distribution of  is normal
    E()=0




                                               - 79 -
STAT 2010, Business Stat                          2006     Jaimie Kwon



    The standard deviation of  is , which is constant regardless of
      the value of x.
    The value of  for different observations are independent
 (Or, people says  i ~ iid N(0,2))

17.4 Assessing the model
 How well does our model fit the data? (There may be no relationship
   at all!)
 “The sum of squares for error (SSE)” can also be computed as
                                          2 s xy 
                                                 2

   SSE    y i  y i 
                   ˆ             (n  1) sy     
                           2
                                               sx 
                                                 2
              i                                   
    The {smaller, larger} SSE suggests more accurate model
 “The standard error (SE) of estimate” is an estimate of , given by
            SSE
   s 
            n2
    The {smaller, larger}  suggests more accurate model
 To formally test whether the slope is non-zero, do the following:
    H0: 1=0 vs. H1: 1≠0
    Test statistic is given by
            b1  0
       t          where
              sb1
                     s
       sb1 
                  (n  1) s x
                            2



    If assumptions regarding error variable hold: Under H0, t follows
      student t distribution with v=n-2 degrees of freedom
    At significance level , reject H0 if |t|>t/2(n-2)




                                                  - 80 -
STAT 2010, Business Stat                           2006     Jaimie Kwon



    100(1-)% C.I. for 1 is given by
       b1  t / 2 (n  2) sb1

 Coefficient of determination is given by
            2
           sxy                SSE
   R2           1                      r2
                             yi  y 
           2 2                         2
          s s
           x y
                            i

   =(explained variation in y)/(total variation in y)
   = (Regression SS)/(Total SS)
    The {higher, lower} value of R2 means better fit of the linear
       model
 Need to be able to extract SOME information from Minitab output
 Typical disclaimer: correlation doesn’t imply causality

17.5 (Optional) Applications in finance

17.6 Using the regression equation
 E.g. Things we really care about: Predicting the final score from the
   midterm score
 Predicting the particular value of y for a given x:
                                  1 ( xg  x )
                                               2

   y  t / 2 (n  2) s
   ˆ                            1 
                                  n (n  1) sx2



 Estimating the expected value of y for a given x:
                                1 ( xg  x )
                                             2

   y  t / 2 (n  2) s
   ˆ                             
                                n (n  1) sx2



 These intervals gets {wider, narrower} as xg moves away from x

18 Multiple regression
 Extension of the simple linear regression to multiple X variables




                                                   - 81 -
STAT 2010, Business Stat        2006                    Jaimie Kwon



 E.g. predicting final from midterm, quiz #1, quiz #2 scores




                                - 82 -
STAT 2010, Business Stat         2006                   Jaimie Kwon




19 Logistics and things you/I really care
 Glossary
 See the syllabus

19.1 Couple of words about quiz #2 and final
19.1.1      Quiz #2
 n=25; Sample mean = 23.64; sampled median = 24; sample
   SD=4.26





 Correlation between quiz#2 and midterm score=0.572

19.1.2      Final
 No need for cheat-sheet for part I (you will be given a formula sheet)
 Make your own cheat-sheet for part II (covers chapter 8~)
 Need Assist form for part I of the final!




                                 - 83 -
STAT 2010, Business Stat           2006                       Jaimie Kwon



 A reminder that effective Winter Quarter 2006, Assist (“Scantron”)
    forms will no longer be provided by the Statistics Department for the
    STAT 1000 and STAT 2010 standardized tests.
 Students can purchase Assist forms at the CSUEB Bookstore for 50
    cents
 INSTRUCTIONS FOR COMPLETING ASSIST FORMS:


    1. Students should enter as much of their names as possible in the
    “Your Last Name” boxes.


    2. The two letters of their Net ID are entered in the "First Initial"
    and "Middle Initial" boxes on that same line.


    3. The four digits of their Net ID should be entered in the first four
    boxes of the "Social Security Number" section.





                                   - 84 -
STAT 2010, Business Stat     2006     Jaimie Kwon




19.2 Practice midterm (50 minutes)
   …




                             - 85 -
STAT 2010, Business Stat          2006                    Jaimie Kwon




19.3 Practice final (50 minutes)
 Is anti-lock brake system (ABS) in cars really effective? If it were
   effective,
    The number of accidents would decrease, and
    The cost of accident repairs would be less
 Data were collected on 500 cars with ABS and 500 cars without.
   The number of cars involved in accidents was recorded, as was the
   cost of repairs.
    42 out of 500 cars without ABS had accident and 38 out of 500
      cars with ABS had accident in a given year. What can we
      conclude?
    For the repair cost for the two groups, we obtain:
         n1 =42, x1 = 2,075 and s1= 671

         n 2 =38, x 2 = 1,714 and s2= 624

 For the two situations above, perform:
    Compute the 95% CI for the parameter of interest
    Set up the null and alternative hypotheses
    Compute test statistic and perform the test at 5% significance
      level
    Compute the P-value for the test (if you can)




                                  - 86 -
STAT 2010, Business Stat                    2006                               Jaimie Kwon



    STAT/MATH 6401, Advanced Probability I, Fall 2005 Course Note


                                      Dr. Jaimie Kwon
                                      August 16, 2012


Table of Contents


1    What is statistics? ...........................................................................3
2    Graphical and tabular descriptive statistics .....................................4
    2.1   Types of data ...........................................................................4
    2.2   Techniques for nominal data ....................................................5
    2.3   Graphical techniques for interval data ......................................5
    2.4   Describing the relationship b/w two variables ...........................7
    2.5   Time series data .......................................................................8
3    Art and science of graphical presentations .....................................8
4    Numerical descriptive techniques ...................................................9
    4.1   Measures of central location .....................................................9
    4.2   Measures of variability ............................................................10
    4.3   Measures of relative standing and box plots ...........................11
    4.4   Measures of linear relationship ...............................................13
    4.5   Comparing graphical and numerical techniques .....................15
    4.6   General guidelines for exploring data .....................................15
5    Data collection and sampling ........................................................16
    5.1   Methods of collecting data ......................................................16




                                            - 87 -
STAT 2010, Business Stat                      2006                              Jaimie Kwon



    5.2     Sampling ................................................................................16
    5.3     Sampling plans .......................................................................17
    5.4     Sampling and nonsampling errors ..........................................18
6    Probability ....................................................................................19
    6.1     Assigning probability to events ...............................................19
    6.2     Joint, marginal, and conditional probability .............................20
    6.3     Probability rules and trees ......................................................23
    6.4     Bayes’ Law .............................................................................24
    6.5     Identifying the correct method ................................................24
7    Random variables and discrete probability distributions ...............25
    7.1     Random variables and probability distributions.......................25
    7.2     Bivariate distributions .............................................................28
    7.3     Binomial distribution ...............................................................31
    7.4     Poisson Distribution ................................................................32
8    Continuous probability distributions ..............................................33
    8.1     Probability density functions ...................................................33
    8.2     Normal distribution .................................................................34
    8.3     Exponential distribution ..........................................................40
    8.4     Other continuous distributions ................................................40
9    Sampling distributions ..................................................................44
    9.1     Sampling distribution of the mean ..........................................44
    9.2     Sampling distribution of a proportion ......................................49
    9.3     Sampling distribution of the difference between two means ...51
10        Intro to estimation ......................................................................53




                                              - 88 -
STAT 2010, Business Stat                    2006                              Jaimie Kwon



 10.1       Concept of estimation ..........................................................54
 10.2       Estimating the population mean when the population SD is
 known 55
 10.3       Selecting the sample size ....................................................57
11    Introduction to hypothesis testing ..............................................58
 11.1       Concepts of hypothesis testing ............................................58
 11.2       Testing the population mean when the population standard
 deviation is known ...........................................................................61
 11.3       What about type II error probability? ....................................65
 11.4       The road ahead ...................................................................66
12    Inference about a population .....................................................67
 12.1       Inference about a population mean when the sd is unknown
            67
 12.2       Inference about a population variance .................................68
 12.3       Inference about a population proportion ..............................68
13    Inference about comparing two population ................................71
 13.1       Inference about the difference between two means:
 independent samples ......................................................................71
 13.2       Observational and experimental (controlled study) data ......73
 13.3       Inference about the difference between two means: matched
 pairs experiment ..............................................................................73
 13.4       Inference about the ratio of two variances ...........................75
 13.5       Inference about the difference between two population
 proportions ......................................................................................75




                                           - 89 -
STAT 2010, Business Stat                   2006                               Jaimie Kwon



14    Statistical inference: review of chapters 12 and 13 ....................76
15    Analysis of variance ..................................................................78
16    Chi-squared tests ......................................................................78
17    Simple linear regression and correlation ....................................79
 17.1      Model ..................................................................................79
 17.2      Estimating the coefficients ...................................................79
 17.3      Required conditions on the error variable ............................79
 17.4      Assessing the model ...........................................................80
 17.5      (Optional) Applications in finance ........................................81
 17.6      Using the regression equation .............................................81
18    Multiple regression ....................................................................81
19    Logistics and things you/I really care .........................................83
 19.1      Couple of words about quiz #2 and final ..............................83
     19.1.1     Quiz #2..........................................................................83
     19.1.2     Final ..............................................................................83
 19.2      Practice midterm (50 minutes) .............................................85
 19.3      Practice final (50 minutes) ...................................................86




                                           - 90 -

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:3
posted:8/16/2012
language:
pages:90