Introduction to Probability and Statistics Eleventh Edition (PowerPoint download)

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Introduction to Probability and Statistics Eleventh Edition (PowerPoint download) Powered By Docstoc
					              Al-Imam Mohammad Ibn Saud University

                  Modeling and Simulation
                        Lecture 16
                Output Analysis
      Large-Sample Estimation Theory

30 May 2009
                           Dr. Anis Koubâa
Goals of Today
 Understand the problem of confidence in
  simulation results
 Learn how to determine of range of value with

  a certain confidence a certain stochastic
  simulation result
 Understand the concept of

     Margin of Error
     Confidence Interval with a certain level of

   Required
           Park, Discrete Event Simulation - A First Course,
     Lemmis
     Chapter 8: Output Analysis

   Optional
     Harry Perros, Computer Simulation Technique - The
     Definitive Introduction, 2007
     Chapter 5
    Problem Statement
   For a deterministic simulation model one run will be
    sufficient to determine the output.
   A stochastic simulation model will not give the same
    result when run repetitively with independent random
     One   run is not sufficient to obtain confident simulation
      results from one sample.
   Statistical Analysis of Simulation Result: multiple runs
    to estimate the metric of interest with a certain
 Stoachatsic Simulation results may vary from one
  run/replication to another
 Simulation results depends on three factors:

       The seed of the RNG
       Number of samples/size of samples
       Simulation time
   Objective
     For  a given large sample output, determine what is the
        mean value with a certain confidence on the result.
What types of parameters to estimate?

 In general, a stochastic variable is described by
  their probability distributions and parameters.
    For quantitative random variables: mean m and
     variance s.
    For a binomial random variables: success
     probability p.
 If the values of parameters are unknown, we make

  inferences about them using sample information.
How to express the confidence?

   Simulation results must be expressed with a certain
   The confidence needs the following parameters
       Confidence Level: 99%, 98%, 95% or 90%
       Variance: the variance of the simulation results
       Sample Size: the number of simulation results under study
   There are two ways:
       Margin of Error: The maximum error of estimation.
       Confidence interval: The interval where most of the simulation
        results lie.
The Margin of Error

      Margin of error: The maximum
       error of estimation, calculated as

       1.96  std error of the estimator
Estimating Means and Proportions

For a quantitative population,
       Point estimator of population mean μ : x
       Margin of error (n  30) :  1.96

  For a binomial population,
 Point estimator of population proportion p : p  x/n
 Margin of error (n  30) :  1.96
      Example 1
   A homeowner randomly samples 64 homes similar to his own and
    finds that the average selling price is 252,000 SAR with a standard
    deviation of 15,000 SAR.
    Question: Estimate the average selling price for all similar homes in
    the city.

      Point estimator of μ : x  252, 000
                                 s          15, 000
      Margin of error :  1.96       1.96          3, 675
                                  n            64
          Example 2
A quality control technician wants to estimate the proportion of soda
bottles that are under-filled. He randomly samples 200 bottles of soda
and finds 10 under-filled cans.
What is the estimation of the proportion of under-filled cans?

n  200    p  proportion of underfilled cans
Point estimator of p : p  x/n  10 / 200  .05
                        pq         (.05)(.95)
Margin of error:  1.96     1.96             .03
                        n             200
   Interval Estimators

Confidence Interval
      Confidence Interval

• “Fairly sure” means “with high probability”, measured
  using the confidence coefficient, 1-a.

            Usually, 1-a = 0.90, 0.95, 0.98, 0.99

 • Suppose 1-a = 0.95 and
   that the estimator has a
   normal distribution.
                     Parameter  1.96SE
To Change the Confidence Level

• To change to a general confidence level, 1-a, pick a value of z
  that puts area 1-a in the center of the z-distribution (i.e. Normal
  Distribution N(0,1).

                                  Tail area       a     Confidence
                                                          Level      za/2
                                  0.05          0.1     90%          1.645
                                  0.025         0.05    95%          1.96
                                  0.01          0.02    98%          2.33
          x  z a /2              0.005         0.01    99%          2.58
    100(1-a)% Confidence Interval: Estimator  za/2SE
Confidence Intervals for Means and Proportions

    For a quantitative population
  Confidence interval for a population mean μ :
                   x  za / 2

   For a binomial population
 Confidence interval for a population proportion p :
                    p  za / 2
     Example 1
A random sample of n = 50 males showed a
 mean average daily intake of dairy products
 equal to 756 grams with a standard deviation of
 35 grams.
 Find a 95% confidence interval for the
 population average m.
           s  756  1.96 35
  x  1.96                       756  9.70
            n               50
       or 746.30  m  765.70 grams.
    Example 1
 Find a 99% confidence interval for m, the
 population average daily intake of dairy
 products for men.

             s                      35
  x  2.58        756  2.58             756  12.77
           n                 50
   or 743.23  m  768.77 grams.
             The interval must be wider to provide for the
             increased confidence that is does indeed
             enclose the true value of m.
      Example 2
 Of a random sample of n = 150 college students, 104 of
  the students said that they had played on a soccer team
  during their K-12 years.
  Estimate the proportion of college students who played
  soccer in their youth with a 98% confidence interval.

             pq     104         .69(.31)
    p  2.33
    ˆ                   2.33
             n      150           150
       .69  .09      or .60  p  .78.
How to Choose the Sample Size?
Choosing the Sample Size
  The total amount of relevant information in a
   sample is controlled by two factors:
   - The sampling plan or experimental design:
   the procedure for collecting the information
   - The sample size n: the amount of information
   you collect.
  In a statistical estimation problem, the accuracy
   of the estimation is measured by the margin of
   error or the width of the confidence interval.
     Choosing the Sample Size

1.   Determine the size of the margin of error, B, that you
     are willing to tolerate.
2.   Choose the sample size by solving for n or n  n 1  n2
     in the inequality: 1.96 SE  B, where SE is a function of
     the sample size n.
3.   For quantitative populations, estimate the population
     standard deviation using a previously calculated value
     of s or the range approximation s  Range / 4.
4.   For binomial populations, use the conservative
     approach and approximate p using the value p  .5.

A producer of PVC pipe wants to survey wholesalers who buy his product in
order to estimate the proportion of wholesalers who plan to increase their
purchases next year.
What sample size is required if he wants his estimate to be within .04 of the
actual proportion with probability equal to .95?

     pq                          .5(.5)
1.96     .04              1.96         .04
     n                             n
     1.96 .5(.5)                           n  24.52  600.25
 n              24.5
         .04                             He should survey at least 601
  Key Concepts

I. Types of Estimators
    1. Point estimator: a single number is calculated to estimate the
    population parameter.
    2. Interval estimator: two numbers are calculated to form an
    interval that contains the parameter.
II. Properties of Good Estimators
    1. Unbiased: the average value of the estimator equals the
    parameter to be estimated.
    2. Minimum variance: of all the unbiased estimators, the best
    estimator has a sampling distribution with the smallest standard
    3. The margin of error measures the maximum distance between
    the estimator and the true value of the parameter.
Key Concepts
III. Large-Sample Point Estimators
    To estimate one of four population parameters when the
    sample sizes are large, use the following point estimators with
    the appropriate margins of error.
Key Concepts

IV. Large-Sample Interval Estimators
   To estimate one of four population parameters when the
   sample sizes are large, use the following interval estimators.
     Key Concepts
1.  All values in the interval are possible values for the unknown
    population parameter.
2.  Any values outside the interval are unlikely to be the value of
    the unknown parameter.
3.  To compare two population means or proportions, look for the
    value 0 in the confidence interval. If 0 is in the interval, it is
    possible that the two population means or proportions are
    equal, and you should not declare a difference. If 0 is not in
    the interval, it is unlikely that the two means or proportions are
    equal, and you can confidently declare a difference.
V. One-Sided Confidence Bounds
    Use either the upper (+) or lower (-) two-sided bound, with the
    critical value of z changed from za / 2 to za.

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