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PADM 7060 Quantitative Methods for Public Administration Unit 3 Chapters 9-10. Fall 2004 Jerry Merwin. Meier & Brudney Part III: Probability. Chapter 7: Introduction to Probability Chapter 8: The Normal Probability Distribution Chapter 9: The Binomial Probability Distribution

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meier brudney part iii probability
Meier & BrudneyPart III: Probability
  • Chapter 7: Introduction to Probability
  • Chapter 8: The Normal Probability Distribution
  • Chapter 9: The Binomial Probability Distribution
  • Chapter 10: Some Special Probability Distributions
meier brudney chapter 9 the binomial probability distribution
Meier & Brudney: Chapter 9 The Binomial Probability Distribution
  • What is the binomial probability distribution?
  • What is a Bernoulli process?
meier brudney chapter 9 the binomial probability distribution page 2
Meier & Brudney: Chapter 9 The Binomial Probability Distribution (Page 2)
  • Let’s discuss the characteristics of a Bernoulli process: (book says two, but can be three)
    • Each trial’s outcome must be mutually exclusive: Success or failure
    • Probability of success (p) remains constant (as does probability of failure, q)
    • Trials are independent (Not affected by outcomes of earlier trials)
    • Examples:
      • Coin flip
      • Roll of die (six or not six)
      • Fire in a community
meier brudney chapter 9 the binomial probability distribution page 3
Meier & Brudney: Chapter 9 The Binomial Probability Distribution (Page 3)
  • More on Bernoulli process - Must how three things to determine probability:
    • Number of trials
    • Number of successes
    • Probability of success in any trial
  • Formula (see page 146)
    • Combination of n things taken r at a time
meier brudney chapter 9 the binomial probability distribution page 4
Meier & Brudney: Chapter 9 The Binomial Probability Distribution (Page 4)
  • Let’s look at some examples:
    • Starting at the bottom of page 146:
      • How many sets of three balls can be selected from four balls (a, b, c, d)?
      • How many combinations of four things two at a time?
      • We can list the possibilities with these examples to come up with solutions.
meier brudney chapter 9 the binomial probability distribution page 5
Meier & Brudney: Chapter 9 The Binomial Probability Distribution (Page 5)
  • What if we do not want to list all the combinations?
    • See the formula on page 147
  • Do you understand how to calculate a factorial?
    • Examples of formula with coin flip (pages 147-149)
meier brudney chapter 9 the binomial probability distribution page 6
Meier & Brudney: Chapter 9 The Binomial Probability Distribution (Page 6)
  • What is the problem with calculating factorials?
    • Cumbersome in larger numbers!
  • So what alternative do we have?
  • How can the normal curve be used?
    • A general rule‑of‑thumb is that once n ≥ 30, the normal distribution can be used to calculate the binomial probability distribution.
meier brudney chapter 9 the binomial probability distribution page 7
Meier & Brudney: Chapter 9 The Binomial Probability Distribution (Page 7)
  • How about the problem on the Republican’s chance of being hired in Chicago? (Page 151)
    • See the information for the binomial distribution
    • The standard deviation of a probability distribution at the bottom of 151
    • Simply convert the number actually hired into a z score and look up the value.
meier brudney chapter 9 the binomial probability distribution page 8
Meier & Brudney: Chapter 9 The Binomial Probability Distribution (Page 8)
  • On page 152, you\'re told that you should use this only when n x p is greater than 10 and n x (1 -p) is greater than 10.
  • Also, keep in mind that the normal curve tells you the probability only that r or fewer (or more) events occurred. It does not tell you the probability of exactly r events occurring.
  • This probability can be determined only by using the binomial distribution.
meier brudney chapter 9 the binomial probability distribution page 9
Meier & Brudney: Chapter 9 The Binomial Probability Distribution (Page 9)
  • Problems: 9.2 & 9.6
meier brudney chapter 10 special probability distributions
Meier & Brudney: Chapter 10 Special Probability Distributions
  • What is the Hypergeometric Probability Distribution and when is it used?
    • When you have a finite population the hypergeometric probability distribution is better than the binomial distribution.
meier brudney chapter 10 special probability distributions page 2
Meier & Brudney: Chapter 10 Special Probability Distributions (Page 2)
  • Let’s look at the Andersonville City Fire department example (pp. 157-158).
    • There are 34 women out of a pool of 100 eligible applicants.
    • The p = .34, and 30 trials (n = 30).
    • Thus your mean is:  = 30 x .34 = 10.2.
    • The key difference is in computing the standard deviation.
  • The formula is shown on page 158 and the answer is on 159.
meier brudney chapter 10 special probability distributions page 3
Meier & Brudney: Chapter 10 Special Probability Distributions (Page 3)
  • What is The Poisson distribution?
    • It is named for the Frenchman, Simeon Dennis Poisson, who developed it in the first half of the 19th century.
    • It is used to describe a pattern of behavior when some event occurs at varying, random intervals over a continuum of time, length, or space.
    • Your text uses muggings & potholes as examples.
    • It has been widely used to describe the probability functions of phenomena such as product demand; demand for services; # of calls through a switchboard; passenger cars at toll stations, etc.
meier brudney chapter 10 special probability distributions page 4
Meier & Brudney: Chapter 10 Special Probability Distributions (Page 4)
  • How is the Poisson distribution different from the Bernoulli process?
    • The number of trials is not known in a Poisson process.
    • Instead, the Poisson distribution is concerned with a discrete, random variable which can take on the values x = 0,1,2,3 . . . where the three dots mean "ad infinitum".
    • The formula is:  = xe-/x! (see page 160)
meier brudney chapter 10 special probability distributions page 5
Meier & Brudney: Chapter 10 Special Probability Distributions (Page 5)
  • The formula is:  = xe-/x!
    • where:
      •  (lambda) = the probability of an event
      • x = # of occurrences
      • e = 2.71828 (a constant that is the base of the Naperian or natural logarithm system)
  • Fortunately, this is not something we need to calculate;
    • we\'ll use Table 2 (pp. 444-445), as we are sane.
meier brudney chapter 10 special probability distributions page 6
Meier & Brudney: Chapter 10 Special Probability Distributions (Page 6)
  • Let’s look at the management example on pages 160-161.
  • Why does the Poisson table only go to 20 for the λvalues?
    • We can use the normal distribution table for those values above 20.
meier brudney chapter 10 special probability distributions page 7
Meier & Brudney: Chapter 10 Special Probability Distributions (Page 7)
  • Problems 10.2, 10.4, 10.6, & 10.16
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