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PADM 7060 Quantitative Methods for Public Administration Unit 3 Chapters 9-10

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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PADM 7060 Quantitative Methods for Public Administration Unit 3 Chapters 9-10

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  1. PADM 7060 Quantitative Methods for Public AdministrationUnit 3 Chapters 9-10 Fall 2004 Jerry Merwin

  2. 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

  3. Meier & Brudney: Chapter 9 The Binomial Probability Distribution • What is the binomial probability distribution? • What is a Bernoulli process?

  4. 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

  5. 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

  6. 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.

  7. 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)

  8. 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.

  9. 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.

  10. 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.

  11. Meier & Brudney: Chapter 9 The Binomial Probability Distribution (Page 9) • Problems: 9.2 & 9.6

  12. 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.

  13. 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.

  14. 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.

  15. 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)

  16. 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.

  17. 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.

  18. Meier & Brudney: Chapter 10 Special Probability Distributions (Page 7) • Problems 10.2, 10.4, 10.6, & 10.16

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