Padm 7060 quantitative methods for public administration unit 3 chapters 9 10
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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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PADM 7060 Quantitative Methods for Public Administration Unit 3 Chapters 9-10

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Padm 7060 quantitative methods for public administration unit 3 chapters 9 10

PADM 7060 Quantitative Methods for Public AdministrationUnit 3 Chapters 9-10

Fall 2004

Jerry Merwin


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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