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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 AdministrationUnit 3 Chapters 9-10

Fall 2004

Jerry Merwin

- Chapter 7: Introduction to Probability
- Chapter 8: The Normal Probability Distribution
- Chapter 9: The Binomial Probability Distribution
- Chapter 10: Some Special Probability Distributions

- What is the binomial probability distribution?
- What is a Bernoulli process?

- 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

- 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

- 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.

- Starting at the bottom of page 146:

- 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)

- 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.

- 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.

- 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.

- Problems: 9.2 & 9.6

- 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.

- 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.

- 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.

- 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)

- 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)

- where:
- Fortunately, this is not something we need to calculate;
- we'll use Table 2 (pp. 444-445), as we are sane.

- 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.

- Problems 10.2, 10.4, 10.6, & 10.16