Statistics

1
Statistics Data Analysis

• Course Number B01.1305
• Course Section 31
• Meeting Time Wednesday 6-850 pm

CLASS 4
2
Class 4 Outline

• Brief review of last class
• Questions on homework
• Chapter 5 Special Distributions

3
Review of Last Class

• Probability trees
• Probability distribution functions
• Expected value
• Standard deviation

4
Chapter 5

• Some Special Probability Distributions

5
Chapter Goals

• Introduce some special, often used distributions
• Understand methods for counting the number of
  sequences
• Understand situations consisting of a specified
  number of distinct success/failure trials
• Understanding random variables that follow a
  bell-shaped distribution

6
Counting Possible Outcomes

• In order to calculate probabilities, we often
  need to count how many different ways there are
  to do some activity
• For example, how many different outcomes are
  there from tossing a coin three times?
• To help us to count accurately, we need to learn
  some counting rules
• Multiplication Rule If there are m ways of
  doing one thing and n ways of doing another
  thing, there are m times n ways of doing both

7
Example

• An auto dealer wants to advertise that for 20G
  you can buy either a convertible or 4-door car
  with your choice of either wire or solid wheel
  covers.
• How many different arrangements of models and
  wheel covers can the dealer offer?

8
Counting Rules

• Recall the classical interpretation of
  probabilityP(event) number of outcomes
  favoring event / total number of outcomes
• Need methods for counting possible outcomes
  without the labor of listing entire sample space
• Counting methods arise as answers to
• How many sequences of k symbols can be formed
  from a set of r distinct symbols using each
  symbol no more than once?
• How many subsets of k symbols can be formed from
  a set of r distinct symbols using each symbol no
  more than once?
• Difference between a sequence and a subset is
  that order matters for a sequence, but not for a
  subset

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Counting Rules (cont)

• Create all k3 letter subsets and sequences of
  the r5 letters A, B, C, D and E
• How many sequences are there?
• How many subsets are there?

10
Counting Rules (cont)
11
Review Sequence and Subset

• For a sequence, the order of the objects for each
  possible outcome is different
• For a subset, order of the objects is not
  important

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Example

• A group of three electronic parts is to be
  assembled into a plug-in unit for a TV set
• The parts can be assembled in any order
• How many different ways can they be assembled?
• There are eight machines but only three spaces on
  the machine shop floor.
• How many different ways can eight machines be
  arranged in the three available spaces?
• The paint department needs to assign color codes
  for 42 different parts. Three colors are to be
  used for each part. How many colors, taken three
  at a time would be adequate to color-code the 42
  parts?

13
Binomial Distribution

• Percentages play a major role in business
• When percentage is determined by counting the
  number of times something happens out of the
  total possibilities, the occurrences might
  following a binomial distribution
• Examples
• Number of defective products out of 10 items
• Of 100 people interviewed, number who expressed
  intention to buy
• Number of female employees in a group of 75
  people
• Of all the stocks trades on the NYSE, the number
  that went up yesterday

14
Binomial Distribution (cont)

• Each time the random experiment is run, either
  the event happens or it doesnt
• The random variable X, defined as the number of
  occurrences of a particular event out of n trials
  has a binomial distribution if
• For each of the n trials, the event always has
  the same probability ? of happening
• The trials are independent of one another

15
Example Binomial Distribution

• You are interested in the next n3 calls to a
  catalog order desk and know from experience that
  60 of calls will result in an order
• What can we say about the number of calls that
  will result in an order?
• Questions
• Create a probability tree
• Create a probability distribution table
• What is the expected number of calls resulting in
  an order?
• What is the standard deviation?

16
Binomial Distribution the Easy Way
Number of Occurrences, X Proportion or Percentage
Mean E(X) n ? E(p) ?
Standard Deviation ?X(n ?(1- ?))0.5 ?p(?(1- ?)/n)0.5
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Finding Binomial Probabilities
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Example Binomial Probabilities

• How many of your n6 major customers will call
  tomorrow?
• There is a 25 chance that each will call
• Questions
• How many do you expect to call?
• What is the standard deviation?
• What is the probability that exactly 2 call?
• What is the probability that more than 4 call?

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Example

• Its been a terrible day for the capital markets
  with losers beating winners 4 to 1
• You are evaluating a mutual fund comprised of 15
  randomly selected stocks and will assume a
  binomial distribution for the number of
  securities that lost value
• Questions
• What assumptions are being made?
• What is the random variable?
• How many securities do you expect to lose value?
• What is the standard deviation of the random
  variable?
• Find the probability that 8 securities lose value
• What is the probability that 12 or more lose
  value?

20
The Normal Distribution
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Normal Distribution

• The normal distribution is sometimes called a
  Gaussian Distribution, after its inventor, C. F.
  Gauss (1777- 1855).
• Well-known bell-shaped distribution
• Mean and standard deviation determine center and
  spread of the distribution curve
• The mathematical formula for the normal f (y) is
  given in HO, p. 157. We won't be needing this
  formula just tables of areas under the curve.
• The empirical rule holds for all normal
  distributions
• Probability of an event corresponds to area under
  the distribution curve

22
Standard Normal Distribution

• Normal Distribution with ?0 and ?1
• Letter Z is used to denote a random variable that
  follows a Standard Normal Distribution

23
Visualization
Symmetrical
Tail
Tail
Mean, Median and Mode
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Characteristics

• Bell-shaped with a single peak at the exact
  center of the distribution
• Mean, median and mode are equal and located at
  the peak
• Symmetrical about the mean
• Falls off smoothly in both directions, but the
  curve never actually touches the X-axis

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Why Its Important

• Many psychological and educational variables are
  distributed approximately normally
• Measures of reading ability, introversion, job
  satisfaction, and memory are among the many
  psychological variables approximately normally
  distributed
• Although the distributions are only approximately
  normal, they are usually quite close.
• It is easy for mathematical statisticians to work
  with
• This means that many kinds of statistical tests
  can be derived for normal distributions
• Almost all statistical tests discussed in this
  text assume normal distributions
• These tests work very well even if the
  distribution is only approximately normally
  distributed.

26
More Visualizations
?3.1 years, Plant A
?3.9 years, Plant B
?5 years, Plant C
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Z-score

• Compute probabilities using tables or computer
• Convert to z-score
• Look up CUMULATIVE PROBABILITY ON TABLE

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Determining Probabilities
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LOOKUP Table
Standard Normal Lookup Table
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Example

• Sales forecasts are assumed to follow a normal
  distribution
• Target, or expected value is 20M with a 3M
  standard deviation
• What is the probability of sales lower than 15M?
• What is the probability sales exceed 25M?
• What is the probability sales are between 15M
  and 25M ??
• What is the value of k such that the sales
  forecast exceeds k is 60 ?

31
Example

• Benefits compensation costs for employees with a
  certain financial services firm are approximately
  normally distributed with a mean of 18,600 and
  standard deviation of 2,700.
• Find the probability that an employee chosen at
  random has an benefits package that costs less
  than 15,000
• Find the probability that an employee chosen at
  random has an benefits package that costs more
  than 21,000
• What is the value of k such that the benefits
  compensation exceeds k is 95 ?

32
Example

• A telephone-sales firm is considering purchasing
  a machine that randomly selects and automatically
  dials telephone numbers
• The firm would be using the machine to call
  residences during the evening calls to business
  phones would be wasted.
• The manufacturer of the machine claims that its
  programming reduces the business-phone rate to
  15
• As a test, 100 phone numbers are to be selected
  at random from a very large set of possible
  numbers
• Are the binomial assumptions satisfied?
• Find the probability that at least 24 of the
  numbers belong to business phones
• If in fact 24 of the 100 numbers turn out to be
  business phones, does that cast series doubt on
  the manufacturers claim?
• Find the expected value and standard deviation of
  the number of business phone numbers in the
  sample

33
Example

• Assumed the stock market closed at 8,000
  yesterday.
• Today you expect the market to rise a mean of 1
  point, with a standard deviation of 34 points.
  Assume a normal distribution.
• What is the probability the market goes down
  tomorrow?
• What is the probability the market goes up more
  than 10 points tomorrow?
• What is the probability the market goes up more
  than 40 points tomorrow?
• What is the probability the market goes up more
  than 60 points tomorrow?
• Find the probability that the market changes by
  more than 20 points in either direction.
• What is the value of k such that the market close
  exceeds k is 75 ?

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Using R

• factorial(n) n!
• dbinom(x, n, p) binomial probability
  distribution function
• pbinom(x, n, p) binomial cumulative
  distribution function
• pnorm(q, mean, sd) normal cumulative
  distribution function
• qnorm(p, mean, sd) inverse CDF

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Homework 4

• Hildebrand/Ott
• 5.2, page 141
• 5.3, page 141
• 5.9, page 150
• 5.14, page 150
• 5.32, page 163
• 5.33, page 163
• 5.34, page 163
• Reading Chapter 6 (all) and 7 (all).

• Verzani
• 6.5