BCOR 1020 Business Statistics

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BCOR 1020Business Statistics

• Lecture 8 February 12, 2007

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Overview

• Chapter 5 Probability
• Contingency Tables
• Tree Diagrams
• Counting Rules

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Chapter 5 Contingency Tables

• What is a Contingency Table?
• A contingency table is a cross-tabulation of
  frequencies into rows and columns.
• It is like a frequency distribution for two
  variables.

Cell
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Chapter 5 Contingency Tables

• Example Salary Gains and MBA Tuition
• Consider the following cross-tabulation table for
  n 67 top-tier MBA programs

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Chapter 5 Contingency Tables

• Example Salary Gains and MBA Tuition
• Are large salary gains more likely to accrue to
  graduates of high-tuition MBA programs?
• The frequencies indicate that MBA graduates of
  high-tuition schools do tend to have large salary
  gains.
• Also, most of the top-tier schools charge high
  tuition.
• More precise interpretations of this data can be
  made using the concepts of probability.

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Chapter 5 Contingency Tables

• Marginal Probabilities
• The marginal probability of a single event is
  found by dividing a row or column total by the
  total sample size.
• For example, find the marginal probability of a
  medium salary gain (P(S2)).
• Conclude that about 49 of salary gains at the
  top-tier schools were between 50,000 and
  100,000 (medium gain).

P(S2)
33/67
.4925
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Chapter 5 Contingency Tables

• Marginal Probabilities
• Find the marginal probability of a low tuition
  P(T1).
• There is a 24 chance that a top-tier schools
  MBA tuition is under 40.000.

.2388
16/67
P(T1)
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Clickers
Consider the overhead of the cross-tabulation of
salary gains and MBA tuitions. Find the
marginal probability of a large salary gain
(P(S3)). A 17/67 B 17/33 C 19/67 D
32/67
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Chapter 5 Contingency Tables

• Joint Probabilities
• A joint probability represents the intersection
  of two events in a cross-tabulation table.
• Consider the joint event that the school has low
  tuition and large salary gains (denoted as P(T1 ?
  S3)).
• There is less than a 2 chance that a top-tier
  school has both low tuition and large salary
  gains.

.0149
1/67
P(T1 ? S3)
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Chapter 5 Contingency Tables

• Conditional Probabilities
• Found by restricting ourselves to a single row or
  column (the condition).
• For example, knowing that a schools MBA tuition
  is high (T3), we would restrict ourselves to the
  third row of the table.
• To find the probability that the salary gains are
  small (S1) given that the MBA tuition is large
  (T3)

.1563
P(S1 T3)
5/32
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Clickers
Consider the overhead of the cross-tabulation of
salary gains and MBA tuitions. Find the
probability that the salary gains are large (S3)
given that the MBA tuition is large (T3). P(S3
T3) ? A 5/15 B 15/32 C 12/32 D
12/15
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Chapter 5 Contingency Tables

• Independence
• To check for independent events in a contingency
  table, compare the conditional to the marginal
  probabilities.
• For example, if large salary gains (S3) were
  independent of low tuition (T1), then P(S3 T1)
  P(S3).
• What do you conclude about events S3 and T1?
  (Clickers)
• A Dependent or B Independent

Conditional Marginal
P(S3 T1) 1/16 .0625 P(S3) 17/67 .2537
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Chapter 5 Contingency Tables

• Relative Frequencies
• Calculate the relative frequencies below for each
  cell of the cross-tabulation table to facilitate
  probability calculations.
• Symbolic notation for relative frequencies

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Chapter 5 Contingency Tables

• Relative Frequencies
• Here are the resulting probabilities (relative
  frequencies). For example,

P(T1 and S1) 5/67
P(T2 and S2) 11/67
P(T3 and S3) 15/67
P(S1) 17/67
P(T2) 19/67
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Chapter 5 Contingency Tables

• Relative Frequencies

• The nine joint probabilities sum to 1.0000 since
  these are all the possible intersections.

• Summing the across a row or down a column gives
  marginal probabilities for the respective row or
  column.

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Chapter 5 Contingency Tables

• How Do We Get a Contingency Table?
• Contingency tables require careful organization
  and are created from raw data.
• Consider the data of salary gain and tuition for
  n 67 top-tier MBA schools.

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Chapter 5 Contingency Tables

• How Do We Get a Contingency Table?
• The data should be coded so that the values can
  be placed into the contingency table.
• Once coded, tabulate the frequency in each cell
  of the contingency table using the appropriate
  menus in our statistical analysis software.

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Chapter 5 Tree Diagrams

• What is a Tree?
• A tree diagram or decision tree helps you
  visualize all possible outcomes.
• Start with a contingency table.
• For example, this table gives expense ratios by
  fund type for 21 bond funds and 23 stock funds.

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Chapter 5 Tree Diagrams

• To label the tree, first calculate conditional
  probabilities by dividing each cell frequency by
  its column total.

.5238

• For example,

11/21
P(L B)

• Here is the table of conditional probabilities

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Chapter 5 Tree Diagrams

• To calculate joint probabilities, use

P(A ? B) P(A B)P(B) P(B A)P(A)

• The joint probability of each terminal event on
  the tree can be obtained by multiplying the
  probabilities along its branch.

• For example, consider the probability of a low
  expense Bond

P(B and L)
(.5238)(.4773)
.2500
Consider the tree on the next slide
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Chapter 5 Tree Diagrams
Tree Diagram for Fund Type and Expense Ratios
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Chapter 5 Counting Rules

• Fundamental Rule of Counting
• If event A can occur in n1 ways and event B can
  occur in n2 ways, then events A and B can occur
  in n1 x n2 ways.
• In general, m events can occur n1 x n2 x x nm
  ways.
• For example, consider the number of different
  possibilities for license plates if each plate
  consists of three letters followed by a
  three-digit number. How many possibilities are
  there?

26 x 26 x 26 x 10 x 10 x 10 17,576,000
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Chapter 5 Counting Rules

• Sampling with or without replacement
• Sampling with replacement occurs when an object
  is selected and then replaced before the next
  object is selected. (i.e. the object can be
  selected again).
• For example, our license plate example.
• Sampling without replacement occurs when an
  object is selected and then not replaced (i.e.
  the object cannot be selected again).
• For example, consider the number of different
  possibilities for license plates if each plate
  consists of three letters followed by a
  three-digit number and no letters or numbers can
  be repeated

26 x 25 x 24 x 10 x 9 x 8 11,232,000
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Chapter 5 Counting Rules

• Factorials
• The number of ways that n items can be arranged
  in a particular order is n factorial.
• n factorial is the product of all integers from 1
  to n.
• n! n(n1)(n2)...1
• By definition, 0! 1
• Factorials are useful for counting the possible
  arrangements of any n items.
• There are n ways to choose the first, n-1 ways to
  choose the second, and so on.

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Chapter 5 Counting Rules

• Permutations and Combinations
• A permutation is an arrangement in a particular
  order of r randomly sampled items from a group of
  n items (i.e., XYZ is not the same as ZYX).
• If r items are randomly selected (with
  replacement) from n items, then the number of
  permutations is
• nr
• If r items are randomly selected (without
  replacement) from n items, then the number of
  permutations, denoted by nPr is

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Chapter 5 Counting Rules

• Permutations and Combinations
• A combination is an arrangement of r items chosen
  at random from n items where the order of the
  selected items is not important (i.e., XYZ is the
  same as ZYX).
• If r items are randomly selected (without
  replacement) from n items, then the number of
  combinations can be determined by dividing out
  the number of distinct orderings of the r items
  (r!) from the number of permutations.
• The number of combinations, denoted by nCr is

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Chapter 5 Counting Rules

• Example Lottery Odds
• Consider the Colorado Lottery drawing
• There are 42 balls, numbered 1 42. (n 42)
• 6 balls are selected at random. (r 6)
• Order is unimportant. (combinations, not
  permutations)
• How many different combinations are possible?

The probability that a single ticket will have
the winning combination of numbers is 1 in
5,245,786!
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Clickers

• Consider a standard deck of playing cards which
  
• consists of 52 cards.
• If five cards are drawn at random and order is of
  
• no importance, how many distinct 5-card poker
• hands are possible?
• A 2,598,960
• B 3,168,367
• C 311,875,200
• D 380,204,032