ECON 2300 LEC

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ECON 2300 LEC 4

• 08/30/06

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Discussion Topics

• Elements and Variables
• Cross sectional and Time series data
• Sample and Population
• Relative and Percent Frequency distribution
• Cumulative distribution Ogive
• Examples Last class topics

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Elements vs. Variables

• Elements Entities on which data are collected
• Variable Characteristics of interest for the
  elements

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Statistical Inference

• Population Set of all elements of interest in a
  particular study
• Sample Subset of population
• Estimating and testing hypotheses about the
  characteristics of a population using data from a
  sample
• Example data collected for height of 10
  students in class 58, 59, 6, 62,
  55, 57, 59,61, 52, 6

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Statistical Inference

• Data can be used to estimate and test hypotheses
  about the height of the whole class
• The average height of the 10 students is equal to
  5 8
• Using this value, we can possibly say that
  average height of the students in class is 58
  with a margin of -2 inches giving an interval
  of 56 and 510

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Example

• In the Fall of 2003, Arnold Schwarzenegger
  challenged Governor Gray Davis for the governor
  of California. A policy institute of California
  survey of registered voters reported Arnold
  Schwarzenegger in the lead with an estimated 54
  of the vote.
• What was the population for this survey?
• What was the sample for this survey?
• Why was a sample used in this situation? Explain

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Types of data

• Cross-Sectional and Time series data
• Cross-Sectional
• Data collected at the same time or approximately
  same point of time
• Stock prices on the same day
• Time series
• Data collected over a number of time periods
• Average sales over six months

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Frequency Distribution
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Relative frequency Percent frequency

• Relative frequency-fraction or proportion of
  items in each class
• Relative freq of classfrequency of class/n
  (total number of items)
• Percent frequencyRelative frequency100

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Relative frequency Percent frequency
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Example
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Frequency distribution

• Number of classes- number of data items 50 - 5
  classes
• Width of the class
• Approximate width(33-12)/54.2
• Round it upto 5
• In practice the above two parameters trial and
  error
• Class limits 10-14, 15-19, 20-24, 25-29, 30-34

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Frequency distribution
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Relative frequency Percent frequency
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Cumulative distributions

• Shows the number of data items with values less
  than or equal to upper class limits
• Can be used to graphically represent frequency,
  relative frequency and percent frequency
  distributions.

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Cumulative distributions
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Ogive

• Graph of cumulative distribution
• Data values on X-axis and frequencies on Y-axis
• A point is plotted corresponding to the
  cumulative frequency of each class
• Gaps between classes eliminated by considering
  points halfway between class limits

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Example

• Sorting through Unsolicited e-mail and spam
  affects the productivity of office workers. An
  Insight/Express survey monitored office workers
  to determine the unproductive time per day
  devoted to unsolicited e-mail and spam. The
  following data show a sample of time in minutes
  devoted to this task.
• 2 4 8 4 8 1 2 32 12 1 5 7 5 5 3 4 24 19 4
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• Frequency distribution (Classes 1-5, 6-10, 11-15
  16-20 and so on)
• A relative frequency distribution
• A cumulative frequency distribution
• A cumulative relative frequency distribution
• An ogive
• What percentage of workers spend 5 minutes or
  less on unsolicited e-mail and spam? What
  percentage of office workers spend more than 10
  minutes a day on this task.

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Detecting Outliers

• Outliers - observations with unusually large or
  small values compared to rest of the data set.
• Could be
• Incorrectly recorded data (can be removed)
• Incorrectly included data (can be removed)
• Correct but unusual data (cannot be removed)
• z-scores can be used to identify outliers
• Any value with z-score less than -3 and greater
  than 3 outlier (Empirical Rule)

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Detecting Outliers

• Such data values need to be evaluated to
  determine their validity in belonging to the data
  set.

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z-scores

• It gives the relative location of values within a
  data set
• z-score (zi) number of standard deviations
• xi is from the mean
• Two observations in two different data sets with
  same z-score same relative location (same number
  of standard deviations from the mean)

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z-score
Consider the data for class sizes 46, 54, 42,
46, 32
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Probability

• Real world problems
• Lot of uncertainties
• Decision making difficult
• Managers decision analysis of uncertainties
• What are the chances that sales will decrease if
  we increase prices
• What is the likelihood the new car design will be
  a success
• How likely is it that the project will be
  finished on time
• Numerical measure of the likelihood that event
  will occur probability
• Measure of degree of uncertainty associated with
  events

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Probability

• Always assigned value on a scale from 0 to 1
• Values near zero event unlikely to occur
• Values near one event almost certain
• Problem A spinner has 4 equal sectors colored
  yellow, blue, green and red. What are the chances
  of landing on blue after spinning the spinner.
  What are the chances of landing on red.
• Solution Any guesses?
• The chances of landing on blue are 1 in 4
• What about red?
• The chances of landing on red are 1 in 4 again

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Definitions

• Experiment Situation involving chance or
  probability that leads to results called
  outcomes.
• Example Spinning the spinner
• Rolling a dice
• Outcome Result of a single trial of an
  experiment
• Example 1, 2, 3, 4, 5 or 6-rolling a dice
• - Yellow, blue, red or
  green-spinning
• spinner
• - Head or tail-tossing a coin

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Probability

• Event One or more outcomes of an experiment
  landing on red
• Sample space Set of all experimental outcomes
• yellow, red, blue, green
• - 1, 2, 3, 4, 5, 6
• Sample point An experimental outcome

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Probability

• Probability of an event A is the number of ways
  event A can occur divided by the total number of
  possible outcomes
• Experiment A spinner has 4 equal sectors colored
  yellow, blue, green and red. After spinning, what
  is the probability of landing on each color.
• Outcomes Possible outcomes are yellow, red, blue
  and green

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Probability

• Probabilities

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Probability

• Experiment A coin is tossed. What is the
  probability of each outcome.
• Outcomes possible outcomes of this experiment
  are head and tail
• Probabilities

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Probability
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Counting Rules

• Identifying and counting experimental
  outcomes-necessary step in assigning
  probabilities
• Multistep experiments-Experiment performed in
  multiple steps
• For example Tossing two coins
• Experimental Outcomes Pattern of
• heads and tails appearing on upward
• faces of the two coins

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Counting Rules

• Experiment of tossing two coins two step
  process
• Sample space for this experiment
• S(H,H), (H,T), (T,H), (T,T)
• (H,H)- Head appearing on first coin
• Head appearing on second coin
• Counting Rule
• For an experiment consisting of k steps with n1
  possible outcomes on first step, n2 outcomes on
  second and so on
• The total number of possible outcomes is given by
  n1.n2..nk

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Counting Rules

• Experiment of tossing two coins, here k-2
• Tossing first coin, n12
• Tossing second coin, n22
• The total number of possible outcomes 4
• Tree diagram Graphical representation that
  helps in visualizing a multi-step experiment

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Combinations

• Useful when experiment involves selecting n
  objects from a set of N objects
• Counting rule for combinations
• The number of combinations of N
• objects taken n at a time is given by

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Combinations
Example Suppose you had to choose 2 coins out of
a total of 4 coins randomly. Then number of ways
to do this are
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Permutations

• Compute number of experimental outcomes when n
  objects are to be selected from a set of N
  objects where order of selection is important
• The number of permutations of N objects taken n
  at a time is given by

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Permutations
Example Suppose you have to select 2 parts out
of 5 for inspection. Let parts be labeled A, B,
C, D, E. The number of permutations
possible AB, BA, AC, CA, AD, DA, AE, EA, BC,
CB, BD, DB, BE, EB, CD, DC, CE, EC, DE, ED
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Assigning Probabilities

• Basic requirements
• Probability assigned to each outcome must be
  between 0 and 1 inclusively
• The sum of probabilities of all the experimental
  outcomes must equal 1.0

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Assigning Probabilities

• For n experimental outcomes

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Assigning Probabilities - Approaches

• Three most commonly used approaches
• Classical method-Appropriate when all
  experimental outcomes equally likely
• Relative frequency method Appropriate when data
  are available to estimate the proportion of the
  time when experimental outcome will occur if the
  experiment is repeated a large number of times
• Subjective method Appropriate when relevant data
  not available and outcomes are not equally likely

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Events and their probabilities

• Event Collection of sample point (outcome)
• Example Kentucky Power Light company is
• starting a project designed to increase the
• generating capacity of one of its plants in
• northern Kentucky. Project is divided into two
• sequential steps stage 1 (design) and stage 2
• (construction)

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Events and their probabilities

• Management cannot predict before hand the exact
  time required to complete each stage of the
  project
• Analysis of similar construction projects
  revealed possible completion times of 2, 3 or 4
  months for design stage and 6, 7 or 8 months for
  construction stage.
• Thus event of completing project in 10 months
  denoted by A would consist of following sample
  points
• A(2,6), (2,7), (2,8), (3,6), (3,7), (4,6)

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Events and their probabilities

• P(A)P(2,6)P(2,7)P(3,6)P(3,7)P
• 4,6).15.15.05.1.2.05.70

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Basic Relationships - Probability

• Complement of an event
• Event consisting of all sample points that
• are not in event under consideration

Set of all possible outcomes
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Basic Relationships - Probability

• P(A)P(Ac)1
• Therefore P(A)1- P(Ac)

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Addition Law
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Mutually Exclusive Events

• Two events are said to be mutually exclusive if
  they have no sample points in common

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Union of events

• Union of A and B is the event containing all
  sample points belonging to A or B or both. It is
  denoted by

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Intersection of events

• Intersection of events A and B is the event
  containing sample points belonging to both A and
  B. It is denoted by

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Conditional Probability

• Sometimes probability of an event is influenced
  by whether a related event already occurred.
• Probability of event A given that B occurred
  conditional probability.
• It is denoted by P(AB)

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Conditional Probability

• Example Consider the situation of promotion
  status of male and female police officers. Police
  force consists of 1200 officers, 960 men and 240
  women. Over the past two years, 324 officers
  received promotions (288-male officers, 36 female
  officers). Argument made that discrimination is
  done to women

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Conditional Probability

• Let
• Mevent an officer is a male
• Wevent an officer is a female
• Aevent an officer is promoted
• Acevent an officer is not promoted

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Conditional Probability
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Conditional Probability
Thus conditional probability does not itself
prove that discrimination exists but it supports
the argument presented by female officers
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Independent Events

• Two events A and B are independent if the fact
  that A occurs does not affect the probability of
  B occuring.
• Mathematical terms Two events A and
• B are independent if

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Multiplication Law

• Used to compute probability of intersection of
  events
• Based on definition of conditional probability

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Multiplication Law
Example In a Ford dealership, manager knows that
60 of customers buy Mustang Aevent that the
first customer buys Mustang Bevent that the
second customer buys Mustang Probability that
next two customers buy Mustang