Review 3401

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Review 3401
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STATISTICS

• Art and Science of Collecting and Understanding
  DATA

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Data and Data Sets

• Data are the facts and figures that are
  collected, summarized, analyzed, and interpreted.
• Why? Because you want To Predict, Estimate and
  Ultimately make business decisions.
• The data collected in a particular study are
  referred to as the data set.

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Elements, Variables, and Observations

• The elements are the entities on which data are
  collected.
• A variable is a characteristic of interest for
  the elements.
• The set of measurements collected for a
  particular element is called an observation.

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Data, Data Sets, Elements, Variables, and
Observations
Stock Annual Earn/ Company
Exchange Sales(M) Sh.() Dataram A
MEX 73.10 0.86 EnergySouth OTC 74.00
1.67 Keystone NYSE 365.70 0.86
LandCare NYSE 111.40
0.33 Psychemedics AMEX 17.60 0.13
Observation
Variables
Elements
Data Set
Datum
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Scales of Measurement

• Scales of measurement include
• Nominal
• Ordinal
• Interval
• Ratio
• The scale determines the amount of information
  contained in the data.
• The scale indicates the data summarization and
  statistical analyses that are most appropriate.

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Scales of Measurement

• Nominal
• Data are labels or names used to identify an
  attribute of the element.
• A nonnumeric label or a numeric code may be used.
  Example
• Students of a university are classified by the
  school in which they are enrolled using a
  nonnumeric label such as Business, Humanities,
  Education, and so on.
• Alternatively, a numeric code could be used
  for the school variable (e.g. 1 denotes Business,
  2 denotes Humanities, 3 denotes Education, and so
  on).

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Scales of Measurement

• Ordinal
• The data have the properties of nominal data and
  the order or rank of the data is meaningful.
• A nonnumeric label or a numeric code may be used.
  Example
• Students of a university are classified by
  their class standing using a nonnumeric label
  such as Freshman, Sophomore, Junior, or Senior.
• Alternatively, a numeric code could be used for
  the class standing variable (e.g. 1 denotes
  Freshman, 2 denotes Sophomore, and so on).

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Scales of Measurement

• Interval Ratio
• Interval Ratio data are always numeric.

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Qualitative and Quantitative Data

• Data can be further classified as being
  qualitative or quantitative.
• The statistical analysis that is appropriate
  depends on whether the data for the variable are
  qualitative or quantitative.
• In general, there are more alternatives for
  statistical analysis when the data are
  quantitative.

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Qualitative Data (Categorical Variables)

• Qualitative data are labels or names used to
  identify an attribute of each element.
• Qualitative data use either the nominal(cannot be
  ordered meaningfully) or ordinal(values can be
  meaningfully ordered) scale of measurement.
• Qualitative data can be either numeric or
  nonnumeric.
• The statistical analysis for qualitative data are
  rather limited.

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Quantitative Data

• Quantitative data indicate either how many or how
  much.
• Quantitative data that measure how many are
  discrete (Number of cars owned by a family, of
  accidents in I-4 day).
• Quantitative data that measure how much are
  continuous because there is no separation between
  the possible values for the data. (take values in
  intervals)
• Quantitative data are always numeric.
• Ordinary arithmetic operations(adding and
  averaging) are meaningful only with quantitative
  data.

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Cross-Sectional and Time Series Data

• Cross-sectional data are collected at the same or
  approximately the same point in time.
• Example data detailing the number of building
  permits issued in June 2000 in each of the
  counties of Florida
• Time series data are collected over several time
  periods.
• Example data detailing the number of building
  permits issued in Volusia County in each of the
  last 36 months

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Example Time-Series
Year Small Business Administration Budget (
Millions)
1991 464 1992 1,891 1993 1,177 1994 2,058 1995 798
1996 749
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Example
Time series
Year Small Business Administration Budget (
Millions)
1991 464 1992 1,891 1993 1,177 1994 2,058 1995 798
1996 749
Elementary unit defined by year
Quantitative data
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Example Cross-Sectional
Firm Sales Industry Group SP Rating
IBM 66,346 Office Equipment A Exxon 59,023 Fuel
A- GE 40,482 Conglomerates A ATT 34,357 Teleco
mmunications A-
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Example
Cross-Sectional
Multivariate Data (3 variables)
Firm Sales Industry Group SP Rating
IBM 66,346 Office Equipment A Exxon 59,023 Fuel
A- GE 40,482 Conglomerates A ATT 34,357 Teleco
mmunications A-
First Observation IBM
Elementary units
Quantitative variable
Nominal Qualitative variable
Ordinal Qualitative variable
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Sources of Data

• Primary Data
• When you control the design of the
  data-collection plan
• More Control, Exactly what you want
• More Expensive, Time consuming
• Secondary Data
• You use data previously collected by others for
  their purposes. (US Government-INTERNET)

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Experimental VS Observational

• When data are not available through existing
  sources (secondary data) data can be obtained by
  conducting statistical studies and are classified
  as
• Experimental study. A variable of interest is
  first identified. Then one or more other
  variables are identified and controlled so that
  data can be obtained about how they influence the
  variable of interest (DATA MINING FDA ). Only
  experiment allows conclusions to be drawn about
  causes and effect.
• Observational Study. No attempt is made to
  control the variable of interest (SURVEY)

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Descriptive Statistics

• Descriptive statistics are the tabular,
  graphical, and numerical methods used to
  summarize data.
• The most common numerical descriptive statistic
  is the average (or mean).

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Statistical Inference

• Statistical inference is the process of using
  data obtained from a small group of elements (the
  sample) to make estimates and test hypotheses
  about the characteristics of a larger group of
  elements (the population).
• The objective of inferential statistics is to
  make (predictions decisions) about certain
  characteristics of the population based on
  information contained in a sample.
• A Population is the set representing all
  observations of interest.
• A Sample is a subset of measurement selected from
  the population of interest

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Descriptive Statistics Numerical Methods

• Measures of Location
• Measures of Variability
• Measures of Relative Location and Detecting
  Outliers
• Exploratory Data Analysis
• Measures of Association Between Two Variables

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Measures of Location

• Mean
• Median
• Mode
• Quartiles

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Example

• Given below is a sample of monthly rent values
  ()
• for one-bedroom apartments. The data is a sample
  of 70
• apartments in a particular city. The data are
  presented
• in ascending order.

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Excel

• Go to tools
• Select Data analysis
• Select Descriptive Statistics
• Select Summary Statistics box
• Select Confidence Level for Mean box
• Select Ok

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Mean

• The mean of a data set is the average of all the
  data values.
• If the data are from a sample, the mean is
  denoted by
• .
• If the data are from a population, the mean is
  denoted by (mu).

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Example Apartment Rents

• Mean

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Median

• The median is the measure of location most often
  reported for annual income and property value
  data.
• A few extremely large incomes or property values
  can inflate the mean.
• The median of a data set is the value in the
  middle when the data items are arranged in
  ascending order.
• If i is not an integer, round up. The p th
  percentile is the value in the i th position.
• If i is an integer, the p th percentile is the
  average of the values in positions i and i 1.

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Example Apartment Rents

• Median Median 50th percentile
• i (p/100)n (50/100)70 35
  Averaging the 35th and 36th data values
• Median (475 475)/2 475

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Mode

• The mode of a data set is the value that occurs
  with greatest frequency.
• The greatest frequency can occur at two or more
  different values.
• If the data have exactly two modes, the data are
  bimodal.
• If the data have more than two modes, the data
  are multimodal.

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Example Apartment Rents

• Mode
• 450 occurred most frequently (7 times)
• Mode 450

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Percentiles

• A percentile provides information about how the
  data are spread over the interval from the
  smallest value to the largest value.
• Admission test scores for colleges and
  universities are frequently reported in terms of
  percentiles.

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Example Apartment Rents

• 90th Percentile

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Using Excel

• Go to Insert
• Select Function
• Select Statistical
• Under Function Name Select Percentile
• Select OK
• Under Array Select A2A71
• Under K Select .90
• Select Ok

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Quartiles

• Quartiles are specific percentiles
• First Quartile 25th Percentile
• Second Quartile 50th Percentile Median
• Third Quartile 75th Percentile

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Excel

• Go to Insert
• Select Function
• Select Statistical
• Under Function Name Select Quartile
• Select OK
• Under Array Select A2A71
• Under Quart Select 3 for the third quartile
• Select Ok

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Measures of Variability

• Range
• Variance
• Standard Deviation

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Range

• The range of a data set is the difference between
  the largest and smallest data values.
• It is the simplest measure of variability.
• It is very sensitive to the smallest and largest
  data values.

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Example Apartment Rents

• Range
• Range largest value - smallest value
• Range 615 - 425 190

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Variance

• The variance is a measure of variability that
  utilizes all the data.
• It is based on the difference between the value
  of each observation (xi) and the mean (x for a
  sample, m for a population).

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Variance

• The variance is the average of the squared
  differences between each data value and the mean.
• If the data set is a sample, the variance is
  denoted by s2.
• If the data set is a population, the variance is
  denoted by ? 2.

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Standard Deviation

• The standard deviation of a data set is the
  positive square root of the variance.
• It is measured in the same units as the data,
  making it more easily comparable, than the
  variance, to the mean.
• If the data set is a sample, the standard
  deviation is denoted s.
• If the data set is a population, the standard
  deviation is denoted ? (sigma).

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Example Apartment Rents

• Variance
• Standard Deviation

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Sample Variance Standard D Using Excel
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Sample Variance Standard D Using Excel

• Go to Insert
• Select Function
• Select Statistical
• Under Function Name Select either VARA (for
  Sample Variance) or STDEVA (for Sample Standard
  Deviation)
• Select OK
• Under Value 1 Select A2A71
• Select Ok

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Measures of Relative Locationand Detecting
Outliers

• Empirical Rule
• Detecting Outliers

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Empirical Rule

• For data having a bell-shaped
  distribution
• Approximately 68 of the data values will be
  within one standard deviation of the mean.

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Empirical Rule

• For data having a bell-shaped distribution
• Approximately 95 of the data values will be
  within two standard deviations of the mean.

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Empirical Rule

• For data having a bell-shaped distribution
• Almost all (99.7) of the items will be within
  three standard deviations of the mean.

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Detecting Outliers

• An outlier is an unusually small or unusually
  large
• value in a data set.

• It might be
• an incorrectly recorded data value
• a data value that was incorrectly included in
  the
• data set
• a correctly recorded data value that belongs in
• the data set

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Detecting Outliers Using IQR

• IQR 3rd Quartile 1st Quartile
• The lower limit is located 1.5(IQR) below Q1.
• The upper limit is located 1.5(IQR) above Q3.
• Data outside these limits are considered outliers.

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Example Apartment for Rent

• IQR 522.5 446.25 76.25
• Lower Limit Q1 - 1.5(IQR) 446.25 -
  1.5(76.25) 331.875
• Upper Limit Q3 1.5(IQR) 522.5 1.5(76.25)
  636.875
• There are no outliers (values less than 332 or
• greater than 637) in the apartment rent
  data.

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Exploratory Data Analysis

• Five-Number Summary

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Five-Number Summary

• Smallest Value
• First Quartile
• Median
• Third Quartile
• Largest Value

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Example Apartment Rents

• Five-Number Summary
• Lowest Value 425 First Quartile 445
• Median 475
• Third Quartile 525 Largest Value 615

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Descriptive Statistics to Summarize a Variable

• Variable Name
• Number of Observations
• Lowest Value
• Mean
• Median
• Standard Deviation
• Standard Error
• Maximum Value
• 1st Quartile
• 3rd Quartile.

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Rent Example
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Measures of Association Between Two Variables

• Covariance
• Correlation Coefficient

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Covariance

• The covariance is a measure of the linear
  association between two variables.
• Positive values indicate a positive relationship.
• Negative values indicate a negative relationship.

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Covariance

• If the data sets are samples, the covariance is
  denoted by sxy.
• If the data sets are populations, the covariance
  is denoted by .

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Using Excel

• Go Tools
• Select Data Analysis
• Select Covariance
• Select OK
• Input Range
• Select Ok

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COVARIANCE

• EX

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Correlation Coefficient

• The coefficient can take on values between -1 and
  1.
• Values near -1 indicate a strong negative linear
  relationship.
• Values near 1 indicate a strong positive linear
  relationship.
• If the data sets are samples, the coefficient is
  rxy.
• If the data sets are populations, the coefficient
  is .

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Using Excel

• Go Tools
• Select Data Analysis
• Select Correlation
• Select OK
• Input Range
• Select Ok

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• -

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Chapter4 Introduction to Probability

• Experiments, Counting Rules, and
• Assigning Probabilities
• Events and Their Probability
• Some Basic Relationships of Probability

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Probability

• Probability is a numerical measure of the
  likelihood that an event will occur.
• Probability values are always assigned on a scale
  from 0 to 1.
• A probability near 0 indicates an event is very
  unlikely to occur.
• A probability near 1 indicates an event is almost
  certain to occur.
• A probability of 0.5 indicates the occurrence of
  the event is just as likely as it is unlikely.

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Probability as a Numerical Measureof the
Likelihood of Occurrence
Increasing Likelihood of Occurrence
0
1
.5
Probability
The occurrence of the event is just as likely
as it is unlikely.
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An Experiment and Its Sample Space

• An experiment is any process that generates
  well-defined outcomes. (Ex Toss a coin, Roll a
  die, Select a part for inspection)
• A procedure that produces an outcome
• Not perfectly predictable in advance
• Head-Tail- 1,2,3,4,5,6 Defective no defective
• The sample space for an experiment is the set of
  all experimental outcomes.
• A sample point is an element of the sample space,
  any one particular experimental outcome.

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Random Experiment

• Event
• Happens or not, each time random experiment is
  run
• Formally a collection of outcomes from sample
  space
• A yes or no situation if the outcome is in the
  list, the event happens
• Each random experiment has many different events
  of interest
• Example tossing a coin - the event Head
• Probability of an Event
• A number between 0 ( NEVER HAPPENS ) and 1 (
  ALWAYS HAPPENS )
• The likelihood of occurrence of an event

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Combinations Vs Permutations

• List all combinations and all permutations of the
  4 letters A,B,C, and D When they are taken 3 at
  a time
• Combinations ABC ABD ACD BCD 4 Combinations
• Permutations ABC ABD ACD BCD
• ACB ADB ADC BDC
• BAC BAD CAD CBD
• BCA BDA CDA CDB
• CAB DAB DAC DBC
• CBA DBA DCA DCB
  24 Permutations

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COUNTING RULE

• Combinations WITHOUT REGARD TO ORDER
• Where
• N!N(N-1)(N-2).(2)(1)
• n!n(n-1)(n-2).(2)(1)
• And 0!1

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• The odds of winning the lottery in Florida are

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Permutations

• Suppose coach XYZ has not settled on who the
  starters are and has a total of 10 team members.
  How many different lineups can he form now?
• 10!/(10-5)! 30240
• Permutations are the possible ordered selections
  of r objects out of a total of n objects. The
  number of permutations of n objects taken r at a
  time

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Assigning Probabilities

• Classical Method
• Assigning probabilities based on the assumption
  of equally likely outcomes.
• Relative Frequency Method
• Assigning probabilities based on experimentation
  or historical data.
• Subjective Method
• Assigning probabilities based on the assignors
  judgment.

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Chapter 5
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Random Variables

• A random Variable is a specification or
  description of a numerical result from a random
  experiment.
• A random variable is the idea or abstraction of
  the values which are collected in a random
  experiment. A number is what is seen when we
  observe a random variable. For example,
  Tomorrows Dow Jones closing Would be a random
  variable, while the number (hopefully) 12,000
  would be the observation of this random Variable

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Random Variables (Continued)

• A random variable can be classified as being
  either discrete or continuous depending on the
  numerical values it assumes.
• A discrete random variable may assume either a
  finite number of values or an infinite sequence
  of values.
• A continuous random variable may assume any
  numerical value in an interval or collection of
  intervals.

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Example JSL Appliances

• Discrete random variable with a finite number of
  values
• Let x number of TV sets sold at the store in
  one day
• where x can take on 5 values (0, 1, 2, 3,
  4)
• Discrete random variable with an infinite
  sequence of values
• Let x number of customers arriving in one day
• where x can take on the values 0, 1, 2, .
  . .
• We can count the customers arriving, but there
  is no finite upper limit on the number that might
  arrive.

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Discrete Probability Distributions

• The probability distribution for a random
  variable describes how probabilities are
  distributed over the values of the random
  variable.
• The probability distribution is defined by a
  probability function, denoted by f(x), which
  provides the probability for each value of the
  random variable.
• The required conditions for a discrete
  probability function are
• f(x) gt 0
• ?f(x) 1
• We can describe a discrete probability
  distribution with a table, graph, or equation.

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Example JSL Appliances

• Using past data on TV sales (below left), a
  tabular representation of the probability
  distribution for TV sales (below right) was
  developed.
• Number
• Units Sold of Days x f(x)
• 0 80 0 .40
• 1 50 1 .25
• 2 40 2 .20
• 3 10 3 .05
• 4 20 4 .10
• 200 1.00

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Example JSL Appliances

• Graphical Representation of the Probability
  Distribution

.50
.40
Probability
.30
.20
.10
0 1 2 3 4
Values of Random Variable x (TV sales)
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Continued

• Table
• X0 P(X0) 2/5 .40
• X1 P(X1) 1/4 .25
• X2 P(X2) 1/5 .20
• X3 P(X3) 1/20 .05
• X4 P(X4) 1/10 .10

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Example

• Determine the probability of
• P(2?X ?3)
• Determine the probability of
• P(Xgt1)

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Example

• Determine the probability of
• P(X?1)

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Expected Value and Variance

• The expected value, or mean, of a random variable
  is a measure of its central location.
• Expected value of a discrete random variable
• E(x) ? ?xf(x)
• The variance summarizes the variability in the
  values of a random variable.
• Variance of a discrete random variable
• Var(x) ? 2 ?(x - ?)2f(x)
• The standard deviation, ?, is defined as the
  positive square root of the variance.

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Example JSL Appliances

• Expected Value of a Discrete Random Variable
• x f(x) xf(x)
• 0 .40 .00
• 1 .25 .25
• 2 .20 .40
• 3 .05 .15
• 4 .10 .40
• E(x) 1.20
• The expected number of TV sets sold in a day is
  1.2

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Example JSL Appliances

• Variance and Standard Deviation
• of a Discrete Random Variable
• x x - ? (x - ?)2 f(x) (x - ?)2f(x)
• 0 -1.2 1.44 .40 .576
• 1 -0.2 0.04 .25 .010
• 2 0.8 0.64 .20 .128
• 3 1.8 3.24 .05 .162
• 4 2.8 7.84 .10 .784
• 1.660 ? ?
• The variance of daily sales is 1.66 TV sets
  squared.
• The standard deviation of sales is 1.2884 TV
  sets.

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Binomial Probability Distribution (Special
Discrete)

• Properties of a Binomial Experiment
• The experiment consists of a sequence of n
  identical trials.
• Two outcomes, success and failure, are possible
  on each trial.
• The probability of a success, denoted by p, does
  not change from trial to trial.
• The trials are independent.

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Binomial Probability Distribution

• Binomial Probability Function
• where
• f(x) the probability of x successes in n
  trials
• n the number of trials
• p the probability of success on any one
  trial
• x number of successes

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Example UCF

• Binomial Probability Distribution
• At UCF, 75 of students live in the
  dormitories. A random sample of 5 students is
  selected. Use the binomial probability formula to
  answer the following questions.
• a. What is the probability that the sample
  contains exactly three students who live in the
  dormitories?
• b. What is the probability that the sample
  contains more than three students who live in the
  dormitories?

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Example UCF

• c) What is the probability that the sample
  contains at least 4 students who do not live in
  the dormitory ?
• d) What is the probability that the sample
  contains less than 2 students who live in the
  dormitory ?
• e). What is the expected number of students (in
  the sample) who do live in the dormitories?

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Example UCF

• Using the Binomial Probability Function
• a. What is the probability that the sample
  contains exactly three students who live in the
  dormitories?
• (a)
• ?

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Example UCF

• Using the Binomial Probability Function
• b. What is the probability that the sample
  contains more than three students who live in the
  dormitories?
• (b) 45
• ?

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Example UCF

• (c )Using the Binomial Probability Function at
  least 4 means 4 and 5. Probability of success is
  .25.

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Example UCF

• (d )Using the Binomial Probability Function less
  than 2 means 1 and 0. Probability of success is
  .75.

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Example UCF
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Binomial Probability Distribution

• Expected Value
• E(x) ? np
• Variance
• Var(x) ? 2 np(1 - p)
• Standard Deviation

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Example UCF

• Binomial Probability Distribution ( c )
• Expected Value
• E(x) ? 5(.75) 3.75 employees out of
  5
• Variance
• Var(x) ? 2 5(.75)(.25) .93
• Standard Deviation

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Chapter 6 Continuous Probability Distributions

• Normal Probability Distribution
• Standard Normal Distribution

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Continuous Random Variables

• A continuous random variable can assume any value
  in an interval on the real line or in a
  collection of intervals.
• It is not possible to talk about the probability
  of the random variable assuming a particular
  value.
• Instead, we talk about the probability of the
  random variable assuming a value within a given
  interval.
• The probability of the random variable assuming a
  value within some given interval from x1 to x2 is
  defined to be the area under the graph of the
  probability density function between x1 and x2.

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Continuous Probability Distributions

• Normal Probability Distribution
• The Standard Normal Probability Distribution

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Normal Probability Distribution

• Graph of the Normal Probability Density Function

f(x)
x
?
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The Characteristics of the Normal Distribution

• Characteristics
• 1. f(x) approaches 0 as x approaches µ
  (infinity)
• 2. symmetric around vertical line at x µ
• 3. area to right of mean is ½ of total area under
  curve area to left of mean is ½ of total area
  under curve
• 4. different values for µ (mean) s2 (variance)
  determine different curves µ determines where
  curve centered s2 determines how spread out
  curve is, see Figure 4-2 in text

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The Normal Distribution
µ
1. f(x) approaches 0 as x approaches infinity
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The Normal Distribution
µ
2. symmetric around vertical line at x µ
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The Normal Distribution
1/2 of total area
1/2 of total area
µ
3. area to left of mean is 1/2 of total area
area to right of mean is 1/2 of total area
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Normal Probability Distribution

• The shape of the normal curve is often
  illustrated as a bell-shaped curve.
• The highest point on the normal curve is at the
  mean, which is also the median and mode.
• The mean can be any numerical value negative,
  zero, or positive.
• continued

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Normal Probability Distribution

• of Values in Some Commonly Used Intervals
• 68.26 of values of a normal random variable are
  within /- 1 standard deviation of its mean.
• 95.44 of values of a normal random variable are
  within /- 2 standard deviations of its mean.
• 99.72 of values of a normal random variable are
  within /- 3 standard deviations of its mean.

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Areas Under Normal Curve
f(x)
x
µ - s
µ
µ s
68
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The Normal Distribution
f(x)
x
µ - 2s
µ 2s
µ
95
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Areas Under Normal Curve
f(x)
x
µ
µ - 3s
µ 3s
99.7
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Example Analysis of Test Scores

• Let X be a random variable whose values are the
  test scores obtained on a nationwide test given
  to high school seniors. Suppose that X is
  normally distributed with a mean (m) of 600 and a
  standard deviation (s) of 65.

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Given that m 600 and s 65

• The probability that X lies within 2 s 2(65)
  130 points of 600 is 95. In other words, 95 of
  all test scores lie between 470 and 730.
• Similarly, 99.7 of the scores are within 3 s
  3(65) 195 points of 600. That is, between 405
  and 795.

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Normal Probability Distribution

• Normal Probability Density Function
• where
• ? mean
• ? standard deviation
• ? 3.14159
• e 2.71828

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What is Standard Normal Random Variable?

• The Standard normal random variable is a normal
  variable with mean(m) 0 and standard deviation
  (s) 1 See next slide
• A continuous random variable Z (Z is special
  designation usually reserved for this type of
  variable) is a standard normal random variable if
  its density function is as shown on next slide.

138
NOTICE Z RATHER THAN X.
mean 0 and standard deviation 1
139
Reading Table the Z Table

• Example Find the area under the Standard Normal
  Curve between z0 and z2.54 using the Standard
  Normal Table of areas.
• For row value of 2.5 column under 0.04,
  meaning Z 2.54, value 0.4945 --means area
  under standard normal curve between z 0 z
  2.54 is 0.4945 (49.45 of total area under curve)

140
Z Table
for row value of 2.5 column under 0.04,
meaning Z 2.54, value 0.4945
141
Using Z Table for Negative z-Values Values less
than the Mean

• Area under standard normal curve between
• z 0 z -2.54 (note minus 2.54) is also
  0.4945 (49.45 of total area under curve).
• Area under curve between z 0 and some z0 is
  same to right and left of z 0.
• Remember that the Normal Curve is symmetrical.
  Each half is a mirror image of the other.

142
Find P(Z gt 1.5)

• This probability is the area to the right of z
  1.5. This area is equal to the difference between
  the total area to the right of z 0 which equals
  ?? and the area between z 0 and z 1.5. The
  value for z 1.5 from Table 2 is_____??

143
Z Table
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.5
SKIPPED 1.0 - 1.4
144
Find P(Z gt 1.5)

• Find P(Z gt 1.5) This probability is the area
  to the right of z 1.5 Area .5000 - .4332
  .0668

.
145
Find P(0.5 lt Z lt 2)

• This probability is the area between z 0.5 z
  2 (draw figure).

146
Z Table
z
.00 .01 .02 .03 .04
.0000 .0398 .0793 .1179 .1554 .1915 .2258 .2580 .
2881 .3159
.0040 .0080 .0120 .0160 .0438 .0478
.0517 .0557 .0832 .0871 .0910 .0948 .1217
.1255 .1293 .1331 .1591 .1628 .1664
.1700 .1950 .1985 .2019 .2054 .2291
.2324 .2357 .2389 .2612 .2642 .2673
.2704 .2910 .2939 .2967 .2996 .3186
.3212 .3238 .3264
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.5
SKIPPED 1.0 - 1.4
.4332 .4345 .4357 .4370 .4382
147
Z Table
z
.00 .01 .02 .03 .04
.4778 .4783 .4788 .4793 .4826 .4830
.4834 .4838 .4864 .4868 .4871 .4875 .4896
.4898 .4901 .4904 .4920 .4922 .4925
.4927 .4940 .4941 .4943 .4945 .4955
.4956 .4957 .4959 .4966 .4967 .4968
.4969 .4975 .4976 .4977 .4977 .4982
.4982 .4983 .4984
.4772 .4821 .4861 .4893 .4918 .4938 .4953 .4965 .
4974 .4981
2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
148
Find P(0.5 lt Z lt 2)
.

• P(0.5 lt Z lt 2)
• 0.4772 - 0.1915 0.2857.

149
. Find P(Z lt 2)

• This probability is the area to the left of z 2
  (draw figure).

150
Find P(Z lt 2)

• This probability is the area to the left of z 2
  (see figure). This area is equal to the sum of
  the area to the left of z 0 and the area
  between z 0 and z 2
• P(Z lt 2) 0.5 0.4772 0.9772

151
Find P(-2 lt Z lt -0.5)

• This probability is the area between z -2 z
  -0.5 (draw figure).

152
Find P(-2 lt Z lt -0.5)

• This probability is the area between z -2 z
  -0.5 (see figure). This area is equal to the area
  between z 0.5 z 2. Why?

153
Find P(-2 lt Z lt -0.5)

• This probability is the area between z -2 z
  -0.5. This area is equal to the area between z
  0.5 z 2. Why? The values for z 2 (0.4772)
  z 0.5 (0.1915) are found in Table 4-2.

154
Find P(-2 lt Z lt -0.5)

• P(-2 lt Z lt -0.5) 0.4772 - 0.1915 0.2857

155
Standard Normal Probability Distribution

• Transforming x to z
• Converting to the Standard Normal Distribution

156
Standardizing x to Z

• Example Suppose X is normally distributed with µ
  4 s 2.
• Find P(0 lt X lt 6) Convert X 0 to a Z-value
• Z (X - µ) / s z1 (0 - 4) / 2 -2
• Convert X 6 to a Z-value
• (Z (X - µ) / s) z2 (6 - 4) / 2 1

157
P(0 lt X lt 6) P(-2 lt Z lt 1)
area .3413
z values
(Zs sigma 1)
0
-2
1
6
4
0
x values
(Xs sigma 2)
area .4774
158
P(0 lt X lt 6) P(-2 lt Z lt 1)
z values
(Zs sigma 1)
0
-2
1
6
4
0
x values
(Xs sigma 2)
Z (X - µ) / s
159
The Normal Distribution (cont.)

• it is true that P(0 lt X lt 6) P(-2 lt Z lt 1)
• P(-2 lt Z lt 1) is area between z -2 and z 1
  this area is sum of area between z -2
  z 0 (area A1) area between z 0 z 1
  (area A2)
• P(-2 lt Z lt 1) area A1 area A2 0.4774
  0.3413 0.8185 means there is 81.85
  probability that X will be between 0 6 (or
  81.85 of all X values fall between 0 6)

160
Example Survey

• Standard Normal Probability Distribution
• According to a survey, subscribers to The WSJ
  interactive edition spend on average 15 hours
  per week using the computer at work. Assume the
  distribution is normally distributed and that the
  standard deviation is 6 hrs. What is the
  probability a randomly selected subscribers spend
  more than 20 hrs using the computer at work ?
  P(x gt 20).

161
Example Survey

• Standard Normal Probability Distribution
• The Standard Normal table shows an area of .2967
  for the region between the z 0 and z .83
  lines below. The shaded tail area is .5 - .2967
  .2033. The probability of more 20 hrs is
  .2033.
• z (x - ?)/?
• (20 - 15)/6
• .83

162
Example Survey

• Using the Standard Normal Probability Table

163
Final Exam

• The time needed to complete a final examination
  in a particular college course is normally
  distributed with a mean of 80 minutes and a
  standard deviation of 10 minutes. Answer the
  following questions
• A. What is the probability of completing the exam
  in one hour or less ?
• B. Assume that the class has 60 students and that
  the examination period is 90 minutes in length.
  How many students do you expect will be unable to
  complete the exam in the allotted time ?