PS400 Quantitative Methods

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PS400 Quantitative Methods

• Dr. Robert D. Duval
• Course Introduction
• Presentation Notes and Slides
• Version of January 9, 2001

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Overview of Course
P Syllabus P Texts P Grading P Assignments P
Software
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The First Two Weeks
P Review and Setting The Logic of Research P
Logic P Microcomputers P Statistics
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Overview of Statistics
P Descriptive Statistics P Frequency
Distributions P Probability P Statistical
Inference P Statistical tests P Contingency
Tables P Regression Analysis
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The Logic of Research
A quick review of the research process
P Theory P Hypothesis P Observation P Analysis
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Sample Theories

• IR - Balance of Power
• Wars erupt when there are shifts in the balance
  of power
• Domestic Policy
• The crime rate is affected by the economy

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Theory
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Theory Hypothesis
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Theory Hypothesis Observatio
n
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Theory Analysis Hypothesis
Observation
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Theory Analysis Hypothesis
Observation
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Theory Deduction Analysis Hypoth
esis Induction Operationalization
Observation
Confirmation/ rejection
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Logic
A short primer on Deduction and Inference
We will look at Symbolic Logic in order to
examine how we employ deduction in cognition.
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Logic
What is Logic?

• Logic
• The study by which arguments are classified into
  good ones and bad ones.
• Comprised of Statements
• "Roses are red
• "Republicans are Conservatives

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Logic
Compound Statements

• Conjunctions (Conjunction Junction)
• Two simple statements may be connected with a
  conjunction
• and
• "Roses are Red and Violets are blue.
• "Republicans are conservative and Democrats are
  liberal.
• or
• "Republicans are conservative or Republicans are
  moderate."

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Operators

• There are three main operators
• And ()
• Or (v)
• Not ()
• These may be used to symbolize complex statements
• The other symbol of value is
• Equivalence (?)
• This is not quite the same as equal to.

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Truth Tables

• Statements have truth value
• For example, take the statement PQ
• This statement is true only if P and Q are both
  true.
• P Q PQ
• T T T
• T F F
• F T F
• F F F

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Truth Tables (cont)

• Hence Republicans are conservative and Democrats
  are liberal. is true only if both parts are
  true.
• On the other hand, take the statement PvQ
• This statement is true only if either P or Q are
  true, but not both. (Called the exclusive or)
• P Q PvQ
• T T F
• T F T
• F T T
• F F F

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The Inclusive or

• Note that or can be interpreted differently.
• Both parts of the conjunction may be true in the
  inclusive or. This statement is true if either
  or both P or Q are true.
• P Q PvQ
• T T T
• T F T
• F T T
• F F F

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The Inclusive or

• Note that or can be interpreted differently.
• Both parts of the conjunction may be true in the
  inclusive or. This statement is true if either
  or both P or Q are true.
• P Q PvQ
• T T T
• T F T
• F T T
• F F F

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Tautologies

• Note that p v p must be true
• Roses are red or roses are not red. must be
  true.
• A statement which must be true is called a
  tautology.
• A set of statements which, if taken together,
  must be true is also called a tautology (or
  tautologous).
• Note that this is not a criticism.

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The Conditional

• The Conditional
• if a (antecedent)
• then b (consequent)
• It is also called the hypothetical, or
  implication.
• This translates to
• A implies B
• If A then B
• A causes B

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The Implication

• We symbolize the implication by
• We use the conditional or implication a great
  deal.
• It is the core statement of the scientific law,
  and hence the hypothesis.

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Equivalency of the Implication

• Note that the Implication is actually equivalent
  to a compound statement of the simpler operators.
• p v q
• Please note that the implication has a broader
  interpretation than common English would suggest

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Rules of Inference

• In order to use these logical components, we have
  constructed rules of Inference
• These rules are essentially how we think.

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Disjunctive Syllogism
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Hypothetical Syllogism
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Modus Ponens
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Modus Tollens
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Logical Systems

• Logic gives us power in our reasoning when we
  build complex sets of interrelated statements.
• When we can apply the rules of inference to these
  statements to derive new propositions, we have a
  more powerful theory.

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Tautologous systems

• Systems in which all propositions are by
  definition true, are tautologous.
• Balance of Power
• Why do wars occur? Because there is a change in
  the balance of power.
• How do you know that power is out of balance? A
  war will occur.
• Note that this is what we typically call circular
  reasoning.
• The problem isnt the circularity, it is the lack
  of utility.

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Paradoxes
P The Liars Paradox lt Epimenedes the Cretan says
that all Cretans are liars. P The ??? Paradox (a
variant) lt The next statement is true. lt The
previous statement is false.
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Microcomputer Architecture
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Using the Computer
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Computers - Basic Architecture
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Basic components
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The CPU
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Some simple binary arithmetic
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Binary numbering
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Binary addition
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Miscellaneous
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Digital Systems

• So, in the end, we can see that computers simply
  move ad add 0s and 1s.
• And out of this, we can build incredibly rich and
  complex experiences
• Such as
• Or

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Statistics
A Philosophical Overview

• Methods as Theory
• Methods as Language

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Principle organizing concepts
P The Nature of the Problem P Measurement P
Standards for comparison
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Mathematical notation Important
mathematical notation the student
needs to know.
n
å
X
PSummation lt For instance, the sum of all Xi
from I1 to n means beginning with the first
number in your data set, add together all n
numbers. lt The 3 is a symbolic representation of
the process of adding up a specified series or
collection of numbers.
i
i

1
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Mathematical notation (cont.)
P Square Roots and Exponents P e - the base of
natural logarithms P Exponential and
Logarithmic Equations
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The Base of Natural Logarithms
Where does e come from?
P e is the base of natural logarithms P It is
derived from
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Demystifying e (sort of)

• So how does this translate to real life?
• Compound interest
• Where
• PV Present Value (amount deposited)
• FV future value (amount accrued)
• i interest rate (e.g .06 for 6 interest)
• k number of periods/year
• n number of years

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Levels of Measurement

• P Nominal
• ltDichotomous
• P Ordinal
• P Interval
• ltRatio

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Levels of Measurement

• Nominal
• Dichotomous
• Ordinal
• Interval
• Ratio
• For instance Levels of Measurement

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Nominal Measurement
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Statistics
Induction about the Observable World
P A statistic is a number that provides
information about some variable of interest. P
Descriptive Statistics lt Numbers that describe
some aspect of the world P Inferential
Statistics lt We use inferential statistics to
take information from a sample and make some
inference about a population.
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Descriptive Statistics

• P There are two main ways we describe collections
  of data.
• Measures of Central Tendency
• Measures of Dispersion

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Statistical Tools for Describing the World -
Distributions
P Intuitive Definition lt A bunch of numbers that
measure a characteristic for a group of cases. lt
May be represented by a set of numbers, a graph
or picture, or even a mathematical equation.
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Measures of Central Tendency

• Measures which provide some indication of the
  typical value or the 'middle' of the distribution

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Measures of Central Tendency The Arithmetic Mean
(or Average)

• The sum of all of the numbers in a set, divided
  by the number in the set
• Most appropriate for symmetric distributions
• Influenced by extreme values

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Measures of Central Tendency The Median

• The middle number in the data set.
• (Sort the Data...
• The Median is the middle value if there are an
  odd number of cases.
• The Median is the average of the two middle
  values if there are an even number of cases.
• Best measure for skewed distributions
• Not very tractable mathematically

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Measures of Central Tendency The Mode

• The most frequently occurring value.
• Used primarily for nominal data.
• The peak value of a frequency distribution is
  also referred to as the mode.

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Measures of Central Tendency Common terms for
this concept.

• We use the idea of measures of central tendency a
  great deal in everyday language.
• Average, accordance, bread-and-butter,
  commonplace, Commensurate, congruent, consistent,
  conventional, customary, day-to-day, everyday,
  frequent, garden variety, general, habitual,
  humdrum, invariably, likeness, mean, median,
  medium, mediocrity, middle, middling,
  nondescript, normal, ordinary, popular,
  prevailing, regular, the same, standard,
  stereotypical, stock, typical, unexceptional,
  uniform, usual
• From The Elementary Forms of Statistical Reason
  by R. P. Cuzzort and James S. Vrettos)

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Measures of Dispersion
The Range Range Highest value - lowest
value Uses only two pieces of information Strongly
influenced by the particular
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Measures of Dispersion
Percentiles
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Measures of Dispersion
The Deviation about the Mean
The Deviation about the Mean Indicates how
far a value is from the center.
X
X
-
i
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Measures of Dispersion
The average deviation.
Can we find the average of the deviations? Alw
ays sums to 0.0!
n
(
)
å
X
X
-
i
i

1
AD

n
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Measures of Dispersion
The average absolute deviation.
Can we find the average of the absolute value of
the deviations? Yes, but difficult to use.
n
å
X
X
-
i
i

1
AD

n
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Measures of DispersionThe Standard Deviation
Square the deviations to remove minus
signs Take the square root to return to the
original scale
n
(
)
2
å
-
X
X
i
-
i
1
s

n
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Measures of DispersionThe Variance
The mean of the squared deviations
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Calculating the Standard Deviation

• The best way to calculate the standard deviation
  is to use a computer.
• If one is not available, try the table method.
• StDevdemo.xls (Excel)
• StDevdemo.wb3 (Quattro Pro)

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Population measures

• OKI lied. The formula for the standard
  deviation is not quite as I described.
• It turns out that the St. dev. Is biased in small
  samples.
• The estimate is a little too small in small
  samples.
• Thus we designate whether we are using population
  or sample data.

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Population vs. Sample Means
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Population vs. Sample Standard Deviations
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Frequency Distribution
A frequency distribution is a graph or chart that
shows the number of observations of a given
value, or class interval.
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The Frequency Histogram

• To create a frequency histogram
• Determine the class interval width.
• Determine the number of intervals desired.
• Tally number of observations in each range.
• Create bar chart from class totals.

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Example Frequency Distribution

• Develop a frequency histogram for the following
  crime rate data for the 50 states.
• Use the data provided in class

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Frequency Polygon

• Same as a frequency histogram except the
  midpoints of the class intervals are used
• Points are connected with a line graph
• A large number of classes will make the
  distribution a smooth curve if there is a large
  sample size.

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Frequency DistributionsShape

• Modality
• The number of peaks in the curve
• Skewness
• An asymmetry in a distribution where values are
  shifted to one extreme or the other.
• Kurtosis
• The degree of Peakedness in the curve

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Frequency DistributionsModality

• Unimodal
• Bimodal
• Multimodal

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Frequency DistributionsSkewness

• The Third Moment about the Mean
• Right Skew (Positive Skew)
• Left Skew (Negative Skew)

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Frequency DistributionsMeasuring Skewness

• Measuring skewness
• Normal distribution has skewness 0.0
• (Normal ranges between 3.0)

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Frequency DistributionsKurtosis

• The Fourth Moment about the Mean
• Platykurtic
• Leptokurtic
• Mesokurtic

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Frequency DistributionsMeasuring Kurtosis

• Measure of kurtosis
• Normal distributions have kurtosis 3.0

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Frequency Distributions - Types

• The Normal
• The Uniform
• The Log-normal
• The Exponential
• Statistical Distributions
• t
• ?-Square
• F

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Freuency Distributions Types (cont.)

• Hyper-geometric
• Poisson
• Binomial
• Gamma
• Weibul
• Beta

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Graphs - Types

• Descriptive Graphs
• Bar Chart
• Pie Graph
• Line Graph
• Distributions
• Histogram
• Box Plot
• Steam and Leaf

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Graphs
Histogram
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Graphs
Pie Graph
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Graphs
Line Graph
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Graphs
Histogram
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Graphs
Box Plot
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Graphs
Stem and Leaf
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Probability Density Functions

• A probability density function is a frequency
  distribution whose area is set equal to 1.0.
• Most distributions are PDFs.
• They let us assess the likelihood or probability
  of cases taking on particular values.

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The Normal Distribution

• The normal distribution is one of the most
• Popular
• Ubiquitous
• Useful
• distributions that we have.
• It gives great predictive ability when we can
  apply it to data.

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The Normal Distribution the Formula

• The normal curve is described by the following
  formula.

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The Normal Distribution (cont.)

• This formula will give us the following
  distribution

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Standard Normal Variables

• A Standard Normal Variable is one that has been
  transformed by the following formula
• All Z-scores, as they are called, will have a
  mean 0.0 and s 1.0

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Standard Normal Distribution

• The Standard Normal distribution is thus one that
  has ? 0.0 and ? 1.0
• We say this symbolically as
• Z ?N(0,?)
• (or Z is normally distributed with a mean of zero
  and a standard deviation of one)

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The Normal PDF

• Because the normal curve is a PDF, we can use it
  to make probability assessments about values in
  the distribution

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Using the Normal PDF

• We know the following facts
• Area under the curve 1.0
• Its symmetric, so the probability of Xi being
  greater that 0.0 is .5
• Symbolically,
• P(Xi gt 0.0) .5

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Using the Normal PDF

• We can use this information in the following
  fashion
• The P(0.0 ? Xi ? 1.0) .3413
• Thus 68 of the Xis will fall between ?1?
• Thus 96 of the Xis will fall between ?2?

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The Central Limit Theorem

• The Normal Distribution pops up in one very
  important context
• The Central Limit Theorem
• This is a fundamental concept is inferring the
  characteristics of a population based upon a
  sample.

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Sampling Distributions

• The probability distribution of a statistic is
  called its sampling distribution.
• If we collect a sample and calculate the mean,
  that is one data point in the sampling
  distribution of the sample mean.
• If we do this many times, we have a sampling
  distribution, which we can then describe.

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The Central Limit Theorem

• The CLT tells us
• As the sample size n gets larger, the sampling
  distribution of the sample mean can be
  approximated by a normal distribution with mean
  of ?, and a standard deviation equal to
• where ? and ? are the population
  characteristics.

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The Implications of the Central Limit Theorem

• We can use the CLT to make probability statements
  about the sample mean because we know its
  distributional characteristics.
• Even if the original variable X is not normally
  distributed, the sampling dist of the sample mean
  is!

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Statistical inference

• We can use information about the way variables
  are distributed to make assessments of
  probability about them.
• Many of these questions are phrased as Is A
  greater (or less) than B?
• This may also be phrased
• Does A belong to the same population as B?

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Assessing probabilities

• Take income
• Would you expect a doctors to have a higher
  income than the population at large?
• Would dog-catchers be lower?
• Would you expect males to have a higher income
  than females
• Is WV income lower than the national average?
  What about Oklahoma?

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Statistical Decision-making

• Many of these questions are best answered with a
  statement of statistical confidence a
  probability assessment.
• This statistical confidence places a decision
  within an objective framework.
• If we define the criteria for making decisions
  according to some reasonable standards, then we
  can remove (or certainly reduce the subjectivity
  of the researcher.

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Statistical D-M (cont.)

• If you collected the following information, would
  you conclude that males had a higher income than
  females?
• MeanMales 55.5K, MeanFemales 54.9K
• MeanMales 55.5K, MeanFemales 50.9K
• MeanMales 55.5K, MeanFemales 34.9K
• Where would you draw the line?
• Does sample size matter?

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Statistical Decision-making

• Many of these questions are best answered with a
  statement of statistical confidence a
  probability assessment.
• This statistical confidence places a decision
  within an objective framework.
• If we define the criteria for making decisions
  according to some reasonable standards, then we
  can remove (or certainly reduce the subjectivity
  of the researcher.

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Statistical Decision-making problem setup

• The Mon river