Probabilities and Expected Values

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Probabilities and Expected Values
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Normal Distribution, Statistical Inference,
Central Limit Theorem
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The Probability of any Set of Events
1 lt Probability gt 0
A B C
Probability of A or B or C happening 1 (This
is true as long as events A, B and C are
exhaustive and mutually exclusive)
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Probability
Probability of drawing a blue balloon 4/12 1/3
Probability of drawing a red balloon 3/12 1/4
Probability of drawing a blue balloon 5/12
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Expected Value
If we did this over and over, we expect to get
blue balloons 1/3 of the time
If we did this over and over, we expect to get
red balloons 1/4 of the time
If we did this over and over, we expect to get
blue balloons 5/12 of the time
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Probability Sampling from Population
Inferential Statistics Making Claims about a
Population
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What are we interested in knowing about?
Point estimates (means, correlations, slopes)
How confident are we about those estimates?
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Question What is the likelihood that this sample
comes from a population with a point estimate of
____.
Answer depends on
Variation around that estimate
Number of cases in the sample
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Population
Low Support for Democracy
Medium Support for Democracy
High Support for Democracy
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For a sample estimate to approach the population
estimate, the sample must be random. This means
that every case has an equal probability of being
selected.
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Support for Democracy
Medium 2
High 3
High 3
Low 1
Low 1
Medium 2
Low 1
Low 1
Medium 2
High 3
High 3
High 3
Sample Mean 2.08 Median? Mode?
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6
5
4
Frequency
Normal curve
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2
1
Std. Dev .90
Mean 2.08
N 12.00
0
3.00
2.50
2.00
1.50
1.00
Support for Democracy
Histogram A Probability Distribution
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Probability of selecting a person with low
support for democracy 4/12 1/3
Probability of selecting a person with medium
support for democracy 1/4
Probability of selecting a person with medium
support for democracy 5/12 .42
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Conditional Probabilities Relationships
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What is the probability of having high support
for democracy if you have a low income?
What is the probability of having high support
for democracy if you have a medium income?
What is the probability of having high support
for democracy if you have a high income?
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Normal Distribution, Statistical Inference,
Central Limit Theorem
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The Normal Distribution
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The Normal Distribution

• Symmetric
• Continuous prob of any one point zero,
  because the area of a line is zero we always
  compute probabilities of lying between some
  designated x and y
• All have same general shape
• Cases more concentrated in the middle than in the
  tails
• Shape determined by mean and standard deviation
• The area under the curve is 1
• The probability of any event under the curve is
  determined by the height of the curve at that
  place

Number of cases y axis
Value of the variable x axis
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• Approximately 68 percent of the area under a
  normal curve lies between the values of the mean
  and the standard deviation and the mean.
• Approximately 95 of the area lies between 2
  standard deviations and the mean.
• Approximately 99.7 lies between 3 standard
  deviations and the mean.

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The Standard Normal Distribution
Same as a normal distribution, but the standard
deviation is 1 and the mean is 0
0
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Any normal distribution can be turned into a
standard normal with a linear transformation
1) Subtract the mean from every observation
2) Divide by the standard deviation
This is called a z-score.
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The Central Limit Theorem
Given a population with ANY distribution
Taking random samples of size n from that
distribution The sample means will be
(approximately) normally distributed.
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Sampling Distribution Illustration
http//www.ruf.rice.edu/7Elane/stat_sim/sampling_
dist/index.html
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Why do we care about the Normal Distribution?
1) Many of the political phenomena that we study
are distributed normally. For
example, Ideology there are lots of people in
the middle and not as many people on the
tails 2) The normal distribution has some cool
properties, like being able to easily compute
percentiles.
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Assume grades on a test are normally distributed
mean of 80 standard deviation of 5 What is
the percentile rank of a person who received a
score of 70 on the test?
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Z table
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To take another example, what is the percentile
rank of a person receiving a score of 90 on the
test?
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If a test is normally distributed with a mean
of 60 and a standard deviation of 10, what
proportion of the scores are above 85?
A z table can be used to calculate that .9938 of
the scores are less than or equal to a score 2.5
standard deviations above the mean. It follows
that only 1-.9938 .0062 of the scores are above
a score 2.5 standard deviations above the mean.
Therefore, only .0062 of the scores are above
85.
Given the sample, what is the probability of
selecting out a test grade higher than 85?
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Now, back to statistical analysis
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Properties of Estimators
Remember that Ordinary Least Squares minimizes
the squared errors from the line.
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Residuals of OLS analysis (errors of the slope)
have a mean of zero, by definition they have
been computed by their minimization. We also
assume that they are distributed
normally. Therefore they are distributed along a
standard normal distribution, mean of zero. (The
standard deviation is not necessarily 1, but it
is assumed to be constant across all values of
x). Foreshadowing if this assumption does not
hold, you cannot use OLS.
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The Null Hypothesis
What is the question that we ask with statistics?
Answer How likely is it that the relationship
is zero? The null hypothesis is that the
relationship is zero. We are trying to reject
the null hypothesis.
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We have a point estimate we call it a slope,
but we cannot be sure about that point because we
do not know anything about the population. So,
we know that there is error in our estimate. So,
neither the upper bound and lower bound of our
estimate can contain zero.
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Generally, we want to be at least 95 confident
that our estimate does not include zero.
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So, to be 95 confident that the true estimate
does not contain zero, then the estimate must be
two standard deviations from the mean of the
standard normal curve, which is zero.
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If a certain interval is a 95 confidence
interval, then we can say that if we repeated the
procedure of drawing random samples and computing
confidence intervals over and over again, 95 of
those confidence intervals include the true value
from the population.
This is not to say that we are 98 confident that
the true value lies between the upper and lower
bound.
Instead, I am 98 confident that a Confidence
interval covers the true value from the
population, based not on this single CI from this
single test, but rather as a result of what would
happen were I to repeat the process of drawing
samples and doing this test over and over again.
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Some Examples
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Effect of Index of Signals on the Number of Cases
on the U.S Supreme Court Agenda, 1953-1995
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7
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4.62
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3.85
Upper bound of the 95 confidence
interval Estimate Lower bound of the 95
confidence interval
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3
2.11
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1.27
1.19
1.34
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0
-1
-2
1
2
3
4
5
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Lag Year
Note Note Estimates are Ordinary Least Square
unstandardized regression coefficients and
confidence intervals are computed using panel
corrected standard errors, calculated according
to Beck and Katz (1995). Controls in this
analysis include policy area dummy variables,
Burger and Rehnquist Court dummies, the
legislative agenda, ideological output of the
Supreme Court, and absolute value change of
median voter. The dependent variable is the
number of cases on the Supreme Courts agenda,
across 11 policy areas from 1953-1995. The
independent variable presented here is an index
of salient cases, as measured by Epstein and
Segal (1996), declarations of constitutionality,
the number of lower court reversals and formal
alterations of precedent.
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The Effect of Supreme Court Signals on Amicus
Briefs at Courts of Appeals
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Upper bound 95 Confidence Interval
Point Estimate - slope
Lower bound 95 Confidence Interval
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Upper bound 95 Confidence Interval
Point Estimate - slope
Lower bound 95 Confidence Interval