Probability cont'

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Probability (cont.)
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Assigning Probabilities

• A probability is a value between 0 and 1 and is
  written either as a fraction or as a proportion.
• For the complete set of distinct possible
  outcomes of a random circumstance, the total of
  the assigned probabilities must equal 1.

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Complementary Events
One event is the complement of another event if
the two events do not contain any of the same
simple events and together they cover the entire
sample space. Notation AC represents the
complement of A.
Note P(A) P(AC) 1
ExampleA Simple Lottery (cont) A player
buying single ticket wins AC player does not
win P(A) 1/1000 so P(AC) 999/1000
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Classical Approach

• A mathematical index of the relative frequency of
  likelihood of the occurrence of a specific event.
• Based on games of chance
• The specific conditions of the game are known.

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Estimating Probabilities from Observed
Categorical Data - Empirical Approach
Assuming data are representative, the probability
of a particular outcome is estimated to be the
relative frequency (proportion) with which that
outcome was observed.
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Mutually Exclusive Events
Two events are mutually exclusive if they do not
contain any of the same simple events (outcomes).
Example A Simple Lottery A all three digits
are the same. B the first and last digits are
different The events A and B are mutually
exclusive.
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Independent and Dependent Events

• Two events are independent of each other if
  knowing that one will occur (or has occurred)
  does not change the probability that the other
  occurs.
• Two events are dependent if knowing that one will
  occur (or has occurred) changes the probability
  that the other occurs.

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Example Independent Events

• Customers put business card in restaurant glass
  bowl.
• Drawing held once a week for free lunch.
• You and Vanessa put a card in two consecutive wks.

Event A You win in week 1. Event B Vanessa
wins in week 2

• Events A and B refer to to different random
  circumstances and are independent.

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Example Dependent Events
Event A Alicia is selected to answer Question
1. Event B Alicia is selected to answer
Question 2.
Events A and B refer to different random
circumstances, but are A and B independent
events?

• P(A) 1/50.
• If event A occurs, her name is no longer in the
  bag P(B) 0.
• If event A does not occur, there are 49 names in
  the bag (including Alicias name), so P(B)
  1/49.

Knowing whether A occurred changes P(B). Thus,
the events A and B are not independent.
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Probability Calculations

• Some Useful Formulas to Keep in Mind (Or in Hand)
• U Union (or)
• n Intersection (and)
• General Formulas
• Adding (or)
• P(A U B) P(A) P(B) P(A n B)
• Non-mutually Exclusive of Overlapping Outcomes.
• P(A U B) P(A) P(B)
• Mutually Exclusive Outcomes

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Probability Calculations (cont.)

• General Formulas
• Multiplying (and/sequential events)
• P(A n B) P(A)(P(BA)
• Nonindependence sampling without replacement
• P(A n B) P(A)P(B)
• Independence sampling with replacement

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Joint and Marginal Probabilities

• These probabilities refer to the proportion of an
  event as a fraction of the total.
• P(30 to 64) 62,689/103,870 .60
• P(30 to 64 n married) 43,308/103,870 .42

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Unions and intersections

• PAÈB ¹ PA PB because A and B do overlap.
• PAÈB PA PB - PAÇB.
• AÇB is the intersection of A and B it includes
  everything that is in both A and B, and is
  counted twice if we add PA and PB.

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PAUB PA PB - PAnB. P(18 to 29 U
Married) .21 .57 - .07 .71
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Conditional Probability

• Consider two events A and B.
• What is the probability of A, given the
  information that B occurred? P(A B) ?
• Example
• What is the probability that a women is married
  given that she is 18 - 29 years old?

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Probability Problems

• P(Married 18-29) 7842/ 22,512

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Conditional probability and independence

• If we know that one event has occurred it may
  change our view of the probability of another
  event. Let
• A rain today, B rain tomorrow, C rain
  in 90 days time
• It is likely that knowledge that A has occurred
  will change your view of the probability that B
  will occur, but not of the probability that C
  will occur.
• We write P(BA) ¹ P(B), P(CA) P(C). P(BA)
  denotes the conditional probability of B, given
  A.
• We say that A and C are independent, but A and B
  are not.
• Note that for independent events P(AÇC)
  P(A)P(C).

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Age and Marital Status

• P(M) 59,920/103,870 .57
• P(18 to 29) 22,512/103,870 .21
• P(M Ç 18 to 29) 7,842/103,870 .07
• P(M U 18 to 29) .57 .21 - .07 .71
• P(M18 to 29) 7,842/22,512 .34
• P(M30 to 64) 43,808/62,689 .69
• Knowledge of the age changes P(M). Age and
  Marital status are not independent.

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Group Practice
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Continuous variables

• A continuous random variable is one which can (in
  theory) take any value in some range, for example
  crop yield, maximum temperature, height, weight,
  etc.

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Probability distributions

• If we measure a random variable many times, we
  can build up a distribution of the values it can
  take.
• Imagine an underlying distribution of values
  which we would get if it was possible to take
  more and more measurements under the same
  conditions.
• This gives the probability distribution for the
  variable.

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Continuous probability distributions

• Because continuous random variables can take all
  values in a range, it is not possible to assign
  probabilities to individual values.
• Instead we have a continuous curve, called a
  probability density function, which allows us to
  calculate the probability a value within any
  interval.
• This probability is calculated as the area under
  the curve between the values of interest. The
  total area under the curve must equal 1.

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Normal (Gaussian) distributions

• Normal (also known as Gaussian) distributions are
  by far the most commonly used family of
  continuous distributions.
• They are bell-shaped and are indexed by two
  parameters
• The mean m the distribution is symmetric about
  this value
• The standard deviation s this determines the
  spread of the distribution. Roughly 2/3 of the
  distribution lies within 1 standard deviation of
  the mean, and 95 within 2 standard deviations.

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The probability of continuous variables

• IQ test
• Mean 100 and sd 15
• What is the probability of randomly selecting an
  individual with a test score of 130 or greater?
• P(X 95)?
• P(X 112)?
• P(X 95 or X 112)?

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The probability of continuous variables (cont.)

• What is the probability of randomly selecting
  three people with a test score greater than 112?
• Remember the multiplication rule for independent
  events.

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Introduction to Statistical Inference

• Chapter 11

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Populations vs. Samples

• Population
• The complete set of individuals
• Characteristics are called parameters
• Sample
• A subset of the population
• Characteristics are called statistics.
• In most cases we cannot study all the members of
  a population

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Inferential Statistics

• Statistical Inference
• A series of procedures in which the data obtained
  from samples are used to make statements about
  some broader set of circumstances.

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Two different types of procedures

• Estimating population parameters
• Point estimation
• Using a sample statistic to estimate a population
  parameter
• Interval estimation
• Estimation of the amount of variability in a
  sample statistic when many samples are repeatedly
  taken from a population.
• Hypothesis testing
• The comparison of sample results with a known or
  hypothesized population parameter

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These procedures share a fundamental concept

• Sampling distribution
• A theoretical distribution of the possible values
  of samples statistics if an infinite number of
  same-sized samples were taken from a population.

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Example of the sampling distribution of a
discrete variable
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Continuous Distributions

• Interval or ratio level data
• Weight, height, achievement, etc.
• JellyBlubbers!!!

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Histogram of the Jellyblubber population
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Repeated sampling of the Jellyblubber population
(n 3)
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Repeated sampling of the Jellyblubber population
(n 5)
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Repeated sampling of the Jellyblubber population
(n 10)
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Repeated sampling of the Jellyblubber population
(n 40)
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For more on this concept

• Visit
• http//www.ruf.rice.edu/lane/stat_sim/sampling_di
  st/index.html

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Central Limit Theorem

• Proposition 1
• The mean of the sampling distribution will equal
  the mean of the population.
• Proposition 2
• The sampling distribution of means will be
  approximately normal regardless of the shape of
  the population.
• Proposition 3
• The standard deviation (standard error) equals
  the standard deviation of the population divided
  by the square root of the sample size. (see 11.5
  in text)

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Application of the sampling distribution

• Sampling error
• The difference between the sample mean and the
  population mean.
• Assumed to be due to random error.
• From the jellyblubber experience we know that a
  sampling distribution of means will be randomly
  distributed with

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Standard Error of the Mean and Confidence
Intervals

• We can estimate how much variability there is
  among potential sample means by calculating the
  standard error of the mean.

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Confidence Intervals

• With our Jellyblubbers
• One random sample (n 3)
• Mean 9
• Therefore
• 68 CI 9 or 1(3.54)
• 95 CI 9 or 1.96(3.54)
• 99 CI 9 or 2.58(3.54)

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Confidence Intervals

• With our Jellyblubbers
• One random sample (n 30)
• Mean 8.90
• Therefore
• 68 CI 8.90 or 1(1.11)
• 95 CI 8.90 or 1.96(1.11)
• 99 CI 8.90 or 2.58(1.11)

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Hypothesis Testing (see handout)

• State the research question.
• State the statistical hypothesis.
• Set decision rule.
• Calculate the test statistic.
• Decide if result is significant.
• Interpret result as it relates to your research
  question.