Stats 242.3(02)

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Stats 242.3(02)

• Statistical Theory and Methodology

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(No Transcript)
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Text

• Dennis D. Wackerly, William Mendenhall III,
  Richard L. Scheaffer, Mathematical Statistics
  with applications, 6th Edition, Duxbury Press

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Course Outline
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Introduction

• Chapter 1

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Sampling Distributions

• Chapter 7
• Sampling distributions related to the Normal
  distribution
• The Central Limit theorem
• The Normal approximation to the Binomial

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Estimation

• Chapter 8
• Properties of estimators
• Interval estimation
• Sample size determination

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Properties and Methods of Estimation

• Chapter 9
• The method of moments
• Maximum Likelihood estimation
• Sufficiency (Sufficient Statistics)

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Hypothesis testing

• Chapter 10
• Elements of a statistical test - type I and type
  II errors
• The Z test - one and two samples
• hypothesis testing for the means of the normal
  distribution with small sample sizes
• Power and the NeymannPearson Lemma
• Likelihood ratio tests

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Linear and Nonlinear Models Least Squares
Estimation

• Chapter 11
• Topics covered dependent on available time

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The Analysis of Variance

• Chapter 13
• Topics covered dependent on available time

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Nonparametric Statistical Methods

• Chapter 15
• Topics covered dependent on available time

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Introduction
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What is Statistics?

• It is the major mathematical tool of scientific
  inference methods for drawing conclusion from
  data.
• Data that is to some extent corrupted by some
  component of random variation (random noise)

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Phenomena
Non-deterministic

• Deterministic

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Deterministic Phenomena

• A mathematical model exists that allows accurate
  prediction of outcomes of the phenomena (or
  observations taken from the phenomena)

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Non-deterministic Phenomena

• Lack of perfect predictability

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Non-deterministic Phenomena
Random

• haphazard

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Random Phenomena

• No mathematical model exists that allows accurate
  prediction of outcomes of the phenomena (or
  observations)
• However the outcomes (or observations) exhibit in
  the long run on the average statistical
  regularity

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Example

• Tossing of a Coin
• No mathematical model exists that allows accurate
  prediction of outcome of this phenomena
• However in the long run on the average
  approximately 50 of the time the coin is a head
  and 50 of the time the coin is a tail

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Haphazard Phenomena

• No mathematical model exists that allows accurate
  prediction of outcomes of the phenomena (or
  observations)
• No exhibition of statistical regularity in the
  long run.
• Do such phenomena exist?

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• In both Statistics and Probability theory we are
  concerned with studying random phenomena

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In probability theory

• The model is known and we are interested in
  predicting the outcomes and observations of the
  phenomena.

outcomes and observations
model
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In statistics

• The model is unknown
• the outcomes and observations of the phenomena
  have been observed.
• We are interested in determining the model from
  the observations

outcomes and observations
model
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Example - Probability

• A coin is tossed n 100 times
• We are interested in the observation, X, the
  number of times the coin is a head.
• Assuming the coin is balanced (i.e. p the
  probability of a head ½.)

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Example - Statistics

• We are interested in the success rate, p, of a
  new surgical procedure.
• The procedure is performed n 100 times.
• X, the number of successful times the procedure
  is performed is 82.
• The success rate p is unknown.

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• If the success rate p was known.
• Then

This equation allows us to predict the value of
the observation, X.
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• In the case when the success rate p was unknown.
• Then the following equation is still true the
  success rate

We will want to use the value of the observation,
X 82 to make a decision regarding the value of
p.
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Some definitions

• important to Statistics

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A population

• this is the complete collection of subjects
  (objects) that are of interest in the study.
• There may be (and frequently are) more than one
  in which case a major objective is that of
  comparison.

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A case (elementary sampling unit)

• This is an individual unit (subject) of the
  population.

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A variable

• a measurement or type of measurement that is made
  on each individual case in the population.

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Types of variables

• Some variables may be measured on a numerical
  scale while others are measured on a categorical
  scale.
• The nature of the variables has a great influence
  on which analysis will be used. .

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• For Variables measured on a numerical scale the
  measurements will be numbers.
• Ex Age, Weight, Systolic Blood Pressure
• For Variables measured on a categorical scale the
  measurements will be categories.
• Ex Sex, Religion, Heart Disease

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Note

• Sometimes variables can be measured on both a
  numerical scale and a categorical scale.
• In fact, variables measured on a numerical scale
  can always be converted to measurements on a
  categorical scale.

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Example

• The following variables were evaluated for a
  study of individuals receiving head injuries in
  Saskatchewan.

• Cause of the injury (categorical)
• Motor vehicle accident
• Fall
• Violence
• other

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• Time of year (date) (numerical or categorical)
• summer
• fall
• winter
• spring

• Sex on injured individual (categorical)
• male
• female

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• Age (numerical or categorical)
• lt 10
• 10-19
• 20 - 29
• 30 - 49
• 50 65
• 65

• Mortality (categorical)
• Died from injury
• alive

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Types of variables

• In addition some variables are labeled as
  dependent variables and some variables are
  labeled as independent variables.

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• This usually depends on the objectives of the
  analysis.
• Dependent variables are output or response
  variables while the independent variables are the
  input variables or factors.

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• Usually one is interested in determining
  equations that describe how the dependent
  variables are affected by the independent
  variables

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Example

• Suppose we are collecting data on
• Blood Pressure
• Height
• Weight
• Age

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• Suppose we are interested in how
• Blood Pressure
• is influenced by the following factors
• Height
• Weight
• Age

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• Then
• Blood Pressure
• is the dependent variable
• and
• Height
• Weight
• Age
• Are the independent variables

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Example Head Injury study

• Suppose we are interested in how
• Mortality
• is influenced by the following factors
• Cause of head injury
• Time of year
• Sex
• Age

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• Then
• Mortality
• is the dependent variable
• and
• Cause of head injury
• Time of year
• Sex
• Age
• Are the independent variables

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dependent
Response variable
independent
predictor variable
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A sample

• Is a subset of the population

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In statistics

• One draws conclusions about the population based
  on data collected from a sample

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Reasons

• Cost

It is less costly to collect data from a sample
then the entire population
Accuracy
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Accuracy
Data from a sample sometimes leads to more
accurate conclusions then data from the entire
population
Costs saved from using a sample can be directed
to obtaining more accurate observations on each
case in the population
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Types of Samples

• different types of samples are determined by how
  the sample is selected.

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Convenience Samples

• In a convenience sample the subjects that are
  most convenient to the researcher are selected as
  objects in the sample.
• This is not a very good procedure for inferential
  Statistical Analysis but is useful for
  exploratory preliminary work.

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Quota samples

• In quota samples subjects are chosen conveniently
  until quotas are met for different subgroups of
  the population.
• This also is useful for exploratory preliminary
  work.

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Random Samples

• Random samples of a given size are selected in
  such that all possible samples of that size have
  the same probability of being selected.

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• Convenience Samples and Quota samples are useful
  for preliminary studies. It is however difficult
  to assess the accuracy of estimates based on this
  type of sampling scheme.
• Sometimes however one has to be satisfied with a
  convenience sample and assume that it is
  equivalent to a random sampling procedure

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Some other definitions
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A population statistic (parameter)

• Any quantity computed from the values of
  variables for the entire population.

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A sample statistic

• Any quantity computed from the values of
  variables for the cases in the sample.