Stats 242.3(02) Statistical Theory and Methodology

Instructor:W.H.Laverty Office:235 McLean Hall Phone: Lectures: M W F 2:30pm - 3:20am Arts 206 Lab: M 3:30 - 4:20 Arts 206 Evaluation:Assignments, Labs, Term tests - 40% Final Examination - 60%

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

Course Outline

Introduction Chapter 1

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

Estimation Chapter 8 Properties of estimators Interval estimation Sample size determination

Properties and Methods of Estimation Chapter 9 The method of moments Maximum Likelihood estimation Sufficiency (Sufficient Statistics)

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

Linear and Nonlinear Models Least Squares Estimation Chapter 11 Topics covered dependent on available time

The Analysis of Variance Chapter 13 Topics covered dependent on available time

Nonparametric Statistical Methods Chapter 15 Topics covered dependent on available time

Introduction

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)

Phenomena Deterministic Non-deterministic

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

Non-deterministic Phenomena Lack of perfect predictability

Non-deterministic Phenomena haphazard Random

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

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

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?

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

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

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 model outcomes and observations

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.)

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.

If the success rate p was known. Then This equation allows us to predict the value of the observation, X.

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.

Some definitions important to Statistics

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.

A case (elementary sampling unit): This is an individual unit (subject) of the population.

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

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..

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

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.

Example The following variables were evaluated for a study of individuals receiving head injuries in Saskatchewan. 1.Cause of the injury (categorical) Motor vehicle accident Fall Violence other

2.Time of year (date) (numerical or categorical) summer fall winter spring 3.Sex on injured individual (categorical) male female

4.Age (numerical or categorical) < – Mortality (categorical) Died from injury alive

Types of variables In addition some variables are labeled as dependent variables and some variables are labeled as independent variables.

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.

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

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

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

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

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

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

dependentResponse variable independentpredictor variable

A sample: Is a subset of the population

In statistics: One draws conclusions about the population based on data collected from a sample

Reasons: Cost It is less costly to collect data from a sample then the entire population 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

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

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.

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.

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.

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

Some other definitions

A population statistic (parameter): Any quantity computed from the values of variables for the entire population.

A sample statistic: Any quantity computed from the values of variables for the cases in the sample.