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Statistical Theory and Methodology Basics

Learn fundamental statistical concepts including sampling distributions, estimation, hypothesis testing, and more. Understand the role of statistics in scientific inference.

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Statistical Theory and Methodology Basics

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  1. Stats 242.3(02) Statistical Theory and Methodology

  2. Instructor: W.H.Laverty Office: 235 McLean Hall Phone: 966-6096 Lectures: M W F 2:30pm - 3:20amArts 206Lab: M 3:30 - 4:20 Arts 206 Evaluation: Assignments, Labs, Term tests - 40%Final Examination - 60%

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

  4. Course Outline

  5. Introduction • Chapter 1

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

  7. Estimation Chapter 8 • Properties of estimators • Interval estimation • Sample size determination

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

  9. 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

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

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

  12. Nonparametric Statistical Methods Chapter 15 • Topics covered dependent on available time

  13. Introduction

  14. 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)

  15. Phenomena Non-deterministic Deterministic

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

  17. Non-deterministic Phenomena Lack of perfect predictability

  18. Non-deterministic Phenomena Random haphazard

  19. 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

  20. 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

  21. 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?

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

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

  24. 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

  25. 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.)

  26. 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.

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

  28. 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.

  29. Some definitions important to Statistics

  30. 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.

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

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

  33. 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. .

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

  35. 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.

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

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

  38. Age (numerical or categorical) • < 10 • 10-19 • 20 - 29 • 30 - 49 • 50 – 65 • 65+ • Mortality (categorical) • Died from injury • alive

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

  40. 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.

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

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

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

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

  45. 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

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

  47. dependent Response variable independent predictor variable

  48. A sample: Is a subset of the population

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

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

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