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An Introduction to Math 419: Probability & Statistics

An Introduction to Math 419: Probability & Statistics. by Marty Spears. Uncertainty, Variability & Disorder Exist. Uncertainty, variability and disorder are unavoidable in the world around us. The notion of chance has been around for centuries. Dice in Egyptian tombs from 2000 B.C.

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An Introduction to Math 419: Probability & Statistics

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  1. An Introduction to Math 419:Probability & Statistics by Marty Spears

  2. Uncertainty, Variability & Disorder Exist • Uncertainty, variability and disorder are unavoidable in the world around us. • The notion of chance has been around for centuries. • Dice in Egyptian tombs from 2000 B.C. • Cards & board games in 14th century • Joseph Bertrand (1822-1900) said,“How dare we speak of the laws of chance? Is not chance the antithesis of all law?”

  3. Probability • Formal mathematical development of the “laws of chance” began in 16th Century • Cardano wrote A Book on Games of Chance 1550 • Pascal & Fermat exchanged letters in 1654 over a gamblers dispute concerning a popular dice game. • Pascal’s work is commonly given credit for the birth of a branch in mathematics called probability theory. • Blaise Pascal (1623-1662) said, • “If God does not exist, one will lose nothing by believing in him, while if he does exist, one will lose everything by not believing.”

  4. Probability • Useful for modeling situations that involve uncertainty. • weather, spread of disease, reliability, etc. • Quantifies uncertainty. • 0 ≤ P(A) ≤ 1 for any event A • Forms the basis of inferential statistics. • Inferential statistics are useful for decision making in the presence of uncertainty.

  5. Statistics • The science of collecting, simplifying, and describing data, as well as making inference based on the analysis of data. • Descriptive Statistics • Simplifying, summarizing, describing, etc. • Inferential Statistics • Making inference, drawing conclusions

  6. Probability & Statistics Probability • Called “the laws of chance” by many • Useful for modeling uncertainty Statistics • Based on principles of probability • Useful for making decisions in the face of uncertainty

  7. Statistics is a Process • Statistics is the process of “making sense of data”. • Gathering Data - a critical step • Summarizing Data – descriptive statistics • Analyzing Data – inferential statistics • Communicating Results - a clear statement of the proper interpretation is important

  8. Population versus Sample • You must be able to distinguish between a population and a sample. Population • The set of all possible outcomes. Sample • A subset of the possible outcomes. • Purpose is to accurately reflect the population. • A large sample size does not guarantee a good sample.

  9. Parameter versus Statistic • You must be able to distinguish between a parameter and a statistic. Parameter • A numerical property of the population. • A fixed value. Statistic • A numerical property of the sample. • A random value.

  10. Overview Statement • Descriptive statistics from a sample can be used to draw conclusions about a population (parameter). • This is how statistics is used to make decisions in the presence of uncertainty.

  11. Why Study Statistics? • One obvious reason is that is required!! • A basic understanding is becoming expected/necessary in today’s world. • Many jobs require it. • Ignorance of statistics can be used against you. • It has application in every field.

  12. Types of Data • Qualitative Data (Categorical) • Nominal Data • No natural Ordering. • Ordinal Data • Natural ordering exists.

  13. Quantitative Data (Numerical) • Discrete Data • Finite (countable) number of outcomes possible. • Typically counting something. • Continuous Data • Infinite number of outcomes possible corresponding to an interval on the number line. • Typically measuring something. • Interval Scale – no meaningful zero • Ratio Scale – included a meaningful zero

  14. Gathering Data • A critical step in the process of making sense out of data. • Specify the problem to make sure you understand it • Identify potentially significant variables and factors • Choose an appropriate design. • Collect data.

  15. Sample Survey • Passive tool for collecting data. • Sampling Techniques • Simple Random Sample – equally likely outcomes • Stratified Random Sample – equally likely outcomes within strata (sub-populations) • Cluster Sample – equally likely outcomes from a subset of strata • Systematic Sample – randomly select first sample and system determines the remaining samples • Convenience Sample – not random

  16. Sample Survey • Data Collection Techniques • Personal Interview • Telephone Interview • Self-Administered Interview • Direct Observation • The outcome of a survey can be dramatically changes by the design of a questionnaire.

  17. Experimental Study • An active tool for collecting data that involves manipulation of variables and observation of the effect on other variables. • Experimental Design • Completely Randomized Design (CRD) • Randomized Block Design (RBD) • Blocking is used to reduce or eliminate the bias or variability due to a known factor.

  18. Observational Study • A study with limited control over the conditions of the experiment.

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