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

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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 14 th century Joseph Bertrand (1822-1900) said, “How dare we speak of the laws of chance? Is not chance the antithesis of all law?”

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Probability Formal mathematical development of the “laws of chance” began in 16 th 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.”

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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.

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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

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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

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Statistics is a Process Statistics is the process of “making sense of data”. 1. Gathering Data - a critical step 2. Summarizing Data – descriptive statistics 3. Analyzing Data – inferential statistics 4. Communicating Results - a clear statement of the proper interpretation is important

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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.

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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.

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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.

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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.

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

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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

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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.

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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

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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.

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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.

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

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