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.  Cards & board games in 14 th century Joseph Bertrand ( ) said, “How dare we speak of the laws of chance? Is not chance the antithesis of all law?”

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 ( ) 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.”

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.

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

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

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

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.

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.

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.

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.

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

 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

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.

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

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.

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.

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