Psyc 235: Introduction to StatisticsReview: ?s about Descriptive StatisticsReview: ProbabilityReview ExampleSlide 5Example: Multiplication PrincipleOk? So what does that have to do with statistics?The Assumption of IndependenceImportant Terms/ConceptsProbabilities in Simple ExperimentsProbability of Draw with ReplacementExample: The proverbial Statistician’s Urn.Probability of Draws without ReplacementSlide 14Brief Psychology Experiment.So, now we’ve moved from talking about a single group to multiple groups.Venn Diagrams & Some TermsLet’s make our own…Venn Diagrams and Set TheorySet Theory and ProbabilitySet TheorySet Theory CalculationsCan you tell me…Some other examplesBack to the Psych ExperimentTo-DoPsyc 235:Introduction to StatisticsDON’T FORGET TO SIGN IN FOR CREDIT!Review: ?s about Descriptive Statistics•Displaying Data•Central Tendency: Mean, Median Mode•Dispersion: Variance (sample, population)Review: Probability•Counting and Sample SpaceAll possible outcomes n!•Permutations (order matters)For n objects taken r at a time: n!(n-r)!•Combinations (order doesn’t matter)For n objects taken r at a time: n!r!(n-r)!Review ExampleSay a little league coach has 14 kids and wants to know how many distinct groups of 9 outfielders he can make.•What do we do?Combination14!9!5!What if the little league coach wants to know how many unique ways he can arrange the kids into each of the 9 field positions?•What do we do?Permutation14! 5!Review: Probability•Multiplication Principle (Fundamental Counting Principle)For 2 independent phenomenon, multiply the # possible outcomes for the two individual phenomenaExample: Multiplication PrincipleSay our coach has 2 pitchers, 1 catcher, 5 outfielders, and 6 infielders.Can you help me set up how many unique teams he has now (keeping in mind those positions)?Ok? So what does that have to do with statistics?•Once we know all the possible outcomes, we can compute what the likelihood is of randomly selecting one (or a set of potential) outcomes out of all of those possible outcomes.•Eventually, we’ll want to do this the other way-- given the set of outcomes that we randomly selected from an unknown group, what can we conclude about the characteristics of all possible outcomes from that group.The Assumption of Independence•Independence: the outcomes of one simple experiment are not linked to the outcome of another•Height and weight are NOT independent•Eye color and IQ are independentImportant Terms/Concepts•Simple experiment: some process that leads to one possible outcome being selected from a set of possibilities•Sample space: the set of possible outcomes in a simple experiment•Sample points: members of the sample space•Event: Any subset of the sample space•Elementary Event: An event that contains a single sample pointHow would these terms relate to rolling a die?Probabilities in Simple Experiments•Assume that each roll of the die (or elementary event) is equally likely. Note: How do we know equally likely?Good assumptionRelative frequencySubjective estimation based on some criteria•What is the probability of any one roll of the die?•What is the probability of getting three 6s in three rolls?Probability of Draw with Replacement•What is replacement?: •All sample points in the sample space are available each time you do a simple experiment.•Like rolling a die.•When you want to know the likelihood of obtaining some specific set of results, use multiplication principle.Example: The proverbial Statistician’s Urn.Probability of Draws without Replacement•Instead of considering each simple experiment in isolation, think about the events.•How many events satisfy your conditions?•How many events are in the sample space?•P(E)= # of events satisfy condition/total # of eventsExample: The proverbial Statistician’s Urn.Brief Psychology Experiment.•Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.Which is more likely?•1.Linda is a bank teller.•2.Linda is a bank teller and is active in the feminist movement.So, now we’ve moved from talking about a single group to multiple groups.•Is there a better way to represent what we’re talking about now?•Yes!•Venn Diagrams.•Let’s draw one that represents what we had in our grecian urn…Venn Diagrams & Some Terms•Universal Set (W): the largest set (sample space)•Mutually Exclusive Events: events that never occur together•Sure Event: event that always occurs•Impossible Event: event that never occurs (empty set)•Exhaustive Events: If events are equal to the universal SetLet’s make our own…•Super bowl Supporters:•Giants, Patriots, Both, NeitherVenn Diagrams and Set Theory•Now, we can count how many sample points fall in each category.•How do we talk about combinations of categories (e.g. how many people favored a team(s) in the super bowl?)•Union: combination of two setsC= A U B•Intersection: contains only objects common to A and BD = A  BSet Theory and Probability•Once we know number of total events and number of events in each category, we can calculate the probability of obtaining a result in any one category.•P(A)= A/W•P(B)= B/W•P(A  B) = A  B/W•P(A U B) = AUB/WSet Theory•Set Compliment A = 1 - A•SubsetsProper Subset (all objects in subset are not only objects in set) C  A Subset (all objects in subset could be only objects in set C  ASet Theory Calculations•P(A  B) = P(A) * P(B)•P(A U B) = P(A) + P(B)- P(A  B)Can you tell me…•A  •A  W•A U •A U W•A  A•A U A= = A= A= W= = WSome other examples•A occurs or B does not (or both)•A occurs without B or B without A•A does not occur or B does not occur (or both)•A occurs without B or A and B both occur•B occurs but A does not•B occurs without A or A and B occurBack to the Psych ExperimentWhich is more likely?•1.Linda is a bank teller.•2.Linda is a bank teller and is active in the feminist movement.If/Then: Logic FallaciesVenn Diagrams can help!To-Do•ALEKS: aim for about 24 hours spent by the end of this week (including class time)•Descriptive Statistics should already be complete•Start work on Probability Sections•Make sure you are up to