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# Class Notes 9-8-15 chapter 5 PY 211

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This 7 page Class Notes was uploaded by Allie Newman on Wednesday September 9, 2015. The Class Notes belongs to PY 211 at University of Alabama - Tuscaloosa taught by Rebecca Allen in Summer 2015. Since its upload, it has received 58 views. For similar materials see Elem Statistical Methods in Psychlogy at University of Alabama - Tuscaloosa.

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Date Created: 09/09/15

PY 211 Class Notes Chapter 5 9815 New Statistical Notation pA probability of eventoutcome A n quotandquot eg pAnB probability that both A and B are true U quotorquot eg pAuB probability that eitherA or B are true quotgiven that eg pAB probability that A is true if we already know that B is true De nitions Fixed event Any event for which the observed outcome is always the same Random event Any event for which observed outcomes can vary Sample Space Total number of possible outcomes for any given event denominator for probability Probability The likelihood that a given outcome will occur symbolized by pevent Probability Frequency of times an outcome occurs divided by the total number of possible outcomes 0 Symbolized as p 0 Used to predict any random event 0 Random event any event where the outcomes observed in that event can vary Unnecessary in a xed event 0 Fixed event any event where the outcomes observed in that event is always the same 0 Operational de nition of probability 0 Frequency that a given outcome occurs divided by the total number of possible outcomes 0 See p0 werpoint for equations Calculating Probability To calculate we need to know 0 Number of total possible outcomes or the sample space 0 How often an outcome of interest occurs 0 The equation for probability is where 0 x equals frequency of times outcome occurs d sanp e space o Probabilities O O O 1 Vary from 0 to 1 Can be written as a fraction decimal or proportion Larger number greater likelihood of outcome 0 Within the sample space for a given event 0 O O O 1 Outcomes are mutually exclusive amp exhaustive only one thing can happen at a time Sum of probabilities of all possible outcomes 10 something has to happen 50 events are binary they happen or not probabilities are relative likelihoods of those events Probability can never be negative An event is either probable probability near 10 or an event is improbable probability near 0 Probability l Examples 0 Of the 69 people who completed our class survey 34 are in a fraternity or sorority lfl randomly select one person from the class what is the probability that person will be in a fraternitysorority o p Greek 3469 493 or 493 0 Of the 69 people who completed our class survey 20 own a brown or black car lfl randomly select one person from the class what is the probability that person will have a brown or black car 0 p Greek 2069 I 0290 or 29 Probability and Relative Frequency 0 Relative frequency of an event is the probability of its occurrence 0 To nd relative frequency 1 Distribute the frequencies Sum of frequencies equals the sample space By distributing frequencies you nd sample space denominator 2 Distribute the relative frequencies Relative frequency re ects probability for each outcome in the distribution 0 Recall relative frequency distribution 0 Scores graphed in terms of proportional frequency 0 Graph number of safety complaints by employees at 45 small businesses IUE I 2 0134 0040 0 011 0000 11 020 81 Eh I 20 02 00 0 00 00 71 011 00 02 0 00 05 50quot 0 0 N 40 1 00 0 Relationship Between Multiple Outcomes Four relationships can exist between two outcomes 0 1 Mutually exclusive 0 2 Independent 0 3 Complementary o 4 Conditional Mutually Exclusive Outcomes Mutually Exclusive when two outcomes cannot occur together 0 dAmBO 0 Where 0 is the symbol for quotandquot 0 Example in one ip of a coin the event it is not possible to ip a head Outcome 1 and a tail Outcome 2 Additive Rule the probability of any one of these outcomes occurring is equal to the sum of their individual probabilities 0 MU B 0Agt MB 0 Where U is the symbol for quotorquot Independent Outcomes Independent when the probability of one outcome does not affect the probability of the second outcome 0 Example if we ip a coin two times the event it is possible to ip a head Outcome 1 on the rst ip and a head Outcome 2 on the second ip Multiplicative Rule the probability that both outcomes occur is equal to the product of their individual probabilities 0 Mn B pltAgtgtlt N3 0 Independent no relationship between events 0 p PU P P o Occurrence probability of one event does not affect occurrence probability of the other Multiplicative Rule the probability that both outcomes occur is equal to the product of their individual probabilities 0 P PnU MN mm o If this isn t true then the events are dependent in some way Complementary Outcomes Complementary sum of probabilities is equal to 100 and outcome is exhaustive of all possible outcomes 0 pA pB 100 0 Example if we ip a coin one time the event the probability of ipping a head Outcome 1 or a tail Outcome 2 is 100 The two outcomes head tail are exhaustive of all possible outcomes for this event Subtracting 1 from probability of one outcome will equal probability of the second outcome Relationship between multiple outcomes oin probability that two outcomes both occur 0 p nU events PmU sample space 0 eg probability of drawing a red face card KQJ x heartsdiamonds 652 115 0 Joint probabilities may be either conditional or independent Conditional Outcomes Conditional the probability of one outcome is dependent on the occurrence of the other outcome 0 Probability of occurrence is changed by the occurrence of the other outcome pPUmU U o a P HP 0 Example You want to determine the probability of drawing two heart cards from a deck A deck has 52 cards of which 13 are hearts 13 On the rst draw p 52 On the second draw the deck has 51 cards only 12 of which are 12 hearts The probability changes p 51 0 Probability of one event A given that another event B occurred 0 Symbolized by quotlquot 0 Probability of selecting uninsured mother U who gave birth in a public hospital P pPn U 80200 040 0 Selecting U given she gave birth in P MU P80130O62 0 Check D 19U P pPnU pP 4065 62f Conditional Probabilities and Bayes39 Theorem Bayes Theorem formula that relates the conditional and marginal unconditional probabilities of two conditional outcomes that occur at random 0 This modi ed formula for conditional probabilities can be used to make inferences concerning parameters in a given population aPUmw U O a P AP 0 Bayes theorem The classic example 0 1 of women aged 40 who have routine screening mammograms actually have breast cancer 0 80 of women with breast cancer will get positive mammograms o 96 of women without breast cancer will also get positive mammograms o A 40yo woman had a positive mammogram What is the probability that she actually has breast cancer o 0 CANCER 0 NO 0 TOTAL 0 CANCER 0 Test 0 80 o 950 o 1030 Pos 0 Test 0 20 o 8950 o 8970 Neg 0 TOTAL 0 100 o 9900 0 10000 0 8001 103010000 078 0 801073 078 Summarizing Bayes theorem allows us to estimate the conditional probability of a given event based on what we know about the probabilities of each of the events involved 0 Also introduces marginal probability 0 Likelihood of an event independent of other events 0 More on marginal probabilities when we talk about hypothesis testing Probability Distributions Random Variable l a variable obtained or measured in a random experiment 0 It is not the actual outcome of a random experiment but describes the possible outcomes in a random experiment 0 Can describe the outcomes for any behavior that varies from person to person or from situation to situation 0 Probability Distribution distribution of probabilities for each outcome of a random variable 0 Probability for obtaining each possible outcome of a random variable 0 Each probability in a distribution ranges from 0 to 1 and can never be negaUve 0 Sum of probabilities in a distribution for a random variable X is equal to 1 ZPx100 o Researcher records number of participants rating certain situation as stressful 5point scale Construct probability distribution for ratings The Frequency fx and Relative Frequency pX TABLE 53910 Distribution of Participant Ratings Ratings I fX I 17X 1 20 250 2 28 350 3 14 175 4 10 125 5 8 100 I 2X 80 ZpX 100 TABLE 511 The Probability Distribution of Participant Ratings pad 250 175 125 100 I The distribution sums to 1 00 Summarizing loint probabilities conditional vs independent 0 Basis for understanding associations among variables 0 One criterion for causation l event P changes likelihood of event U lnferential statistics using sample characteristics to make probability statements about the population Hypothesis testing quothow likely is it that my data would look like this if there were n0 association between the variables ie if probabilities were independenUquot Mean of a Probability Distribution and Expected Value Expected Value the mean or average expected outcome for a given random variable 0 To compute the mean value 1 Multiply each possible outcome X times the probability of its occurrence p 2 Sum each product FEW Expected Outcome the sum of the products for each random outcome times the probability of its occurrence It is necessary to determine the distribution of all other outcomes for a random variable 0 Expected value gives only the average outcome of a random variable 0 This is determined by computing 0 Variance of a probability distribution 0 Standard deviation of a probability distribution 0 Expected value average outcome of a random variable Also need to describe the distribution of all other outcomes for a random variable 0 Same statistics as with any distribution 0 Variance 0 Standard deviation Expected Value and the Binomial Distribution Binomial Probability Distribution distribution of probabilities for each outcome of a bivariate random variable only 2 possible outcomes 0 Can occur by natural occurrence Example outcomes for ipping a coin are heads or tails No other ways to de ne the outcomes for this random variable 0 Can occur by manipulation Example outcomes for selfesteem among children could be high or low Could de ne as having more outcomes but manipulated data to only have two possible outcomes Mean of a Binomial Distribution The product of the number of times the random variable is observed n times the probability of the outcome of interest on an individual observation p 0 it p Homework for Chapter 5 56 12 14 18 20 22 24 26 28 30 32

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