4. Probability DistributionsBasic probability rulesSlide 3Slide 4Probability distribution of a variableSlide 6Slide 7Slide 8Like frequency distributions, probability distributions have descriptive measures, such as mean and standard deviationSlide 10Slide 11Slide 12Normal distributionSlide 14Slide 15Slide 16Slide 17Slide 18Slide 19Notes about z-scoresSlide 21Sampling distributionsSlide 23Sampling distribution of a statistic is the probability distribution for the possible values of the statisticSampling distribution of sample meanSlide 26Slide 27Slide 28Central Limit Theorem: For random sampling with “large” n, the sampling dist. of the sample mean is approximately a normal distributionSlide 30Slide 32Note:Slide 34Slide 35By the Central Limit Theorem (multiple choice)4. Probability DistributionsProbability: With random sampling or a randomized experiment, the probability an observation takes a particular value is the proportion of times that outcome would occur in a long sequence of observations.Usually corresponds to a population proportion (and thus falls between 0 and 1) for some real or conceptual population.Basic probability rulesLet A, B denotes possible outcomes•P(not A) = 1 – P(A)•For distinct possible outcomes A and B, P(A or B) = P(A) + P(B)•P(A and B) = P(A)P(B given A)•For “independent” outcomes, P(B given A) = P(B), so P(A and B) = P(A)P(B).Happiness (2008 GSS data)Income Very Pretty Not too Total ------------------------------- Above Aver. 164 233 26 423 Average 293 473 117 883 Below Aver. 132 383 172 687 ------------------------------ Total 589 1089 315 1993Let A = average income, B = very happyP(A) estimated by (a “marginal probability”), P(not A) = 1 – P(A) = P(B given A) estimated by (a “conditional probability”)P(A and B) = P(A)P(B given A) est. by (which equals , a “joint probability”)B1: randomly selected person is very happyB2: second randomly selected person is very happyP(B1), P(B2) estimated by P(B1 and B2) = P(B1)P(B2) estimated by If instead B2 refers to partner of person for B1, B1 and B2 probably not independent and this formula is inappropriateProbability distribution of a variableLists the possible outcomes for the “random variable” and their probabilities Discrete variable: Assign probabilities P(y) to individual values y, with 0 ( ) 1, ( ) 1P y P y� � S =Example: Randomly sample 3 people and ask whether they favor (F) or oppose (O) legalization of same-sex marriagey = number who “favor” (0, 1, 2, or 3)For possible samples of size n = 3, Sample y Sample y(O, O, O) 0 (O, F, F) 2(O, O, F) 1 (F, O, F) 2(O, F, O) 1 (F, F, O) 2(F, O, O) 1 (F, F, F) 3If population equally split between F and O, these eight samples are equally likely and probability distribution of y is y P(y) 0 1 2 3 (special case of “binomial distribution,” introduced in Chap. 6). In practice, probability distributions are often estimated from sample data, and then have the form of frequency distributionsExample: GSS results on y = number of people you knew personally who committed suicide in past 12 months (variable “suiknew”).Estimated probability distribution is y P(y) 0 .895 1 .084 2 .015 3 .006Like frequency distributions, probability distributions have descriptive measures, such as mean and standard deviation•Mean (expected value) -( ) ( )E Y yP ym= =�µ = represents a “long run average outcome”(median = mode = 0)Standard Deviation - Measure of the “typical” distance of an outcome from the mean, denoted by σ (We won’t need to calculate this formula.) If a distribution is approximately bell-shaped, then:•all or nearly all the distribution falls between µ - 3σ and µ + 3σ•Probability about 0.68 falls between µ - σ and µ + σ 2 = ( ) ( )y P ys mS -Example: From result later in chapter, if n people are randomly selected from population with proportion  favoring legal same-sex marriage (1-, Oppose), then y = number in sample who favor it has a bell-shaped probability distribution withe.g, with n = 1000,  = 0.50, get µ = , σ = Nearly all the distribution falls between about i.e., almost certainly between about 45% and 55% of sample say they favor it.( ) , (1 )E y n nm p s p p= = = -Continuous variables: Probabilities assigned to intervals of numbersEx. When y takes lots of values, as in last example, it is continuous for practical purposes. Then, if probability distribution is approx. bell-shaped, Most important probability distribution for continuous variables is the normal distribution( ) 0.68, ( 2 2 ) 0.95P y P ym s m s m s m s- � � + � - � � + �In previous example, ( ) (484 516) 0.68P y P ym s m s- � � + = � � �Normal distribution•Symmetric, bell-shaped (formula in Exercise 4.56)•Characterized by mean () and standard deviation (), representing center and spread•Probability within any particular number of standard deviations of  is same for all normal distributions•An individual observation from an approximately normal distribution has probability 0.68 of falling within 1 standard deviation of mean 0.95 of falling within 2 standard deviations  0.997 of falling within 3 standard deviationsTable A (inside back cover of text) gives probability in right tail above µ + zσ for various values of z. Second Decimal Place of zz .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 0.0 .5000 .4960 .4920 .4880 .4840 .4801 .4761 .4721 .4681 .4641 … ….1.4 .0808 .0793 .0778 .0764 .0749 .0735 .0722 .0708 .0694 .0681 1.5 .0668 .0655 .0643 .0630 .0618 .0606 .0594 .0582 .0571 .0559 …….……..Example: What is probability falling between µ - 1.50σ and µ + 1.50σ ?•z = 1.50 has right tail probability = •Left tail probability = by symmetry•Two-tail probability =•Probability within µ - 1.50σ and µ + 1.50σ is Example: z = 2.0 gives two-tail prob. = 2(0.0228) = 0.046, probability within µ ± 2σ is 1 - 0.046 = 0.954Example: What z-score corresponds to 99th percentile (i.e., µ + zσ =