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# Class Note for STAT 528 at OSU 12

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Date Created: 02/06/15
Statistics 528 Data Analysis Overview of Today s Lecture Lecture 6 July 11 2006 Christopher Holloman The Ohio State University Summer 2006 o IPS Sections 45 51 General Probability Rules Sampling Distributions for Counts and Proportions Christopher Holloman The Ohio State University Summer 2006 General Probability Rules General Addition Rule for Union of Two Events For any two events A and B PA or B PA PB PA and B Christopher Holloman The Ohio State University Summer 2006 Christopher Holloman The Ohio State University Summer 2006 What is PA and B 0 If A and B are independent then by the multiplication rule PA and B PAPB If A and B are dependent then we need a general multiplication rule PA and B PAPBA where PBlA is the conditional probability that B occurs given the information that A occurs chnstopher hollomen The 0hlo State UrilvErSlty Summer 2005 0 Definition of independence Two events A and B are Independent If PAlB PA and PBlA PB So our old multi lication rule for independent events is a speCIal case 0 the general multiplication rule Definition of conditional probability when PA gt 0 the conditional probability of B given A is PBlA PA and B PA chrlstopher hollom an The 0hlo State UrilvErSlty Summer 2005 Bayes Rule o Bayes Rule provides a formula for calculating a conditional probability ofA B when only information about B A and A unconditional is available PAlB PBlAPA PB I APAPB I ACPAC chnstopher hollomen The 0hlo State UrilvErSlty Summer 2005 Example The osu footpall staolum management purchases a new oeylce to detect whether a person mlght be carrylng alcohol lnto a game The manufacturer suggests that the oey ce ls yery accurate returnln a false pos tlye Sayan alcohol ls present when lt ls not ln only 1 ou of 5000 tests and a false negatlye Sayln alcohol ls not present when tls ln 1 out oflOO tests The sta lurn offlclals estlmate that ln the future the chance of any lnolyloual attemptlng to carry alcohol lnto a game ls 1 ln 10 what ls the prooaolllty a person ls carrylng alcoho glyen that the machlne says that they are chrlstopher hollom an The 0hlo State UrilvErSlty Summer 2005 Sampling Distributions Sampling Distributions A statistic from a random sample or randomized experiment is a random variable The probability distribution of the statistic is its sampling distribution W The population distribution of a variable is the distribution of its values for all members of the population The population distribution is also the probability distribution of the variable when we choose one individual from the population at random The Binomial Distribution The Binomial Setting 1 There are a fixed number n of observations 2 The n observations are all independent 3 Each of the observations falls into one of just two categories which for convenience we call success and failure 4 The probability of a success call it p is the same for each observation Chrlstbpher hellemeh The ohle State Unlverslty Summer zuu Sampling Distribution for Counts and Proportions Count X number of occurrences of some outcome In a fixed number of observations Sam le Proportion number of occurrences out of he number of observations Xn Chrlstbpher hellem eh The ohle State Unlverslty Summer zuu Binomial Distribution The distribution of the count X of successes in the binomial setting is called the binomial distribution with parameters n and p o n is the number of observations 0 p is the probability of a success on any one observation 0 The possible values of X are whole numbers from 0 to n Notation X is Bn p or X N Bn P Chrlstbpher hellem eh The ohle State Unlverslty Summer zuu For each of the following situations indicate whether the binomial distribution is a reasonable probability model for the random variable X a You observe the sex of the next 50 children born at a local hospital X is the number of girls among them b A couple decides to continue to have children until their first girl is born X is the total number of children the couple has Christnpher hellemeh The ohle State University Summer znn c A chemist repeats a solubility test 10 times on the same substance Each test is conducted at a temperature 10 degrees higher than the previous test She counts the number of times that the substance dissolves d Fifty students are tau ht about the binomial distribution in class T e are the given a short quiz on the Subject The num er of stu ents who pass is counte Christnpher helium eh The ohle State University Summer znn The Binomial Distribution as a Sampling Distribution We use the binomial distribution in statistical inference when we have count data in the binomial setting and we want to make inferences about the total proportion of successes In a population When the population is much lager than the sample the count of successes in a S of size n has approxq39natey a Bnp distribution if the population proportion of successes is p Rule of thumb use the binomial sampling distribution for counts when the population is at least 10 times larger than the sample Christnpher hellemeh The ohle State University Summer znn Finding Binomial Probabilities Assume X Bnp if k lt n then n PltX kgtkpk1epgtrk nl n where k7klnkl Note a aa1a2 land 0 1 gt This is the binomial formula Example FreeThrows Lecture 5 X is the number of shots made in three attempts PXx 008 01 01 0103 014014014 042 02 OOMIOX Christupher heiiemen The Ohm State University Summer znn X N B3 058 so we can use the binomial formula to get the probability distribution of X For example PX 2 058217058 042 Christupher heiiem en The Ohm State University Summer znn Finding Binomial Probabilities Using Minitab Imagine that we are interested in knowing the probability that a 058 percent shoomr makes 10 out 20 freermrows 1 PX 10 Emma 17 0587 quot Christupher heiiemen The Ohm State University Summer znn 2 Using Minitab Calc Probability Distributions Binomial Check Probability Number of Trials 20 Probability of Success 058 Check Input Constant 10 gt PX10 01359 Christupher heiiem en The Ohm State University Summer znn Using Minitab to find cumulative probabilities eg PX lt 8 PX lt 7 Instead of findin PXlt72 PX0PX P 7 directly use the cumulative probability feature Calc Probability Distributions Binomial Check Cumulative Proba lity instead of Probability Number of Trials 20 Probability of Success 058 Check Input Constant 7 gt PXlt8 00324 Chnstupher HDHDman The ome State Umversrty Summer zuu Binomial Mean and Standard Deviation If X Bnp what are ugtlt and 0X X is the number of successes in n independent observations that each have probability of success p Let S be the random variable that indicates whether the i 11 observation was a success S 1 or a failure S 0 The distribution of each is Outcome 0 Probability p lp cmrsmpher HDHDm an The ome State Umversrty Summer zuu Using the definitions of the mean and variance of discrete random variables ue 103 01p p 625 1p2p DID2 1D p 1D Chnstupher HDHDman The ome State Umversrty Summer zuu Smce gtlt s 52 s then anp 52x Plrp or 5x V ml r m cmrsmpher HDHDm an The ome State Umversrty Summer zuu What about the sample proportion Xn Using the rule for linear functions of random variables T P Christopher Holloman The Ohio State University Summer zuus Question What does the sampling distribution of X look like Use Minitab enerate lots of samples from a B20058 distribution Calc Random Data Binomial Generate 1000 rows of data Store in Columns C1 Number of Trials 20 Probability of Success 058 Look at the histogram of C1 Christopher helium eh The Ohio State University Summer zuus Question What does the sampling distribution of 1 look like Create a new column with the proportions Calc Calculator Store result in C2 Expression C120 Look at the histogram of C2 Christopher Holloman The Ohio State University Summer zuus Normal Approximation for Counts and Proportions o X is approximately NMP np m P NULU o is approximatelyNp 7p1pNpa Rule of Thumb This approximation is only valid for values of n and p that satisfy np gt10 and n1pgt10 Christopher helium eh The Ohio State University Summer zuus

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