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UW-Madison STAT 371 - Simple Random Sampling

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ProbabilityBret LargetDepartment of StatisticsUniversity of Wisconsin - MadisonSeptember 9, 2004Statistics 371, Fall 2003Probability and BiologyWhy should we know something about probability?• Some biological processes seem to be directly affected bychance outcomes. Examples include formation of gametesand occurrence of genetic mutations.• Formal statistical analysis of biological dataassumes thatvariation not explained by measured variables is caused bychance.• Chance might be used in thedesign of an experiment, suchas the random allocation of treatments or random samplingof individuals.• Probability is the language with which weexpress andinterpret assessment of uncertaintyin a formal statisticalanalysis.• Formal statistical analysis depends onmodeling observeddata as the realization of a random process.Statistics 371, Fall 2004 1Probability and Biology• Probability comes up in everyday life — predicting theweather, lotteries or sports betting, strategies for cardgames, understanding risks of passing genetic diseases tochildren, assessing your own risks of diseases associated inpart with genetic causes.Statistics 371, Fall 2004 1Random Sampling• Most of the formal methods of statistical inference we willuse in this class are based on the assumption that theindividual units in the sample aresampled at random fromthe population of interest.• (Ignore for the present that in practice, individuals are almostnever sampled at random, in a very formal sense, from thepopulation of interest.)• Taking asimple random sample of size n is equivalent to theprocess of:1. representing every individual from a population with asingle ticket;2. putting the tickets into large box;3. mixing the tickets thoroughly;4. drawing out n tickets without replacement.•Stratified random sampling and cluster sampling are exam-ples of random sampling processes that are notsimple. Dataanalysis for these types of sampling strategies go beyond thescope of this course.Statistics 371, Fall 2004 2Simple Random Sampling• The defining characteristic of the process of simple randomsamplingis that every possible sample of size n has the samechance of being selected.• In particular, this means that (a) every individual has thesame chance of being included in the sample; and that (b)members of the sample arechosen independently of eachother.• Note thatpoint (a) above is insufficient to define a simplerandom sample. As an example, consider sampling onecouple at random from a set of ten couples. Each personwould have a one in ten chance of being in the s ample, butthe sam pling is not independent. Possible samples of twopeople from the population who are not in a couple have nochance of being sampled while each couple has a one in tenchance of being sampled.Statistics 371, Fall 2004 3Using R to Take a Random SampleSuppose that you have a numbered set of individuals, numberedfrom 1 to 98, and that I wanted to sample ten of these. Here issome R code that will do just that.> sample(1:98, 10)[1] 19 74 3 51 70 75 14 31 76 86In the sample function, the first argument is the set from whichto sample (in this case the integers from 1 to 98) and the secondargument is the sample size.In the output, the[1] is R’s way of saying that that row of outputbegins with the first element.The same code executed again results in a different randomsample.Statistics 371, Fall 2004 4Inference from Samples to Populations• Statistical inference involves making statements about pop-ulations on the basis of analysis of sampled data.• TheSimple random sampling model is useful because itallows precise mathematical description of the random distri-bution of the discrepancy betweenstatistical estimates andpopulation parameters. This is known as chance error dueto random sampling.• When using the random sampling model, it is importantto askwhat is the population to which the results willbe generalized? The use statistical methods that assumerandom sampling on data that is not collected as a randomsample is prone tosampling bias, in which individuals do nothave the same chance of being sampled.•Sampling bias can lead to incorrect statistical inferencesbecause the sample is unrepresentative of the population inimportant ways.Statistics 371, Fall 2004 5Probability• Probability is a numerical measure of the likelihood of anevent.• Probabilities are alwaysbetween 0 and 1, inclusive.• Notation: The probability of an event E is written Pr{E}.Examples:If a fair coin is tossed, the probability of a head isPr{Heads} = 0.5If bucket contains 34 white balls and 66 red balls and a ball isdrawn at random, the probability that the drawn ball is white isPr{white} = 34/100 = 0.34Statistics 371, Fall 2004 6Interpretations of Probability• The frequency interpretation of probability defines the proba-bility of an event E as the relative frequency with which eventE would occur in an indefinitely long sequence of independentrepetitions of a chance operation.• Asubjective interpretation of probability defines probabilityas an individual’s degree of belief in the likelihood of anoutcome. This school of thought allows the use of probabilityto discuss events that are not hypothetically repeatable.• The textbook follows a frequency interpretation of probabil-ity.• Statistical methods based on subjective probability are calledBayesian, named after the Reverend Thomas Bayes who firstproved a mathematical theorem we will encounter later. Inthe Bayesian approach to statistics, everything unknown isdescribed with a probability distribution. Bayes’ Theoremdescribes the proper way to modify a probability distributionin light of new information.Statistics 371, Fall 2004 7Interpretations of Probability• In particular, Bayesian methods treat population parametersas random variables, requiring a probability distribution basedonprior knowledge and not on data.• Frequency methods treat population parameters asfixed, butunknown.• Methods of statistical analysis based on the frequencyinterpretation of probability are in most common use in thebiological science, but Bayesian a pproaches are becomingmore accepted and more prevalent.• It is my desire to teach you the frequentist approach tostatistical inference while leaving youopen-minded aboutlearning Bayesian statisticsat a future encounter withstatistics.• This requires education in the calculus of probability.Statistics 371, Fall 2004 7Examples of Interpretations ofProbability•


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UW-Madison STAT 371 - Simple Random Sampling

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