IntroductionSamplingRInferenceDefinitionsInterpretationsConditional ProbabilityRulesExampleRandom VariablesBinomialProbabilityBret LargetDepartments of Botany and of StatisticsUniversity of Wisconsin—MadisonStatistics 37115th September 2005Probability and BiologyQuestion: Why should biologists know about probability?Answer (1): Some biological processes seem to be directlyaffected by chance outcomes. Examples:Iformation of gametes;Irecombination events;Ioccurance of genetic mutations;Probability and BiologyQuestion: Why should biologists know about probability?Answer (2): Formal statistical analysis of biological data modelsunexplained variation as caused by chance.Probability and BiologyQuestion: Why should biologists know about probability?Answer (3): In many designed experiments, probability is usedfor:Ithe random allocation of treatments; orIthe random sampling of individuals.Probability and BiologyQuestion: Why should biologists know about probability?Answer (4): Probability is the language with which we express andinterpret assessment of uncertaintyin a formal statistical analysis.Probability and BiologyQuestion: Why should biologists know about probability?Answer (5): Probability comes up in everyday lifeIpredicting the weather;Igambling;Istrategies for games;Iunderstanding risks of passing genetic diseases to children;Iassessing your own risk of disease associated in part withgenetic causes.Random SamplingIThe formal methods of statistical inference taught in thiscourse assumerandom sampling from the population ofinterest.I(Ignore for the present that in practice, individuals are almostnever sampled at random, in a very formal sense, from thepopulation of interest.)Simple Random SamplesIThe process of taking a simple random sample of size n isequivalent to:1. representing every individual from a population with a singleticket;2. putting the tickets into large box;3. mixing the tickets thoroughly;4. drawing out n tickets without replacement.Other Random Sampling StrategiesIStratified random sampling and cluster sampling are examplesof random sampling processes that are notsimple.IData analysis for these types of sampling strategies go beyondthe scope of this course.Simple Random SamplingDefinitionA simple random sample of size n is a random sample taken sothatevery possible sample of size n has the same chance of beingselected.In a simple random sample:Ievery individual has the same chance of being included in thesample;Ievery pair of individuals has the same chance of beingincluded in the sample;Iin fact, every set of k individuals has the same chance ofbeing included in the sample.Insufficient criterion for SRSIThe condition that every individual has the same chance ofbeing included in the sampleis insufficient to imply a simplerandom sample.IFor example, consider sampling one couple at random from aset of ten couples.1. Each person would have a one in ten chance of being in thesample;2. However,each possible set of two people does not have thesame chance of being sampled.3. Pairs of people from the population who are not coupled haveno chance of being sampled;4. while each pair of people in a couple has a one in ten chanceof being sampled.Using R to Take a Random SampleSuppose that you have a numbered set of individuals, numberedfrom 1 to 104, and that I wanted to sample ten of these. Here issome R code that will do just that.> sample(1:104, 10)[1] 9 11 55 100 67 62 68 25 19 54Ithe first argument is the set from which to sample (in thiscase the integers from 1 to 104)Ithe second argument is the sample size;Ithe [1] is R’s way of saying that that row of output beginswith the first element.Iexecuting the same R code again results in a different randomsample.Inference from Samples to PopulationsIStatistical inference involves making statements aboutpopulations on the basis of analysis of sampled data.IThe simple random sampling model is useful because it allowsprecise mathematical description of the random distribution ofthe discrepancy betweenstatistical estimates and populationparameters.IThis is known as chance error due to random sampling.IWhen using the random sampling model, it is important toaskwhat is the population to which the results will begeneralized?Sampling BiasIUsing methods based on random sampling on data notcollected as a random sample is prone tosampling bias, inwhich individuals do not have the same chance of beingsampled.ISampling bias can lead to incorrect statistical inferencesbecausethe sample is unrepresentative of the population inimportant ways.Random ExperimentsDefinitionA random experiment is a process with outcomes that areuncertain.Example: Rolling a single six-sided die once.The outcome (which number lands on top) is uncertain before thedie roll.Outcome SpaceDefinitionThe outcome space is the set of possible simple outcomes from arandom experiment.Example: In a single die roll, the set of possible outcomes is:Ω = {1, 2, 3, 4, 5, 6}EventsDefinitionAn event is a set of possible outcomes.Example: In a single die roll, possible events include:IA =“the die roll is even”;IB =“the die roll is a ‘6’ ”;IC =“the die roll is ‘4 or less’ ”.ProbabilityDefinitionThe probability of an event E , denoted P{E}, is a numericalmeasure between 0 and 1 that represents the likelihood of theevent E in some probability model. Probabilities assigned to eventsmust follow a number of rules.Example: The probability P{the die roll is a ‘6’} equals 1/6 undera probability model that gives equal probability to each possibleresult, but could be different under a different model.ExamplesIf a fair coin is tossed, the probability of a head isP{Heads} = 0.5If bucket contains 34 white balls and 66 red balls and a ball isdrawn uniformly at random, the probability that the drawn ball iswhite isP{white} = 34/100 = 0.34Frequentist Interpretation of ProbabilityThe frequentist interpretation of probability defines theprobability of an event E as the relative frequency withwhich event E would occur in an indefinitely longsequence of independent repetitions of a chanceoperation.Subjective Interpretation of ProbabilityA subjective interpretation of probability definesprobability as an individual’s degree of belief in thelikelihood of an outcome. This school of thought allowsthe use of probability to discuss events that are nothypothetically repeatable.Frequentist StatisticsIThe textbook follows a frequency interpretation of probability.IFrequentist
View Full Document
Unlocking...