Simple Random Sampling Statistics 371 Fall 2003 The defining characteristic of the process of simple random sampling is that every possible sample of size n has the same chance of being selected In particular this means that a every individual has the same chance of being included in the sample and that b members of the sample are chosen independently of each other Note that point a above is insufficient to define a simple random sample As an example consider sampling one couple at random from a set of ten couples Each person would have a one in ten chance of being in the sample but the sampling is not independent Possible samples of two people from the population who are not in a couple have no chance of being sampled while each couple has a one in ten chance of being sampled Statistics 371 Fall 2004 3 Probability and Biology Suppose that you have a numbered set of individuals numbered from 1 to 98 and that I wanted to sample ten of these Here is some R code that will do just that Why should we know something about probability The same code executed again results in a different random sample Statistics 371 Fall 2004 4 Inference from Samples to Populations Statistical inference involves making statements about populations on the basis of analysis of sampled data The Simple random sampling model is useful because it allows precise mathematical description of the random distribution of the discrepancy between statistical estimates and population parameters This is known as chance error due to random sampling When using the random sampling model it is important to ask what is the population to which the results will be generalized The use statistical methods that assume random sampling on data that is not collected as a random sample is prone to sampling bias in which individuals do not have the same chance of being sampled Sampling bias can lead to incorrect statistical inferences because the sample is unrepresentative of the population in important ways Statistics 371 Fall 2004 5 Probability Probability and Biology 1 Probability comes up in everyday life predicting the weather lotteries or sports betting strategies for card games understanding risks of passing genetic diseases to children assessing your own risks of diseases associated in part with genetic causes Statistics 371 Fall 2004 1 Random Sampling Probability is a numerical measure of the likelihood of an event Probabilities are always between 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 is Pr Heads 0 5 If bucket contains 34 white balls and 66 red balls and a ball is drawn at random the probability that the drawn ball is white is Pr white 34 100 0 34 Statistics 371 Fall 2004 Statistics 371 Fall 2004 Probability In the output the 1 is R s way of saying that that row of output begins with the first element Bret Larget September 9 2004 In the sample function the first argument is the set from which to sample in this case the integers from 1 to 98 and the second argument is the sample size Some biological processes seem to be directly affected by chance outcomes Examples include formation of gametes and occurrence of genetic mutations Formal statistical analysis of biological data assumes that variation not explained by measured variables is caused by chance Chance might be used in the design of an experiment such as the random allocation of treatments or random sampling of individuals Probability is the language with which we express and interpret assessment of uncertainty in a formal statistical analysis Formal statistical analysis depends on modeling observed data as the realization of a random process Department of Statistics sample 1 98 10 1 19 74 3 51 70 75 14 31 76 86 University of Wisconsin Madison Using R to Take a Random Sample 6 Most of the formal methods of statistical inference we will use in this class are based on the assumption that the individual units in the sample are sampled at random from the population of interest Ignore for the present that in practice individuals are almost never sampled at random in a very formal sense from the population of interest Taking a simple random sample of size n is equivalent to the process of 1 representing every individual from a population with a single 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 examples of random sampling processes that are not simple Data analysis for these types of sampling strategies go beyond the scope of this course Statistics 371 Fall 2004 2 Conditional Probability and Probability Interpretations of Probability Trees It is a common setting in biological probability problems for an event to consist of the outcomes from a sequence of possibly dependent chance occurrences In this case a probability tree is a very useful device for guiding the appropriate calculations We have already discussed definitions of probability and events The following example will illustrate definitions of conditional probability independence of events and several rules for calculating probabilities of complex events Statistics 371 Fall 2004 10 Example The frequency interpretation of probability defines the probability of an event E as the relative frequency with which event E would occur in an indefinitely long sequence of independent repetitions of a chance operation A subjective interpretation of probability defines probability as an individual s degree of belief in the likelihood of an outcome This school of thought allows the use of probability to discuss events that are not hypothetically repeatable The textbook follows a frequency interpretation of probability Statistical methods based on subjective probability are called Bayesian named after the Reverend Thomas Bayes who first proved a mathematical theorem we will encounter later In the Bayesian approach to statistics everything unknown is described with a probability distribution Bayes Theorem describes the proper way to modify a probability distribution in light of new information Statistics 371 Fall 2004 7 Interpretations of Probability The following relative frequencies are known from review of literature on the subject of strokes and high blood pressure in the elderly 1 Ten percent of people aged 70 will suffer a stroke within five years 2 Of those individuals who had their first stroke within five
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