Stat 5101 Lecture Notes - School of Statistics

7.2. SAMPLES AND POPULATIONS 185There is no end to this game. Every notion that is defined for generalprobability models, we can specialize to empirical distributions. We can defineempirical moments and central moments of all orders, and so forth and so on.But we won’t do that in gory detail. What we’ve done so far is enough for now.7.2 Samples and Populations7.2.1 Finite Population SamplingIt is common to apply statistics to a sample from a population. The populationcan be any finite set of individuals. Examples are the population ofMinnesota today, the set of registered voters in Minneapolis on election day, theset of wolves in Minnesota. A sample is any subset of the population. Examplesare the set of voters called by an opinion poll and asked how they intendto vote, the set of wolves fitted with radio collars for a biological experiment.By convention we denote the population size by N and the sample size by n.Typically n is much less than N. For an opinion poll, n is typically about athousand, and N is in the millions.Random SamplingA random sample is one drawn so that every individual in the population isequally likely to be in the sample. There are two kinds.Sampling without Replacement The model for sampling without replacementis dealing from a well-shuffled deck of cards. If we deal n cards from adeck of N cards, there are (N) n possible outcomes, all equally likely (here weare considering that the order in which the cards are dealt matters). Similarlythere are (N) n possible samples without replacement of size n from a populationof size N. If the samples are drawn in such a way that all are equally likely wesay we have a random sample without replacement from the population.Sampling with Replacement The model for sampling with replacement isspinning a roulette wheel. If we do n spins and the wheel has N pockets, thereare N n possible outcomes, all equally likely. Similarly there are N n possiblesamples with replacement of size n from a population of size N. If the samplesare drawn in such a way that all are equally likely we say we have a randomsample with replacement from the population.Lindgren calls this a simple random sample, although there is no standardmeaning of the word “simple” here. Many statisticians would apply “simple” tosampling either with or without replacement using it to mean that all samplesare equally likely in contrast to more complicated sampling schemes in whichthe samples are not all equally likely.