Case StudyExampleScientific LiteraturePopular LiteratureThe Big PictureSamples and PopulationsRandom SamplingCautionsWhat you should knowSamples and PopulationsBret HanlonFall 2011Samples and Populations 1 / 22Samples and PopulationsThe techniques discussed in 2.1-2.6 (exploratory data analysis)are useful for describing a data set, or a sample of data.It is often of interest to generalize findings from a sample to alarger group that statisticians call a population.This generalization is called statistical inference.Statistical inference is often concerned with using statistics,characteristics that can be calculated from sample data, toestimateparameters, characteristics of populations.Examples:Ip = population proportion,ˆp = sample proportionIµ = population mean,¯y = sample meanIσ = population standard deviation, s = sample standard deviationSamples and Populations 2 / 22The Cartoon of Statistical InferenceSee the board.Samples and Populations 3 / 22How Representative is the Sample?Our goal is to infer from the sample information about thepopulation.To do this we need the sample to be representative of thepopulation.Key ideas very well stated in the book.It is important to read Sections 2.8 and 2.9 (very likely you’ll betested on this concept).Samples and Populations 4 / 22Big Picture WarningThis is one of the big picture issues that I hope you remember 20years from now, reading a newspaper (the internet?).These issues should be in the back of your mind for the rest of thecourse. Is the sample truly a random sample? Is it representativeof the population?In practice, samples are almost never truly random (hopefully theyare close). But we have to proceed with what is available.Samples and Populations 5 / 22Sex and Older WomenExampleFertility declines in women as they age until ending at menopause.Younger women may become pregnant relatively easier than olderpre-menopausal women. A hypothesis rooted in evolution andpsychology states that as women age, they may experience increasesin sexual motivation and seek sex more frequently to overcomedecreasing fertility.How can data be collected to examine this hypothesis?Samples and Populations Case Study Example 6 / 22The Scientific LiteratureThe abstract of a recent (2010) article in the journal Personality andIndividual Differences titled REPRODUCTION EXPEDITING: SEXUALMOTIVATIONS, FANTASIES, AND THE TICKING BIOLOGICAL CLOCKbegins as follows:Beginning in their late twenties, women face the uniqueadaptive problem of declining fertility eventually terminating atmenopause. We hypothesize women have evolved areproduction expediting psychological adaptation designed tocapitalize on their remaining fertility.Samples and Populations Case Study Scientific Literature 7 / 22The Scientific Literature (cont.)The abstract continues to report the results as follows:The present study tested predictions based on thishypothesis—these women will experience increased sexualmotivations and sexual behaviors compared to women notfacing a similar fertility decline. Results from college andcommunity samples (N = 827) indicated women withdeclining fertility think more about sex, have more frequentand intense sexual fantasies, are more willing to engage insexual intercourse, and report actually engaging in sexualintercourse more frequently than women of other age groups.These findings suggest women’s “biological clock” mayfunction to shift psychological motivations and actualbehaviors to facilitate utilizing remaining fertility.Samples and Populations Case Study Scientific Literature 8 / 22The Popular LiteratureTime magazine wrote about the scientific publication with anarticle titled THE SCIENCE OF COUGAR SEX: WHY OLDERWOMEN LUST.Somewhat surprisingly, the Time article is more careful than thearticle in the primary literature in discussion of the importance inhow the data is collected when interpreting the results.Samples and Populations Case Study Popular Literature 9 / 22The Big PictureMany of the statistical methods we will encounter this semesterare based on the premise that the data we have at hand (thesample) is representative of some larger group (the population).We often wish to make statistical inferences about one or morepopulations on the basis of sampled data.Statistical methods often assume that samples are randomlyselected from populations of interest, although in practice, this isfrequently not the case.We need to understand:Ihow to take random samples; andIhow to understand how non-random sampling may affectinferences.Samples and Populations The Big Picture 10 / 22Samples and PopulationsDefinitionA population is all the individuals or units of interest; typically, there isnot available data for almost all individuals in a population.DefinitionA sample is a subset of the individuals in a population; there is typicallydata available for individuals in samples.Samples and Populations Samples and Populations 11 / 22Samples and Populations (cont.)Examples:In the cow data set:Ithe sample is the 50 cows;Ithe population is cows of the same breed on dairy farms.In the plantation example:Ithe sample is the three sites where data was collected;Ithe population is all plantations in Costa Rica where one mightconsider restoration to native forest.In the older women sex example:Ithe sample is the 827 women included in the study;Ithe population is American women aged 18+.Samples and Populations Samples and Populations 12 / 22Properties of Representative SamplesEstimates calculated from sample data are often used to makeinferences about populations.If a sample is representative of a population, then statisticscalculated from sample data will be close to corresponding valuesfrom the population.Samples contain less information than full populations, soestimates from samples about population quantities alwaysinvolve some uncertainty.Random sampling, in which every potential sample of a given sizehas the same chance of being selected, is the best way to obtaina representative sample.However, it often impossible or impractical to obtain a randomsample.Samples and Populations Samples and Populations 13 / 22Random SamplingDefinitionA simple random sample is a sample chosen in such a manner thateach possible sample of the same size has the same chance of beingselected.In a simple random sample, all individuals are equally likely to beincluded in the sample.Samples and Populations Random Sampling 14 / 22Random SamplingEstimates from simple random samples
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