Samples and Populations Bret Hanlon Fall 2011 Samples and Populations 1 22 Samples and Populations The 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 a larger 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 to estimate parameters characteristics of populations Examples I I I p population proportion p sample proportion population mean y sample mean population standard deviation s sample standard deviation Samples and Populations 2 22 The Cartoon of Statistical Inference See the board Samples and Populations 3 22 How Representative is the Sample Our goal is to infer from the sample information about the population To do this we need the sample to be representative of the population Key ideas very well stated in the book It is important to read Sections 2 8 and 2 9 very likely you ll be tested on this concept Samples and Populations 4 22 Big Picture Warning This is one of the big picture issues that I hope you remember 20 years from now reading a newspaper the internet These issues should be in the back of your mind for the rest of the course Is the sample truly a random sample Is it representative of the population In practice samples are almost never truly random hopefully they are close But we have to proceed with what is available Samples and Populations 5 22 Sex and Older Women Example Fertility declines in women as they age until ending at menopause Younger women may become pregnant relatively easier than older pre menopausal women A hypothesis rooted in evolution and psychology states that as women age they may experience increases in sexual motivation and seek sex more frequently to overcome decreasing fertility How can data be collected to examine this hypothesis Samples and Populations Case Study Example 6 22 The Scientific Literature The abstract of a recent 2010 article in the journal Personality and Individual Differences titled R EPRODUCTION EXPEDITING S EXUAL MOTIVATIONS FANTASIES AND THE TICKING BIOLOGICAL CLOCK begins as follows Beginning in their late twenties women face the unique adaptive problem of declining fertility eventually terminating at menopause We hypothesize women have evolved a reproduction expediting psychological adaptation designed to capitalize on their remaining fertility Samples and Populations Case Study Scientific Literature 7 22 The Scientific Literature cont The abstract continues to report the results as follows The present study tested predictions based on this hypothesis these women will experience increased sexual motivations and sexual behaviors compared to women not facing a similar fertility decline Results from college and community samples N 827 indicated women with declining fertility think more about sex have more frequent and intense sexual fantasies are more willing to engage in sexual intercourse and report actually engaging in sexual intercourse more frequently than women of other age groups These findings suggest women s biological clock may function to shift psychological motivations and actual behaviors to facilitate utilizing remaining fertility Samples and Populations Case Study Scientific Literature 8 22 The Popular Literature Time magazine wrote about the scientific publication with an article titled T HE S CIENCE OF C OUGAR S EX W HY O LDER W OMEN L UST Somewhat surprisingly the Time article is more careful than the article in the primary literature in discussion of the importance in how the data is collected when interpreting the results Samples and Populations Case Study Popular Literature 9 22 The Big Picture Many of the statistical methods we will encounter this semester are based on the premise that the data we have at hand the sample is representative of some larger group the population We often wish to make statistical inferences about one or more populations on the basis of sampled data Statistical methods often assume that samples are randomly selected from populations of interest although in practice this is frequently not the case We need to understand I I how to take random samples and how to understand how non random sampling may affect inferences Samples and Populations The Big Picture 10 22 Samples and Populations Definition A population is all the individuals or units of interest typically there is not available data for almost all individuals in a population Definition A sample is a subset of the individuals in a population there is typically data available for individuals in samples Samples and Populations Samples and Populations 11 22 Samples and Populations cont Examples In the cow data set I I the sample is the 50 cows the population is cows of the same breed on dairy farms In the plantation example I I the sample is the three sites where data was collected the population is all plantations in Costa Rica where one might consider restoration to native forest In the older women sex example I I the sample is the 827 women included in the study the population is American women aged 18 Samples and Populations Samples and Populations 12 22 Properties of Representative Samples Estimates calculated from sample data are often used to make inferences about populations If a sample is representative of a population then statistics calculated from sample data will be close to corresponding values from the population Samples contain less information than full populations so estimates from samples about population quantities always involve some uncertainty Random sampling in which every potential sample of a given size has the same chance of being selected is the best way to obtain a representative sample However it often impossible or impractical to obtain a random sample Samples and Populations Samples and Populations 13 22 Random Sampling Definition A simple random sample is a sample chosen in such a manner that each possible sample of the same size has the same chance of being selected In a simple random sample all individuals are equally likely to be included in the sample Samples and Populations Random Sampling 14 22 Random Sampling Estimates from simple random samples are unbiased there is no systematic discrepency between sample estimates and corresponding population values For random samples larger samples are
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