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Mathematical Statistics İST 252 EMRE KAÇMAZ B4 / 14.00-17.00 Mathematical Statistics • In probability theory we set up mathematichal models of processes that are affected by ‘chance’. • In mathematical statistics or statistics, we check these models against the observable reality. • This is called statistical inferance. Mathematical Statistics • It is done by sampling, that is , by drowing random samples. • These are sets of values from a much larger set of values that could be studied at, called the population. • An example is 10 diameters of screws drawn from a large lot of screws. • Sampling is done in order to see whether a model of the population is accurate enough for practical purposes. If this is the case, the model can be used for predictions , decisions, and actions , for instance, in planning produciton, buying equipment, investing in a business projects, and so on. Mathematical Statistics • Most important methods of statistical inference are estimation of parameters, determination of confidence intervals, and hypothesis testing, with application to quality control and acceptance sampling. • In the last section we give an introduction to regression and correlation analysis, which concern experiments involving two variables. Random Sampling • Mathematical statistics consists of methods for designing and evaluating random experiments to obtain information about practical problems. • Such as exploring the relation between iron content and density of iron ore, the quality raw material or manufactured products, the efficiency of air conditioning systems, the performance of certain cars, the effect of advertising, the reaction of consumers to a new product, etc. Random Sampling • Random variables occur more frequently in engineering than one would think. • For example, properties of mass-produced articles (screws, lightbulbs,etc.) always Show random variation, due to small differences in raw material or manufacturing processes. • Samples are selected from populations – 20 screws from a lot of 1000, 100, 5000 voters, 8 beavers in a wildlife conservation project – because inspecting the entire population would be too expensive, timeconsuming, impossible or even senseless • To obtain meaningful conclusions, samples must be random selections. Random Sampling • Each of the 1000 screws must have the same chance of being sampled at least approximately. Only then will the sample mean of a sample of size n = 20 be a good approximation of the population mean μ; and the accuracy of the approximation will generally improve with increasing n, as we shall see. Similarly for other parameters (standard deviation, variance, etc.) Random Sampling • Independent sample values will be obtained in experiments with an infinite sample space S, certainly for the normal distribution. • This is also true in sampling with replacement. • It is approximately true in drawing small samples from a large finite population(for instance, 5 or 10 of 1000 items). • However, if we sample without replacement from a small population, the effect of dependence of smaple values may be considerable. Random Sampling • Random numbers help in obtaining samples that are in fact random selections. • This is sometimes not easy to accomplish because there are many subtle factors that can bias sampling (by personal interviews, by poorly working machines, by the choice of nontypical observation conditions, etc.). • Random numbers can be obtained from a random number generator in Maple, Mathemtica, or other systems. Example • To select a sample of size n = 10 from 80 given ball bearings, we number the bearings from 1 to 80. • We then let the generator randomly produce 10 of the integers from 1 to 80 and include the bearings with the numbers obtained in our sample, for example or whatever. • Random numbers are also contained in (older) statistical tables. Example • Representing and processing data were considered in the first chapter. İn connection with frequency distributions. • These are empirical counterparts of probability distributions and helped motivating axioms and properties in properties in probability theory. Example • The new aspect int his chapter is randomness: the data are samples selected randomly from a population. • Accordingly, we can immediately make the connection to first chapter, using stem-and-leaf plots, box plots, and histograms for representing samples graphically. • Also, we now call the mean 𝑥 the sample mean Example • We call n the sample size, the variance s², the sample variance • and its positive square root s the sample standard deviation. • 𝑥, s², and s are called parameters of a sample; they will be needed throughout this chapter.