DOCSLIB.ORG

Chapter 5 Random Sampling Population and Sample Definition: Population a Population Is the Totality of the Elements Under Study

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Chapter 5 Random Sampling Population and Sample Definition: Population a Population Is the Totality of the Elements Under Study Chapter 5 Random Sampling Population and Sample Definition: Population A population is the totality of the elements under study. We are interested in learning something about this population. Examples: number of alligators in Texas, percentage of unemployed workers in cities in the U.S., the stock returns of IBM. A RV X defined over a population is called the population RV. Usually, the population is large, making a complete enumeration of all the values in the population impractical or impossible. Thus, the descriptive statistics describing the population –i.e., the population parameters- will be considered unknown. Population and Sample • Typical situation in statistics: we want to make inferences about an unknown parameter θ using a sample –i.e., a small collection of observations from the general population- { X1,…Xn}. • We summarize the information in the sample with a statistic, which is a function of the sample. That is, any statistic summarizes the data, or reduces the information in the sample to a single number. To make inferences, we use the information in the statistic instead of the entire sample. Population and Sample Definition: Sample The sample is a (manageable) subset of elements of the population. Samples are collected to learn about the population. The process of collecting information from a sample is referred to as sampling. Definition: Random Sample A random sample is a sample where the probability that any individual member from the population being selected as part of the sample is exactly the same as any other individual member of the population In mathematical terms, given a random variable X with distribution F, a random sample of length n is a set of n independent, identically distributed (iid) random variables with distribution F. Sample Statistic • A statistic (singular) is a single measure of some attribute of a sample (for example, its arithmetic mean value). It is calculated by applying a function (statistical algorithm) to the values of the items comprising the sample, which are known together as a set of data. Definition: Statistic A statistic is a function of the observable random variable(s), which does not contain any unknown parameters. Examples: sample mean, sample variance, minimum, (x1+xn)/2, etc. Note: A statistic is distinct from a population parameter. A statistic will be used to estimate a population parameter. In this case, the statistic is called an estimator. Sample Statistics • Sample Statistics are used to estimate population parameters Example: X is an estimate of the population mean, μ. Notation: Population parameters: Greek letters (μ, σ, θ, etc.) ∧ Estimators: A hat over the Greek letter (θ ) • Problems: – Different samples provide different estimates of the population parameter of interest. – Sample results have potential variability, thus sampling error exits Sampling error: The difference between a value (a statistic) computed from a sample and the corresponding value (a parameter) computed from a population. Sample Statistic • The definition of a sample statistic is very general. There are potentially infinite sample statistics. For example, (x1+xn)/2 is by definition a statistic and we could claim that it estimates the population mean of the variable X. However, this is probably not a good estimate. We would like our estimators to have certain desirable properties. • Some simple properties for estimators: ∧ ∧ - An estimator θ is unbiased estimator of θ if E[θ ] =θ - An estimator is most efficient if the variance of the estimator is minimized. - An estimator is BUE, or Best Unbiased Estimate, if it is the estimator with the smallest variance among all unbiased estimates. Sampling Distributions • A sample statistic is a function of RVs. Then, it has a statistical distribution. • In general, in economics and finance, we observe only one sample mean (from our only sample). But, many sample means are possible. • A sampling distribution is a distribution of a statistic over all possible samples. • The sampling distribution shows the relation between the probability of a statistic and the statistic’s value for all possible samples of size n drawn from a population. Example: The sampling distribution of the sample mean Let X1, … , Xn be n mutually independent random variables each having mean µ and standard deviation σ (variance σ2). 11n 1 X=∑ XXin =1 ++ X Let nni=1 n 1111 µ = = ++ =µ ++ µµ = Then X EX EX[ 1 ] EX[ n ] nnnn 22 11 σ 2 = = ++ Also X Var X Var[ X1 ] Var[ X n ] nn 2222 1122 σσ =σσ ++ =n = nn nn2 σ Thus µµ= and σ = XXn Example: The sampling distribution of the sample mean σ Thus µµ= and σ = XXn Hence the distribution of X is centered at µ and becomes more and more compact about µ as n increases. 2 If the Xi’s are normally distributed, then X ~ N(μ,σ /n). Sampling Distributions • Sampling Distribution for the Sample Mean X ~ N(µ, σ2/n) f ( X ) μ X Note: As n→∞, X →μ –i.e., the distribution becomes a spike at μ! Distribution of the sample variance: Preliminaries n 2 ∑()xxi − s2 = i=1 n −1 Use of moment generating functions (Again) 1. Using the moment generating functions of X, Y, Z, …determine the moment generating function of W = h(X, Y, Z, …). 2. Identify the distribution of W from its moment generating function. This procedure works well for sums, linear combinations etc. Theorem (Summation) Let X and Y denote a independent random variables each having a gamma distribution with parameters (λ,α1) and (λ,α2). Then W = X + Y has a gamma distribution with parameters (λ, α1 + α2). Proof: αα λλ12 mXY( t) = and mt( ) = λλ−−tt Therefore mXY+ ( t) = m X( tm) Y( t) α α αα+ λλ1 2 λ12 = = λλ−−tt λ − t Recognizing that this is the moment generating function of the gamma distribution with parameters (λ, α1 + α2) we conclude that W = X + Y has a gamma distribution with parameters (λ, α1 + α2). Theorem (extension to n RV’s) Let x1, x2, … , xn denote n independent random variables each having a gamma distribution with parameters (αi, λ ), i = 1, 2, …, n. Then W = x1 + x2 + … + xn has a gamma distribution with parameters (α1 + α2 +…+ αn, λ). Proof: α λ i mtx ( ) = i=1,2..., n i λ − t m( t) = m( tm) ( t)... m( t) xx12+ ++... xnn x1 x 2 x α α α αα+ ++... α λλ1 2 λnn λ12 = ... = λλ−−tt λ − t λ − t This is the moment generating function of the gamma distribution with parameters (α1 + α2 +…+ αn, λ). we conclude that W = x1 + x2 + … + xn has a gamma distribution with parameters (α1 + α2 +…+ αn, λ). Theorem (Scaling) Suppose that X is a random variable having a gamma distribution with parameters (α, λ). Then W = ax has a gamma distribution with parameters (α, λ/a) Proof: α λ mtx ( ) = λ − t α α λ λ a then max ( t) = mx ( at) = = λ − at λ − t a Special Cases 1. Let X and Y be independent random variables having a χ2 distribution with n1 and n2 degrees of freedom, respectively. Then, 2 X + Y has a χ distribution with degrees of freedom n1 + n2. 2 2 Notation: X + Y ∼ χ(n1+n2) (a χ distribution with df= n1 + n2.) 2 2. Let x1, x2,…, xn, be independent RVs having a χ distribution with n1 , n2 ,…, nN degrees of freedom, respectively. 2 Then, x1+ x2 +…+ xn ∼ χ(n1+n2+...+nN) Note: Both of these properties follow from the fact that a χ2 RV with n degrees of freedom is a Gamma RV with λ = ½ and α = n/2. 2 Two useful χn results 1. Let z ~ N(0,1). Then, 2 2 z ∼ χ1 . 2. Let z1, z2,…, zν be independent random variables each following a N(0,1) distribution. Then, 22 2 2 Uz=12 + z ++... zν ∼ χν Theorem Suppose that U1 and U2 are independent random variables and that U = 2 U1 + U2. Suppose that U1 and U have a χ distribution with degrees of freedom ν1and ν, respectively. (ν1 < ν). Then, 2 U2 ~ χν2 , where ν2 = ν - ν1. Proof: v1 v 2 1 2 1 Now mt( ) = 2 and mt( ) = 2 U1 1 U 1 2 − t 2 − t Also m( t) = m( tm) ( t) U UU12 mt( ) Hence mt( ) = U U2 mt( ) U1 v 1 2 2 v − v1 22 1 1 2 − t 2 = v1 = 2 1 1 2 − t 2 1 2 − t Q.E.D. Distribution of the sample variance n 2 ∑()xxi − s2 = i=1 n −1 Properties of the sample variance We show that we can decompose the numerator of s2 as: nn 2 22 ∑∑(xii−= x ) ( x −− a )( nx − a ) ii=11= Proof: nn 22 ∑∑()xii− x = ( x −+− aax ) ii=11= n = −22 − − −+− ∑ (xii a )2()()() x axa xa i=1 nn 22 =∑∑(xii − a )2()()() − xa − x −+ a nxa − ii=11= n 2 22 =∑(xi −− a )2()() nx −+ a nx − a i=1 Properties of the sample variance n 2 22 =∑(xi −− a )2()() nx −+ a nx − a i=1 nn 2 22 ∑∑(xii−= x ) ( x −− a )( nx − a ) ii=11= Properties of the sample variance nn 2 22 ∑∑(xii−= x ) ( x −− a )( nx − a ) ii=11= Special Cases 1. Setting a = 0. (It delivers the computing formula.) 2 n nnn∑ xi 2222i=1 ∑∑∑()xii−= x x −= nx x i − iii=111= = n 2. Setting a = µ. nn 2 22 ∑∑(xii−= x ) ( x −−−µµ )( nx ) ii=11= nn 2 22 or ∑∑(xii−µµ ) = ( x −+− x ) nx ( ) ii=11= Distribution of the s2 of a normal variate Let x1, x2, …, xn denote a random sample from the normal distribution µ σ2 n with mean and variance . Then, 2 ∑()xxi − 2 2 (n-1)s2/σ2 ∼ χ , where s = i=1 n-1 n −1 Proof: x − µ x − µ Let zz= 1 ,, = n 1 σσn n 2 ∑( xi − µ ) 22i=1 2 Then zz++= ∼ χn 1 n σ 2 Distribution of the s2 of a normal variate Now, recall Special Case 2: nn ()x−−µ 22 () xx ∑∑iinx()− µ 2 ii=11= = + σ2 σσ 22 or U = U2 + U1, where n 2 ∑()xi − µ = 2 U = i 1 ~ χ σ 2 n 2 We also know that x follows a N(µ, σ /n) Thus, 2 x − µ nx( − µ ) = 2 2 z ~ N(0,1) => Uz1 = = ~ χ .
Load more

Recommended publications
  • Sampling Methods It’S Impractical to Poll an Entire Population—Say, All 145 Million Registered Voters in the United States
    Sampling Methods It’s impractical to poll an entire population—say, all 145 million registered voters in the United States. That is why pollsters select a sample of individuals that represents the whole population. Understanding how respondents come to be selected to be in a poll is a big step toward determining how well their views and opinions mirror those of the voting population. To sample individuals, polling organizations can choose from a wide variety of options. Pollsters generally divide them into two types: those that are based on probability sampling methods and those based on non-probability sampling techniques. For more than five decades probability sampling was the standard method for polls. But in recent years, as fewer people respond to polls and the costs of polls have gone up, researchers have turned to non-probability based sampling methods. For example, they may collect data on-line from volunteers who have joined an Internet panel. In a number of instances, these non-probability samples have produced results that were comparable or, in some cases, more accurate in predicting election outcomes than probability-based surveys. Now, more than ever, journalists and the public need to understand the strengths and weaknesses of both sampling techniques to effectively evaluate the quality of a survey, particularly election polls. Probability and Non-probability Samples In a probability sample, all persons in the target population have a change of being selected for the survey sample and we know what that chance is. For example, in a telephone survey based on random digit dialing (RDD) sampling, researchers know the chance or probability that a particular telephone number will be selected.
    [Show full text]
  • Chapter 8 Fundamental Sampling Distributions And
    CHAPTER 8 FUNDAMENTAL SAMPLING DISTRIBUTIONS AND DATA DESCRIPTIONS 8.1 Random Sampling pling procedure, it is desirable to choose a random sample in the sense that the observations are made The basic idea of the statistical inference is that we independently and at random. are allowed to draw inferences or conclusions about a Random Sample population based on the statistics computed from the sample data so that we could infer something about Let X1;X2;:::;Xn be n independent random variables, the parameters and obtain more information about the each having the same probability distribution f (x). population. Thus we must make sure that the samples Define X1;X2;:::;Xn to be a random sample of size must be good representatives of the population and n from the population f (x) and write its joint proba- pay attention on the sampling bias and variability to bility distribution as ensure the validity of statistical inference. f (x1;x2;:::;xn) = f (x1) f (x2) f (xn): ··· 8.2 Some Important Statistics It is important to measure the center and the variabil- ity of the population. For the purpose of the inference, we study the following measures regarding to the cen- ter and the variability. 8.2.1 Location Measures of a Sample The most commonly used statistics for measuring the center of a set of data, arranged in order of mag- nitude, are the sample mean, sample median, and sample mode. Let X1;X2;:::;Xn represent n random variables. Sample Mean To calculate the average, or mean, add all values, then Bias divide by the number of individuals.
    [Show full text]
  • MRS Guidance on How to Read Opinion Polls
    What are opinion polls? MRS guidance on how to read opinion polls June 2016 1 June 2016 www.mrs.org.uk MRS Guidance Note: How to read opinion polls MRS has produced this Guidance Note to help individuals evaluate, understand and interpret Opinion Polls. This guidance is primarily for non-researchers who commission and/or use opinion polls. Researchers can use this guidance to support their understanding of the reporting rules contained within the MRS Code of Conduct. Opinion Polls – The Essential Points What is an Opinion Poll? An opinion poll is a survey of public opinion obtained by questioning a representative sample of individuals selected from a clearly defined target audience or population. For example, it may be a survey of c. 1,000 UK adults aged 16 years and over. When conducted appropriately, opinion polls can add value to the national debate on topics of interest, including voting intentions. Typically, individuals or organisations commission a research organisation to undertake an opinion poll. The results to an opinion poll are either carried out for private use or for publication. What is sampling? Opinion polls are carried out among a sub-set of a given target audience or population and this sub-set is called a sample. Whilst the number included in a sample may differ, opinion poll samples are typically between c. 1,000 and 2,000 participants. When a sample is selected from a given target audience or population, the possibility of a sampling error is introduced. This is because the demographic profile of the sub-sample selected may not be identical to the profile of the target audience / population.
    [Show full text]
  • Permutation Tests
    Permutation tests Ken Rice Thomas Lumley UW Biostatistics Seattle, June 2008 Overview • Permutation tests • A mean • Smallest p-value across multiple models • Cautionary notes Testing In testing a null hypothesis we need a test statistic that will have different values under the null hypothesis and the alternatives we care about (eg a relative risk of diabetes) We then need to compute the sampling distribution of the test statistic when the null hypothesis is true. For some test statistics and some null hypotheses this can be done analytically. The p- value for the is the probability that the test statistic would be at least as extreme as we observed, if the null hypothesis is true. A permutation test gives a simple way to compute the sampling distribution for any test statistic, under the strong null hypothesis that a set of genetic variants has absolutely no effect on the outcome. Permutations To estimate the sampling distribution of the test statistic we need many samples generated under the strong null hypothesis. If the null hypothesis is true, changing the exposure would have no effect on the outcome. By randomly shuffling the exposures we can make up as many data sets as we like. If the null hypothesis is true the shuffled data sets should look like the real data, otherwise they should look different from the real data. The ranking of the real test statistic among the shuffled test statistics gives a p-value Example: null is true Data Shuffling outcomes Shuffling outcomes (ordered) gender outcome gender outcome gender outcome Example: null is false Data Shuffling outcomes Shuffling outcomes (ordered) gender outcome gender outcome gender outcome Means Our first example is a difference in mean outcome in a dominant model for a single SNP ## make up some `true' data carrier<-rep(c(0,1), c(100,200)) null.y<-rnorm(300) alt.y<-rnorm(300, mean=carrier/2) In this case we know from theory the distribution of a difference in means and we could just do a t-test.
    [Show full text]
  • Sampling Distribution of the Variance
    Proceedings of the 2009 Winter Simulation Conference M. D. Rossetti, R. R. Hill, B. Johansson, A. Dunkin, and R. G. Ingalls, eds. SAMPLING DISTRIBUTION OF THE VARIANCE Pierre L. Douillet Univ Lille Nord de France, F-59000 Lille, France ENSAIT, GEMTEX, F-59100 Roubaix, France ABSTRACT Without confidence intervals, any simulation is worthless. These intervals are quite ever obtained from the so called "sampling variance". In this paper, some well-known results concerning the sampling distribution of the variance are recalled and completed by simulations and new results. The conclusion is that, except from normally distributed populations, this distribution is more difficult to catch than ordinary stated in application papers. 1 INTRODUCTION Modeling is translating reality into formulas, thereafter acting on the formulas and finally translating the results back to reality. Obviously, the model has to be tractable in order to be useful. But too often, the extra hypotheses that are assumed to ensure tractability are held as rock-solid properties of the real world. It must be recalled that "everyday life" is not only made with "every day events" : rare events are rarely occurring, but they do. For example, modeling a bell shaped histogram of experimental frequencies by a Gaussian pdf (probability density function) or a Fisher’s pdf with four parameters is usual. Thereafter transforming this pdf into a mgf (moment generating function) by mgf (z)=Et (expzt) is a powerful tool to obtain (and prove) the properties of the modeling pdf . But this doesn’t imply that a specific moment (e.g. 4) is effectively an accessible experimental reality.
    [Show full text]
  • Arxiv:1804.01620V1 [Stat.ML]
    ACTIVE COVARIANCE ESTIMATION BY RANDOM SUB-SAMPLING OF VARIABLES Eduardo Pavez and Antonio Ortega University of Southern California ABSTRACT missing data case, as well as for designing active learning algorithms as we will discuss in more detail in Section 5. We study covariance matrix estimation for the case of partially ob- In this work we analyze an unbiased covariance matrix estima- served random vectors, where different samples contain different tor under sub-Gaussian assumptions on x. Our main result is an subsets of vector coordinates. Each observation is the product of error bound on the Frobenius norm that reveals the relation between the variable of interest with a 0 1 Bernoulli random variable. We − number of observations, sub-sampling probabilities and entries of analyze an unbiased covariance estimator under this model, and de- the true covariance matrix. We apply this error bound to the design rive an error bound that reveals relations between the sub-sampling of sub-sampling probabilities in an active covariance estimation sce- probabilities and the entries of the covariance matrix. We apply our nario. An interesting conclusion from this work is that when the analysis in an active learning framework, where the expected number covariance matrix is approximately low rank, an active covariance of observed variables is small compared to the dimension of the vec- estimation approach can perform almost as well as an estimator with tor of interest, and propose a design of optimal sub-sampling proba- complete observations. The paper is organized as follows, in Section bilities and an active covariance matrix estimation algorithm.
    [Show full text]
  • Examination of Residuals
    EXAMINATION OF RESIDUALS F. J. ANSCOMBE PRINCETON UNIVERSITY AND THE UNIVERSITY OF CHICAGO 1. Introduction 1.1. Suppose that n given observations, yi, Y2, * , y., are claimed to be inde- pendent determinations, having equal weight, of means pA, A2, * *X, n, such that (1) Ai= E ai,Or, where A = (air) is a matrix of given coefficients and (Or) is a vector of unknown parameters. In this paper the suffix i (and later the suffixes j, k, 1) will always run over the values 1, 2, * , n, and the suffix r will run from 1 up to the num- ber of parameters (t1r). Let (#r) denote estimates of (Or) obtained by the method of least squares, let (Yi) denote the fitted values, (2) Y= Eai, and let (zt) denote the residuals, (3) Zi =Yi - Yi. If A stands for the linear space spanned by (ail), (a,2), *-- , that is, by the columns of A, and if X is the complement of A, consisting of all n-component vectors orthogonal to A, then (Yi) is the projection of (yt) on A and (zi) is the projection of (yi) on Z. Let Q = (qij) be the idempotent positive-semidefinite symmetric matrix taking (y1) into (zi), that is, (4) Zi= qtj,yj. If A has dimension n - v (where v > 0), X is of dimension v and Q has rank v. Given A, we can choose a parameter set (0,), where r = 1, 2, * , n -v, such that the columns of A are linearly independent, and then if V-1 = A'A and if I stands for the n X n identity matrix (6ti), we have (5) Q =I-AVA'.
    [Show full text]
  • Lecture 14 Testing for Kurtosis
    9/8/2016 CHE384, From Data to Decisions: Measurement, Kurtosis Uncertainty, Analysis, and Modeling • For any distribution, the kurtosis (sometimes Lecture 14 called the excess kurtosis) is defined as Testing for Kurtosis 3 (old notation = ) • For a unimodal, symmetric distribution, Chris A. Mack – a positive kurtosis means “heavy tails” and a more Adjunct Associate Professor peaked center compared to a normal distribution – a negative kurtosis means “light tails” and a more spread center compared to a normal distribution http://www.lithoguru.com/scientist/statistics/ © Chris Mack, 2016Data to Decisions 1 © Chris Mack, 2016Data to Decisions 2 Kurtosis Examples One Impact of Excess Kurtosis • For the Student’s t • For a normal distribution, the sample distribution, the variance will have an expected value of s2, excess kurtosis is and a variance of 6 2 4 1 for DF > 4 ( for DF ≤ 4 the kurtosis is infinite) • For a distribution with excess kurtosis • For a uniform 2 1 1 distribution, 1 2 © Chris Mack, 2016Data to Decisions 3 © Chris Mack, 2016Data to Decisions 4 Sample Kurtosis Sample Kurtosis • For a sample of size n, the sample kurtosis is • An unbiased estimator of the sample excess 1 kurtosis is ∑ ̅ 1 3 3 1 6 1 2 3 ∑ ̅ Standard Error: • For large n, the sampling distribution of 1 24 2 1 approaches Normal with mean 0 and variance 2 1 of 24/n 3 5 • For small samples, this estimator is biased D. N. Joanes and C. A. Gill, “Comparing Measures of Sample Skewness and Kurtosis”, The Statistician, 47(1),183–189 (1998).
    [Show full text]
  • Categorical Data Analysis
    Categorical Data Analysis Related topics/headings: Categorical data analysis; or, Nonparametric statistics; or, chi-square tests for the analysis of categorical data. OVERVIEW For our hypothesis testing so far, we have been using parametric statistical methods. Parametric methods (1) assume some knowledge about the characteristics of the parent population (e.g. normality) (2) require measurement equivalent to at least an interval scale (calculating a mean or a variance makes no sense otherwise). Frequently, however, there are research problems in which one wants to make direct inferences about two or more distributions, either by asking if a population distribution has some particular specifiable form, or by asking if two or more population distributions are identical. These questions occur most often when variables are qualitative in nature, making it impossible to carry out the usual inferences in terms of means or variances. For such problems, we use nonparametric methods. Nonparametric methods (1) do not depend on any assumptions about the parameters of the parent population (2) generally assume data are only measured at the nominal or ordinal level. There are two common types of hypothesis-testing problems that are addressed with nonparametric methods: (1) How well does a sample distribution correspond with a hypothetical population distribution? As you might guess, the best evidence one has about a population distribution is the sample distribution. The greater the discrepancy between the sample and theoretical distributions, the more we question the “goodness” of the theory. EX: Suppose we wanted to see whether the distribution of educational achievement had changed over the last 25 years. We might take as our null hypothesis that the distribution of educational achievement had not changed, and see how well our modern-day sample supported that theory.
    [Show full text]
  • Speech by Honorary Degree Recipient
    Speech by Honorary Degree Recipient Dear Colleagues and Friends, Ladies and Gentlemen: Today, I am so honored to present in this prestigious stage to receive the Honorary Doctorate of the Saint Petersburg State University. Since my childhood, I have known that Saint Petersburg University is a world-class university associated by many famous scientists, such as Ivan Pavlov, Dmitri Mendeleev, Mikhail Lomonosov, Lev Landau, Alexander Popov, to name just a few. In particular, many dedicated their glorious lives in the same field of scientific research and studies which I have been devoting to: Leonhard Euler, Andrey Markov, Pafnuty Chebyshev, Aleksandr Lyapunov, and recently Grigori Perelman, not to mention many others in different fields such as political sciences, literature, history, economics, arts, and so on. Being an Honorary Doctorate of the Saint Petersburg State University, I have become a member of the University, of which I am extremely proud. I have been to the beautiful and historical city of Saint Petersburg five times since 1997, to work with my respected Russian scientists and engineers in organizing international academic conferences and conducting joint scientific research. I sincerely appreciate the recognition of the Saint Petersburg State University for my scientific contributions and endeavors to developing scientific cooperations between Russia and the People’s Republic of China. I would like to take this opportunity to thank the University for the honor, and thank all professors, staff members and students for their support and encouragement. Being an Honorary Doctorate of the Saint Petersburg State University, I have become a member of the University, which made me anxious to contribute more to the University and to the already well-established relationship between Russia and China in the future.
    [Show full text]
  • Richard Von Mises's Philosophy of Probability and Mathematics
    “A terrible piece of bad metaphysics”? Towards a history of abstraction in nineteenth- and early twentieth-century probability theory, mathematics and logic Lukas M. Verburgt If the true is what is grounded, then the ground is neither true nor false LUDWIG WITTGENSTEIN Whether all grow black, or all grow bright, or all remain grey, it is grey we need, to begin with, because of what it is, and of what it can do, made of bright and black, able to shed the former , or the latter, and be the latter or the former alone. But perhaps I am the prey, on the subject of grey, in the grey, to delusions SAMUEL BECKETT “A terrible piece of bad metaphysics”? Towards a history of abstraction in nineteenth- and early twentieth-century probability theory, mathematics and logic ACADEMISCH PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam op gezag van de Rector Magnificus prof. dr. D.C. van den Boom ten overstaan van een door het College voor Promoties ingestelde commissie in het openbaar te verdedigen in de Agnietenkapel op donderdag 1 oktober 2015, te 10:00 uur door Lukas Mauve Verburgt geboren te Amersfoort Promotiecommissie Promotor: Prof. dr. ir. G.H. de Vries Universiteit van Amsterdam Overige leden: Prof. dr. M. Fisch Universitat Tel Aviv Dr. C.L. Kwa Universiteit van Amsterdam Dr. F. Russo Universiteit van Amsterdam Prof. dr. M.J.B. Stokhof Universiteit van Amsterdam Prof. dr. A. Vogt Humboldt-Universität zu Berlin Faculteit der Geesteswetenschappen © 2015 Lukas M. Verburgt Graphic design Aad van Dommelen (Witvorm)
    [Show full text]
  • Lecture 8: Sampling Methods
    Lecture 8: Sampling Methods Donglei Du ([email protected]) Faculty of Business Administration, University of New Brunswick, NB Canada Fredericton E3B 9Y2 Donglei Du (UNB) ADM 2623: Business Statistics 1 / 30 Table of contents 1 Sampling Methods Why Sampling Probability vs non-probability sampling methods Sampling with replacement vs without replacement Random Sampling Methods 2 Simple random sampling with and without replacement Simple random sampling without replacement Simple random sampling with replacement 3 Sampling error vs non-sampling error 4 Sampling distribution of sample statistic Histogram of the sample mean under SRR 5 Distribution of the sample mean under SRR: The central limit theorem Donglei Du (UNB) ADM 2623: Business Statistics 2 / 30 Layout 1 Sampling Methods Why Sampling Probability vs non-probability sampling methods Sampling with replacement vs without replacement Random Sampling Methods 2 Simple random sampling with and without replacement Simple random sampling without replacement Simple random sampling with replacement 3 Sampling error vs non-sampling error 4 Sampling distribution of sample statistic Histogram of the sample mean under SRR 5 Distribution of the sample mean under SRR: The central limit theorem Donglei Du (UNB) ADM 2623: Business Statistics 3 / 30 Why sampling? The physical impossibility of checking all items in the population, and, also, it would be too time-consuming The studying of all the items in a population would not be cost effective The sample results are usually adequate The destructive nature of certain tests Donglei Du (UNB) ADM 2623: Business Statistics 4 / 30 Sampling Methods Probability Sampling: Each data unit in the population has a known likelihood of being included in the sample.
    [Show full text]