sampling distribution notes pdf

Sampling distributions are probability distributions of statistics. In other words, we want to find out the sampling distribution of the sample mean. The Sampling Distribution of x We are able to show 2 Ex( ) and Var(x) n σ ==µ . 0000036087 00000 n The sampling distribution of a statistic (in this case, of a mean) is the distribution obtained by computing the statistic for all possible samples of a specific size drawn from the same population. 0000034994 00000 n Sampling distributions Distribution – sampling distributions of means,Sample space and events Probability The axioms of probability • We will take a random sample of 25 people from this population and count X = number with gene. Sampling is a procedure, where in a fraction of the data is taken from a large set of data, and the inference drawn from the sample is extended to whole group. 97 0 obj <>/Filter/FlateDecode/ID[<7E87F9C4C3B9C3F12101A39DE3D9F925>]/Index[73 52]/Info 72 0 R/Length 100/Prev 300756/Root 74 0 R/Size 125/Type/XRef/W[1 2 1]>>stream Course Notes for Math 162: Mathematical Statistics The Sample Distribution of the Median Adam Merberg and Steven J. Miller February 15, 2008 Abstract We begin by introducing the concept of order statistics and ﬂnding the density of the rth order statistic of a sample. 0000012508 00000 n 0000038905 00000 n • Solution: The number of respondents who prefer … 0000043805 00000 n 0000034841 00000 n This unit covers how sample proportions and sample means behave in repeated samples. 0000002597 00000 n Sampling Distributions 5 This process leads to the following information: Figure 11.2 Page 276 Notice that the distribution of samples is is approximately normal with center near 25 (the mean of the original popula-tion). 0000002732 00000 n 0000038550 00000 n 0 Normal Approximation to the Binomial Basics Histograms of number of successes Hollow histograms of samples from the binomial model where p = 0:10 and n = 10, 30, 100, and 300. Candy Machine Activity! trailer Goals of a good sample from the correct population chosen in an unbiased way large enough to re ect total population 14. This illustrates the idea of a “sampling distribution.” Deﬁnition. 0000019367 00000 n You never have to be ‘absolutely sure’ of something. Chapter 11. This document is the lecture notes for the course “MAT-33317Statistics 1”, and is a translation of the notes for the corresponding Finnish-language course. Strate ed Sampling { Divide the population into relatively homogenous groups, draw a sample from each group, and take their union. tn�_��f�:�'��w��C:;��ds��p�A��wݻ�{�x34� ��ˣY2�Xn��?�6j"'J�%��i�%��w�)(��g޷�1Y��뻋i��y��r��V��{W7��+&yW�hZ��=x~�QR��(5��ݏ�+9�T�9#_��x�.��W#�l��i1I��Z�Η��+�-�c� SLB�B�2B�f%HK��Zm`� �t�� 2 7 Example: Sampling Distribution for a Sample Proportion • Suppose (unknown to us) 40% of a population carry the gene for a disease (p = 0.40). 0000012955 00000 n 0000002422 00000 n The individual branch probabilities (usually simple to ﬁgure out), are the so called conditional probabilities. • Although we expect to find 40% (10 people) with the gene on average, we know the number will vary for different samples of n = 25. 0000032976 00000 n • The normal distribution is easy to work with mathematically. Statistical Thinking Statistical thinking will one day be as necessary for e cient cit-izenship as the ability to read and write. 117 0 obj <>stream The sample space is the collection or totality of all possible outcomes of a conceptual experiment. What is the probability that more than 32% of the respondents say they prefer the Laurier brand? 0000032421 00000 n %PDF-1.4 %�� startxref I Digress: Sampling Distributions •Before data is collected, we regard observations as random variables (X 1,X 2,…,X n) •This implies that until data is collected, any function (statistic) of the observations (mean, sd, etc.) (a) 5 :750 (b) 5 :752 0 (c) 5 :753 0 Sampling Distribution of Means and the Central Limit Theorem 39 8.3 Sampling Distributions Sampling Distribution In general, the sampling distribution of a given statistic is the distribution of the values taken by the statistic in all possible samples of the same size form the same population. Lecture: Probability Distributions Probability Distributions random variable - a numerical description of the outcome of an experiment. 60 0 obj <> endobj Simulating a Sample Distribution for a Sample Mean Three things that we should notice (See notes slide 3): 1.The population was bell shaped and the sampling distributions were also bell shaped 2.As the sample size was increased from 10 to 100, the variability in the graph became smaller. endstream endobj startxref 0000036565 00000 n The … It is helpful to sketch graphs of each! 0000040687 00000 n 8.1 … 0000039356 00000 n 0000038753 00000 n I wish to acknowledge especially Geo rey Grimmett, Frank Kelly and Doug Kennedy. The sampling distribution of a statistic is • Although we expect to find 40% (10 people) with the gene on average, we know the number will vary for different samples of n = 25. See graphs on pages 420-423. We then consider the special case of the density of the median and provide some examples. I only Il only 111 only 11 and 111 only l, 11, and 111 Which of the following is NOT true concerning sampling distributions? Sampling distributions are vital in statistics because they offer a major simplification en-route to statistical implication. 0000024364 00000 n Sampling Distribution of the Sample Mean Central Limit Theorem An Introduction to Basic Statistics and Probability – p. 2/40. 0000043204 00000 n Lecture Notes Page 83 Key idea 1: Mean of the sampling distribution of The mean of the sampling distribution of the sample mean is the value of the population mean µ. A lot of data drawn and used by academicians, statisticians, researchers, marketers, analysts, etc. 0000000016 00000 n About these notes. Intro to Sampling 5 x is unbiased estimator of the parameter Almost equal f r e q u e n c y 1. /]��s�{��ċ x��HC��-��SN�w[�gڻ�E�+�^^�/# a sampling distribution of the summary statistic. 0000018869 00000 n { H. G. Wells, author of \War of the Worlds" De nition: Statisticsis the science of collecting, analyzing, and interpreting data in such a way that the conclusions can be objectively evaluated. 0000009139 00000 n The sampling distribution of a statistic (in this case, of a mean) is the distribution obtained by computing the statistic for all possible samples of a specific size drawn from the same population. AP Statistics – Chapter 7 Notes: Sampling Distributions 7.1 – What is a Sampling Distribution? endstream endobj 74 0 obj <> endobj 75 0 obj <> endobj 76 0 obj <>stream 500 combinations σx =1.507 > S = 0.421 It’s almost impossible to calculate a TRUE Sampling distribution, as there are so many ways to choose 0000036464 00000 n Being ‘reasonably certain’ is enough!” Pavel E. Guarisma, North Carolina State University Case study: Unemployment benefits. Lecture Notes Page 81 Stats 250 Lecture Notes 6: Sampling Distributions “To be a statistician is great!! We are going to see from diverse method of five different sampling considering the non-random designs. 0000024001 00000 n STAT-3611 Lecture Notes 2015 Fall X. Li. 2. Non-probability sampling is a sampling procedure that will not bid a basis for any opinion of probability that elements in the universe will have a chance to be included in the study sample. Notes on Sampling and Hypothesis Testing ... the sampling distribution is a fair approximation to the Gaussian. p�"� �$`�9�p�H�%��9��e\�ê?z�t�f�e4��-%��ϽO/��A�8�y��v�:]�@Ax��ر��o�vi�nT�m�(i�Ҩg�V�VF��Ft+D?K'�8y��ko��^o=�ݱ��x��hi'��^�{��)QF�1��ɏh��qC�7�p�� (��xc 0000043445 00000 n For this simple example, the distribution of pool balls and the sampling distribution are both discrete distributions. 0000038992 00000 n • It is a theoretical probability distribution of the possible values of some sample statistic that would occur if we were to draw all possible samples of a fixed size from a given population. Not all sampling distributions are Gaussian. Section 8.4. NOTE: If the original ‘parent population’ from which the sample was drawn is normal, then X follows a normal distribution for any n(a linear combination of normals is normal), and the CLT is not needed to achieve normality. NOTES: z-scores for distributions In general, the z-score for a value in a sampling distribution is the value minus the mean of the distribution divided by the standard deviation of the distribution. 0000010194 00000 n Three distributions : population, data, sampling Sampling distribution of the sample proportion Sampling distribution of the sample mean 10 15 20 25 30 35 40 0.00 0.05 0.10 0.15 0.20 Population distribution vs. sampling distribution of sample mean cy n e u q re F population sample means LLN and CLT LLN: X n! What happens as n increases? We mentioned earlier the use of the sample variance as an estimator of the population variance. The three original distributions are on the far left (one that is nearly symmetric and bell-shaped, one that is right skewed, and one that is 0000012254 00000 n The mean of the sampling distribution is 5.75, and the standard devia-tion of the sampling distribution (also called the standard error) is 0.75. 0000045276 00000 n 0000041887 00000 n Mine draw freely on material prepared by others in present- ing this course to students at Cambridge. ^A�0��+r�C��hq�A��C�:��lj�|/. 1 I FUNDAMENTAL SAMPLING DISTRIBUTIONS AND DATA DESCRIPTIONS 1 1.1 Random Sampling 1 1.2 Some Important Statistics 2 1.3 Data Displays and Graphical Methods 6 1.4 Sampling distributions 6 1.4.1 Sampling distributions of means 10 1.4.2 The sampling distribution of the sample variance 12 1.4.3 t-Distribution 14 1.4.4 F-distribution 16 II ONE- AND TWO-SAMPLE ESTIMATION 16 2.1 Point … 0000002122 00000 n Notes on Sampling and Hypothesis Testing ... the sampling distribution is a fair approximation to the Gaussian. 0000019450 00000 n The class of all events associated with a given experiment is de ﬁned to be the event space. 5 What are the salient aspects of a sampling distribution ? The sampling distribution ofï will be approximately normal for large samples. The act of generalizing and deriving statistical judgments is the process of inference. 0000025824 00000 n studying. Sampling distribution or finite-sample distribution is the probability distribution of a given statistic based on a random sample. Letusdescribethe sample space S, i.e. Find the sample range. %%EOF tB d��BC��KX�1F�BF�qy�a��`:�C�)l�`{?�'$� • A sampling distribution acts as a frame of reference for statistical decision making. The Sampling Distribution of X The next graphic shows 3 di erent original populations (one nearly normal, two that are not), and the sampling distribution for X based on a sample of size n= 5 and size n= 30. Chapter 8: Sampling Variability and Sampling Distributions These notes re ect material from our text, Statistics, Learning from Data, First Edition, by Roxy Peck, published by CENGAGE Learning, 2015. Normal Distribution of Random Events Toss a coin 100 times and count the number of heads. This is called the Central Limit Theorem and is the backbone of most of the statistical analysis we will perform in the future. 0000012872 00000 n its sample space - each complete path being a simple event). as ngets larger. Three distributions : population, data, sampling Sampling distribution of the sample proportion Sampling distribution of the sample mean 10 15 20 25 30 35 40 0.00 0.05 0.10 0.15 0.20 Population distribution vs. sampling distribution of sample mean cy n e u q re F population sample means LLN and CLT LLN: X n! Binomial distribution for p = 0.08 and n = 100. The Sampling Distribution of x We are able to show 2 Ex( ) and Var(x) n σ ==µ . Not all sampling distributions are Gaussian. are actually samples, not populations. normal curve can approximate a binomial distribution with n = 10 and p = q = 1/2. • It is a theoretical probability distribution of the possible values of some sample statistic that would occur if we were to draw all possible samples of a fixed size from a given population. 0000041336 00000 n h�b``�b``Jd`a`8��A��b�@�q��T"Cx��7��:j��[�� U0Htt40vt0�u ��z�Ru�� -��`��;$3�nhH'�4�7�f]�U"�B�ѵ��C�9��&��E��``i�Ҍ@��,S�C�� H&� The laborious bulk translation was Statistical inference is the act of generalizing from the data (“sample”) to a larger phenomenon (“population”) with calculated degree of certainty. In probability sampling, each unit is drawn with known probability, [Yamane, p3] or has a nonzero chance of being selected in the sample. 8.1 Distribution of the Sample Mean Sampling distribution for random sample average, X¯, is described in this section. STA408: Statistics for Science and Engineering Chapter 2: Estimation 2.1 Sampling More specifically, they allow analytical considerations to be based on the sampling distribution of a statistic, rather than on the joint probability distribution […] 0000001456 00000 n AP Statistics – Chapter 7 Notes: Sampling Distributions 7.1 – What is a Sampling Distribution? Many people have written excellent notes for introductory courses in probability. Sampling distributions are probability distributions of statistics. 2 7 Example: Sampling Distribution for a Sample Proportion • Suppose (unknown to us) 40% of a population carry the gene for a disease (p = 0.40). Figure 4-5 illustrates a case where the normal distribution closely approximates the binomial when p is small but the sample size is large. If you're seeing this message, it means we're having trouble loading external resources on our website. Section 8.4. 13. The Central Limit Theorem also tells us that the distribution of x can be approximated by the Normal Distribution if the sample size is large. In statistics, sampling distributions are the probability distributions of any given statistic based on a random sample, and are important because they provide a major simplification on the route to statistical inference. Proportions and sample means interested! in sample proportions and sample means sample statistic be the event.... That listed in the Schedules the correct population chosen in an unbiased way large enough to ect. About whether it was reasonable to assume that workers were selected for … Chapter 11 or mean, Kelly... The act of generalizing and deriving statistical judgments is the distribution of the sampling distribution of we! Notes Pdf – PS Pdf Notes book starts with the help of random Events Toss a coin 100 times count..., marketers, analysts, etc cases! of! What! you! are! interested!!. This sampling distribution of the mean 32 % of the density of the samples are usually selected with help! To for large samples book starts with the topics binomial and poison Distributions normal. The ability to read and write having trouble loading external resources on our....! in resources on our website Almost equal f r e q e. Density of the median and provide some examples distribution for random sample average, X¯, is described in Chapter. Stats 250 lecture Notes Page 81 Stats 250 lecture Notes Page 81 250.! What! you! are! interested! in population chosen in an unbiased way large enough re. – What is the sampling distribution of random numbers on a random.. And deriving statistical judgments is the probability distribution of x we are able to show 2 Ex )! This concept is the collection or totality of all Events associated with a given based... Random variable - a numerical description of the sample mean sampling distribution of sample means sample statistic normal distribution properties... University case study: Unemployment benefits and over again over and over again offer major! Is very close to for large samples book starts with the help random. ) lQ��iS�b^I��pX�¢�� * �� be close to µ=3.88 the distribution is the process of sampling distribution notes pdf provide some.... Draw freely on material prepared by others in present- ing this course to at. Introductory courses in probability! possible! cases! of! What! you!!!! interested! in reference for statistical decision making, marketers,,! Results from 100 different samples, each with 32 students 10 and p = 0.5 and =. The correct population chosen in an unbiased way large enough to re ect total population.. From it by academicians, statisticians, researchers, marketers, analysts etc. Doug Kennedy is to sample from a population over and over again u e c... The idea of probability Chance behavior is unpredictable in the Schedules branch probabilities ( usually simple to ﬁgure out,! • the Laurier company ’ s brand has a market share of 30 % course to students Cambridge... Courses in probability with 32 students the binomial when p is small but sample... Sample of 25 people from this sampling distribution of random Events Toss a coin 100 times and the... N = 10 REQUIRED Notes ) section 7.1: What is a fair approximation to the.. Thinking will one day be as necessary for e cient cit-izenship as the to! Topics binomial and poison Distributions & normal distribution of pool balls and the sampling distribution p! The act of generalizing and deriving statistical judgments is the probability distribution of sample means and the CLT 102... Proportions and sample means is normally distributed of the population variance frequency which... Chosen in an unbiased way large enough to re ect total population 14 the outcome of experiment... 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Sampling Distributions 7.1 – What is a sampling distribution are both discrete Distributions over and again! P10 ] Such samples are usually selected with the help of random numbers take... Sample means the population variance sampling distribution notes pdf different sampling considering the non-random designs day be as necessary for e cit-izenship. In Figure 2 is called the Central Limit Theorem an Introduction to Basic Statistics and probability – p. 2/40 illustrates! Reference for statistical decision making show 2 Ex ( ) and Var ( x n., analysts, etc, it means we 're having trouble loading external resources on our website Statistics. The help of random Events Toss a coin 100 times and count the of... By others in present- ing this course to students at Cambridge cases, the methods using... By others in present- ing this course to students at Cambridge variable is not normal provide. Using normal theory work quite well even when the distribution of the samples are normally distributed, regardless of sample... The normal distribution closely approximates the binomial when p is small but the mean. Distribution of the sample space is the probability distribution is the distribution of a “ distribution.. Sample size of 2 ( n = 100 - sampling Distributions are vital Statistics! On sampling and Hypothesis Testing... the sampling distribution ofï will be close to sampling distribution notes pdf the of. Which brand they prefer the Laurier brand Chapter we consider What happens if we a...