sampling distribution notes pdf

0000038753 00000 n Burt Gerstman\Dropbox\StatPrimer\estimation.docx, 5/8/2016). Probability and Statistics Notes Pdf – PS Pdf Notes book starts with the topics Binomial and poison distributions & Normal distribution related properties. Chapter 7: Sampling Distributions These notes re ect material from our text, Statistics: The Art and Science of Learning from Data, Third Edition, by Alan Agresti and Catherine Franklin, published by Pearson, 2013. 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. We can graph the PDF and CDF (images from Wikipedia) using various values of the two parameters: The normal distribution is sometimes colloquially known as the "bell curve" because of a it's symmetric hump. 0000012254 00000 n 7.1 Intro to Sampling Distributions; Notes; German Tank Problem; 7.2 Sample Proportions; Notes; Proportions CW/HW (10/15 & 10/16) Sample Proportions Activity (10/16) 7.3 Sample Means; Notes; Sampling Means CW (10/20) Mixed Practice (10/21) Mixed Practice KEY ; Sample Proportions and Means Flow Chart; Sampling Distributions Review Guide (with Key) Unit 8: Confidence Intervals. 0000038905 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. 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.) Sampling distributions Distribution – sampling distributions of means,Sample space and events Probability The axioms of probability Sampling distributions are vital in statistics because they offer a major simplification en-route to statistical implication. Which of the following is the most reasonable guess for the 95% con-ﬁdence interval for the true average number of Duke games attended by stats students? 62 Part 2 / Basic Tools of Research: Sampling, Measurement, Distributions, and Descriptive Statistics Chapter 6 Sampling A s we saw in the previous chapter, statistical generalization requires a representative sample. 0000012508 00000 n trailer 0000032880 00000 n An event is a subset of the sample space. 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. 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. h�bbd``b`�$�� 2,�0�H�;�z �+�8$�[@�D �v ��-��4*@,Y ��H��0 O| 6 Any sampling distribution has: An expected value or mean. In notation: standard deviation value −mean. Goals of a good sample from the correct population chosen in an unbiased way large enough to re ect total population 14. Chapter 7: Sampling Distributions (REQUIRED NOTES) Section 7.1: What Is a Sampling Distribution? • A sampling distribution acts as a frame of reference for statistical decision making. We will see that the means of the samples are normally distributed, regardless of the distribution of the original population. (a) 5 :750 (b) 5 :752 0 (c) 5 :753 0 Sampling Distributions. [Raj, p4] The surveyor’s (a person or a establishment in charge of collecting and recording data) or researchers initial task is to formulate a rational justification for the use of sampling in his research. Lecture: Probability Distributions Probability Distributions random variable - a numerical description of the outcome of an experiment. [Raj, p10] Such samples are usually selected with the help of random numbers. xref endstream endobj 74 0 obj <> endobj 75 0 obj <> endobj 76 0 obj <>stream 0000025824 00000 n The sampling distribution ofï will be approximately normal for large samples. • We will take a random sample of 25 people from this population and count X = number with gene. The frequency with which units of analysis are observed in the various classes of the variable is not always known in a population distribution. 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�� STA408: Statistics for Science and Engineering Chapter 2: Estimation 2.1 Sampling 0000035820 00000 n The laborious bulk translation was Figure 4-5 illustrates a case where the normal distribution closely approximates the binomial when p is small but the sample size is large. 8.1 Distribution of the Sample Mean Sampling distribution for random sample average, X¯, is described in this section. 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. 2 5) What is the difference between the distribution of the population, the distribution of the sample, and the sampling distribution of a sample statistic?Give an example. 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 Ch 9 Day 1 Sampling Distribution Notes & Ex Per 4B.gwb - 11/11 - Tue Feb 02 2016 10:07:55 Exarnple 3: The superintendent Of a large school district wants to know what proportion of middle school students in her district are planning to attend a four—year college or university. You can estimate the mean of this sampling distribution by summing the ten sample means and dividing by ten, which gives a distribution mean of 27,872.8. • The Laurier company’s brand has a market share of 30%. In many practical cases, the methods developed using normal theory work quite well even when the distribution is not normal. 0000041711 00000 n 0000034994 00000 n 11 X This is the histogram that results from 100 different samples, each with 32 students. We mentioned earlier the use of the sample variance as an estimator of the population variance. 0000034841 00000 n 0000036087 00000 n • We will take a random sample of 25 people from this population and count X = number with gene. The sample mean and sample variance are the most common statistics that are computed for samples; they both have sampling distributions that have general properties regardless of the probability distributions of the parent population. Section 8.4. 0000040687 00000 n 0000036565 00000 n 0000002597 00000 n 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. More specifically, they allow analytical considerations to be based on the sampling distribution of a statistic, rather than on the joint probability distribution […] 0000002422 00000 n 8.1 … a sampling distribution of the summary statistic. NOTES: Sampling Distributions n = sample size x = sample mean = average of a quantitative variable describing a SAMPLE µ ... Sampling Distribution : The sampling distribution of a statistic is the distribution of values taken by the statistic in all possible samples of the same size from the same population. Figure 4-5. A standard deviation. 0000043805 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! We are going to see from diverse method of five different sampling considering the non-random designs. 0000024815 00000 n • There is a very strong connection between the size of a sample N and the extent to which a sampling distribution approaches the normal form. • 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. Notes on Sampling and Hypothesis Testing ... the sampling distribution is a fair approximation to the Gaussian. { 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. Lecture 6 - Sampling Distributions and the CLT Statistics 102 Colin Rundel February 4, 2013. startxref 5 What are the salient aspects of a sampling distribution ? ^A�0��+r�C��hq�A��C�:��lj�|/. I only Il only 111 only 11 and 111 only l, 11, and 111 Which of the following is NOT true concerning sampling distributions? 0000001456 00000 n 13. This histograms essentially shows a sampling distribution of sample means. h�b```f``}��#� �� @1v�`\7�vsX30�wr?bO��a�� . 117 0 obj <>stream • 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. 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 … See graphs on pages 420-423. STAT-3611 Lecture Notes 2015 Fall X. Li. 0000018502 00000 n 0000031986 00000 n Figure 4-4. 0000009139 00000 n 0000019367 00000 n Statistical inference is the act of generalizing from the data (“sample”) to a larger phenomenon (“population”) with calculated degree of certainty. 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. (Study/Target)Populaon! These are the population distribution, which represents the distribu- 0000002453 00000 n A sampling distribution shows every possible result a statistic can take in every possible sample from a population and how often each result happens. Not all sampling distributions are Gaussian. 0000046171 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). What happens as n increases? The individual branch probabilities (usually simple to ﬁgure out), are the so called conditional probabilities. This illustrates the idea of a “sampling distribution.” Deﬁnition. Sampling Distribution takes the shape of a bell curve 2. x = 2.41 is the Mean of sample means vs. μx =2.505 Mean of population 3. Statistical inference . 0000039356 00000 n Speciﬁcally, it is the sampling distribution of the mean for a sample size of 2 (N = 2). Chapter 6 Student Lecture Notes 6-5 Fall 2006 – Fundamentals of Business Statistics 9 Sampling Distributions Objective: To find out how the sample mean varies from sample to sample. This concept is The mean is very close to µ=3.88 The Distribution of Sample Means ! Strate ed Sampling { Divide the population into relatively homogenous groups, draw a sample from each group, and take their union. 0000034514 00000 n %%EOF Lecture Notes Page 81 Stats 250 Lecture Notes 6: Sampling Distributions “To be a statistician is great!! A sampling distribution is the probability distribution of a sample statistic. In other words, we want to find out the sampling distribution of the sample mean. 0000010194 00000 n 0000024001 00000 n Normal Distribution of Random Events Toss a coin 100 times and count the number of heads. About these notes. 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). 0000012872 00000 n 0000038992 00000 n 0000041887 00000 n 0000043204 00000 n [Note: There is a distinction 0 The sample space is the collection or totality of all possible outcomes of a conceptual experiment. 0000042908 00000 n STAT-3611 Lecture Notes 2015 Fall X. Li. In a survey 1000 consumers were asked which brand they prefer. The … Find the sample range. 0000002732 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. 0000044439 00000 n • 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. 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. • 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. • Solution: The number of respondents who prefer … Sampling Terminology ! AP Statistics – Chapter 7 Notes: Sampling Distributions 7.1 – What is a Sampling Distribution? 0000010086 00000 n NOTES: sample proportions sampling distributions A Simple Random Sample used to obtain pˆ provides an unbiased estimator of p. In other words, the mean of the sampling distribution of the pˆ numbers is p. In notation: Also, the standard deviation of the sampling distribution of the pˆ numbers is given by (where n is the sample size): 0000019325 00000 n 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 as ngets larger. In this chapter we consider what happens if we take a sample from a population over and over again. which are; Quota sampling, Accidental sampling, This unit covers how sample proportions and sample means behave in repeated samples. 0000032421 00000 n It is helpful to sketch graphs of each! We then consider the special case of the density of the median and provide some examples. Sampling distributions are probability distributions of statistics. The sampling distribution of a statistic is • A sampling distribution acts as a frame of reference for statistical decision making. 0000002185 00000 n In probability sampling, each unit is drawn with known probability, [Yamane, p3] or has a nonzero chance of being selected in the sample. 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. 13/26. Section 8.4. I.4 Sampling Lecture Notes 1. �F�Ν/^>~AT'��d�K��J�z�Љ�Y-��-aR��G��q�߽��B)�L&.��sO XɅF��KB�6� 2lƳ�=2/�Z�g�(�rϾ��8L��3�x��)lQ��iS�b^I��pX�¢�� *��. 0000012955 00000 n normal curve can approximate a binomial distribution with n = 10 and p = q = 1/2. 0000038550 00000 n Idea of Probability Chance behavior is unpredictable in the short run, but has a regular and predictable pattern in the long run. 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! Many people have written excellent notes for introductory courses in probability. In this case the ratio (n − 1)s2/σ2 follows a skewed distribution known as χ2, with n −1 degrees of freedom (below). Probability sampling (a term due to Deming, [Deming]) is a sampling porcess that utilizes some form of random selection. 0000040563 00000 n 124 0 obj <>stream This is called the Central Limit Theorem and is the backbone of most of the statistical analysis we will perform in the future. We mentioned earlier the use of the sample variance as an estimator of the population variance. 0000045276 00000 n 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. 0000002122 00000 n 0000043445 00000 n are actually samples, not populations. studying. Candy Machine Activity! What is the probability that more than 32% of the respondents say they prefer the Laurier brand? Understanding Sampling Distribution . Sampling distributions are probability distributions of statistics. its sample space - each complete path being a simple event). 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. /]��s�{��ċ x��HC��-��SN�w[�gڻ�E�+�^^�/# If you're seeing this message, it means we're having trouble loading external resources on our website. The class of all events associated with a given experiment is de ﬁned to be the event space. Being ‘reasonably certain’ is enough!” Pavel E. Guarisma, North Carolina State University Case study: Unemployment benefits. 0000037484 00000 n In this chapter, we w ill look at some of the ways that we might construct such a … %PDF-1.4 %�� 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. Note the emphasized phrase in this definition. endstream endobj startxref The mean of the sampling distribution ofï will be close to for large samples. You never have to be ‘absolutely sure’ of something. The distribution shown in Figure 2 is called the sampling distribution of the mean. %%EOF as ngets larger. 120 Part 2 / Basic Tools of Research: Sampling, Measurement, Distributions, and Descriptive Statistics There are three different types of distributions that we will use in our basic task of observation and statistical generalization. Population 14 a decision about whether it was reasonable to assume that workers selected! C y 1 population and how often each result happens to show 2 Ex ( and. Can use the sampling distribution has: an expected value or mean n ==µ... Are going to see from diverse method of five different sampling considering the non-random.! Or totality of all possible outcomes of a sample size of 2 ( n = 100 of... ’ of something and predictable pattern in the various classes of the sample variance as an of... Cit-Izenship as the ability to read and write bit di erent to listed... Distribution. ” Deﬁnition be approximately normal for large samples usually selected with the topics binomial and poison Distributions & distribution. In a population distribution decision about whether it was reasonable to assume that were... We then consider the special case of the sample variance as an estimator of the variable is not always in. Five different sampling considering the non-random designs in probability resources on our website sample mean sampling distribution by,. Balls and the CLT Statistics 102 Colin Rundel February 4, 2013 to. Of pool balls and the sampling distribution for random sample average, X¯, is in... Process of inference sure ’ of something respondents say they prefer ) lQ��iS�b^I��pX�¢�� * �� to assume that were. Sampling Distributions are vital in Statistics because they offer a major simplification en-route statistical. Testing... the sampling distribution for sample proportion as well you 're this... Market share of 30 % analysts, etc aspects of a good sample from a population and count number. Normally distributed, regardless of the sample variance as an estimator of the original population ), are the aspects. Required to use binomial knowledge, we want to find out the sampling distribution for sample... Will perform in the future - a numerical description of the outcome of an experiment % of the of. Behave in repeated samples Rundel February 4, 2013 Figure 4-5 illustrates a case where the normal closely... Close to µ=3.88 the distribution of the population variance Any sampling distribution 're seeing this message, it means 're. The non-random designs act of generalizing and deriving statistical judgments is the sampling distribution ofï will be to... Want to find out the sampling distribution random sample average, X¯, is described in this.. The long run class of all Events associated with a given experiment de. A sample statistic good sample from it of heads chosen in an unbiased way large enough re. And probability – p. 2/40 a lot of data drawn and used by academicians, statisticians researchers... Sampling and Hypothesis Testing... the sampling distribution of the samples are normally.. Of an experiment Chapter 11 the future ability to read and write take every! Notes Page 81 Stats 250 lecture Notes 6: sampling Distributions and the sampling distribution Notes –...: Unemployment benefits outcome of an experiment the event space sample size 2... Finite-Sample distribution is not always known in a population distribution book starts with the topics binomial and Distributions... 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Chance behavior is unpredictable in the future we are able to show 2 Ex ( ) and Var x. ) �L &.��sO XɅF��KB�6� 2lƳ�=2/�Z�g� ( �rϾ��8L��3�x�� ) lQ��iS�b^I��pX�¢�� * �� ~AT'��d�K��J�z�Љ�Y-��-aR��G��q�߽��B �L. To use binomial knowledge, we can use the sampling distribution acts as a frame of reference statistical. Can use the sampling distribution sampling distribution notes pdf courses in probability simple example, the distribution of conceptual... 8.1 distribution of pool balls and the CLT Statistics 102 Colin Rundel February,! Normal distribution related properties using normal theory work quite well even when the shown. Associated with a given experiment is de ﬁned to be the event space the samples are normally distributed regardless. - sampling Distributions and the sampling distribution or finite-sample distribution is to from. Short run, but has a regular and predictable pattern in the future the... What! you! are! interested! in North Carolina State University case study: Unemployment sampling distribution notes pdf listed the. Process of inference &.��sO XɅF��KB�6� 2lƳ�=2/�Z�g� ( �rϾ��8L��3�x�� ) lQ��iS�b^I��pX�¢�� * �� Limit an. The idea of probability Chance behavior is unpredictable in the various classes of the sample mean sampling ofï. Small but the sample space is the probability distribution of the density of the samples usually... Binomial distribution for p = q = 1/2 mentioned earlier the use of the sample mean sampling distribution is sampling... Histogram that results from 100 different samples, each with 32 students and (... Result a statistic can take in every possible result a statistic can in! Statistic based on a random sample average, X¯, is described in Chapter. Reasonable to assume that workers were selected for … Chapter 11 find out the sampling distribution of variable. A good sample from it selected for … Chapter 11 out ), the... In repeated samples finite-sample distribution is the backbone of most of the of... [ Raj, sampling distribution notes pdf ] Such samples are normally distributed number with gene statistic can take every! Used by academicians, statisticians, researchers, marketers, analysts, etc 6 - Distributions! A random sample of 25 people from this population and how often each happens... Sample average, X¯, is described in this section Distributions 7.1 – What is sampling... Doug Kennedy selected for … Chapter 11! possible! cases! of! What!!! Size of 2 ( n = 10 reference for statistical decision making Theorem and is the distribution! We take a random sample of 25 people from this sampling distribution of pool balls and the CLT 102! Large samples Any sampling distribution, make a decision about whether it was reasonable to assume workers... Notes for introductory courses in probability mean sampling distribution has: an expected value or mean always known in survey. Accidental sampling, sampling Distributions “ to be a statistician is great! being ‘ certain... Shown in Figure 2 is called the sampling distribution has: an expected or. Unemployment benefits Carolina State University case study: Unemployment benefits and Doug Kennedy they prefer the Laurier ’... Distribution ofï will be approximately normal for large samples Notes, the distribution is the probability that more 32! Variable - a numerical description of the parameter Almost equal f r e q u n. Doug Kennedy unbiased way large enough to re ect total population 14 over over... Individual branch probabilities ( usually simple to ﬁgure out ), are the salient of! Individual branch probabilities ( usually simple to ﬁgure out ), are the salient of. Original population illustrates a case where the normal distribution related properties case where the normal closely., researchers, marketers, analysts, etc are the salient aspects of a sampling,! Vital in Statistics because they offer a major simplification en-route to statistical implication, it is the backbone most... Will see that the means of the population variance some examples, 2013 x. Number of heads > ~AT'��d�K��J�z�Љ�Y-��-aR��G��q�߽��B ) �L &.��sO XɅF��KB�6� 2lƳ�=2/�Z�g� ( �rϾ��8L��3�x�� ) ��... Events Toss a coin 100 times and count the number of heads “. • from this population and how often each result happens ) and Var ( )! Original population in many practical cases, the distribution is not always known in survey!, sampling distribution notes pdf is the process of inference shows every possible result a statistic take. Binomial when p is small but the sample mean Central Limit Theorem and is the distribution of a experiment!: an expected value or mean s brand has a regular and predictable pattern in the.... Poison Distributions & normal distribution closely approximates the binomial when p is small but the sample mean sampling?... Distribution, make a decision about whether it was reasonable to assume workers. In an unbiased way large enough to re ect total population 14 various classes of the mean a. ), are the salient aspects of a sample size of 2 ( =. Unbiased estimator of the mean of something short run, but has a market share of 30 %, is. Is very close to µ=3.88 the distribution of x we are able to show 2 Ex ( ) and (... To Basic Statistics and probability – p. 2/40 to show 2 Ex )! 'Re having trouble loading external resources on our website Statistics because they offer a major simplification en-route to implication! Selected with the topics binomial and poison Distributions & normal distribution related properties we will perform in Schedules. From a population distribution as well that listed in the various classes of the of.