Lecture 5: Sampling Distributions
|
|
- Alexia Black
- 5 years ago
- Views:
Transcription
1 Lecture 5: Sampling Distributions Taeyong Park Washington University in St. Louis February 15, 2017 Park (Wash U.) U25 PS323 Intro to Quantitative Methods February 15, / 23
2 Today... Review of normal distributions and the standard normal distribution. Sampling distribution. Lab: Review the online assignment; generating random numbers; normal distribution; central limit theorem. Problem set 1 will be assigned after class. Covers lecture 1 - lecture 5. Due the beginning of next class. A hard copy for the first part and upload.r file to Blackborad for the second part. Weekly online assignment will also be assigend. Park (Wash U.) U25 PS323 Intro to Quantitative Methods February 15, / 23
3 Normal Distribution For a normal distribution with µ = 45 and σ = 5, find the probability that an observation falls: Above the value of 35 Below the value of 40 Between the values of 45 and 55 Between the values of 30 and 40 Park (Wash U.) U25 PS323 Intro to Quantitative Methods February 15, / 23
4 Normal Distribution - Z-score z-score How many standard deviations is my value away from the mean? z = y µ σ Example: z = Example: z = = = 2 2 Park (Wash U.) U25 PS323 Intro to Quantitative Methods February 15, / 23
5 The Standard Normal Distribution and Z-score The standard normal distribution The normal distribution with mean µ = 0 and standard deviation σ = 1. z-score and the standard normal distribution If a variable has a normal distribution, and if its values are converted to z-scores by subtracting the mean and dividing by the standard deviation, then the z-scores have the standard normal distribution. Park (Wash U.) U25 PS323 Intro to Quantitative Methods February 15, / 23
6 z-table Park (Wash U.) U25 PS323 Intro to Quantitative Methods February 15, / 23
7 Learning goals Sampling Distribution Central Limit Theorem Park (Wash U.) U25 PS323 Intro to Quantitative Methods February 15, / 23
8 Probability: Why do we care? Park (Wash U.) U25 PS323 Intro to Quantitative Methods February 15, / 23
9 Makinga prediction using a sample data Population sample?: Straightforward We know that 50% of the entire Californian voters (7 mil.) vote for Republican. What will be the proportion of voting for Republican in the sample of 2,705? Sample population? We don t know about the entire Californian voters. What will be the proportion of voting for Republican among the entire Californian voters given a result from the sample of 2,705? Park (Wash U.) U25 PS323 Intro to Quantitative Methods February 15, / 23
10 Makinga prediction using a sample data Sample population? We don t know about the entire Californian voters. What will be the proportion of voting for Republican among the entire Californian voters given a result from the sample of 2,705 (56.5%)? Instead, Suppose only half the population voted for S. Would it then be surprising that 56.5% of the sampled individuals voted for him? If very unlikely, we can infer that S will win. What if we suppose only 40% voted for S? Park (Wash U.) U25 PS323 Intro to Quantitative Methods February 15, / 23
11 Three types of distributions Population distribution Sample data distribution Sampling distribution Park (Wash U.) U25 PS323 Intro to Quantitative Methods February 15, / 23
12 Population distribution: Example The distribution from which we select the sample. Park (Wash U.) U25 PS323 Intro to Quantitative Methods February 15, / 23
13 Sample data distribution: Example The distrubution of data we actually observe. Park (Wash U.) U25 PS323 Intro to Quantitative Methods February 15, / 23
14 Sampling distribution Sampling Distributions A sampling distribution is the distribution of a statistic given repeated sampling. Repeated sampling probabilities for the possible values the statistic can take. Example: a sampling distribution of a sample mean. A sampling distribution specifies probabilities not for individual observations but for possible values of a statistic computed from the observations. Park (Wash U.) U25 PS323 Intro to Quantitative Methods February 15, / 23
15 Sampling distribution: Example An example of a sampling distribution? Population: American voters Several surveys - several samples Statistic: proportion of respondents that voted for Obama Park (Wash U.) U25 PS323 Intro to Quantitative Methods February 15, / 23
16 Sampling distribution: Example Density Percentage of Obama Voters Park (Wash U.) U25 PS323 Intro to Quantitative Methods February 15, / 23
17 Sampling distribution: What is the point? In most cases we do not have several samples. However... The form of sampling distributions if often known theoretically. We can then derive a distribution of the sample statistics for one sample of the given size n. This allows us to make inferences about population parameters. The sample mean is the most frequently used statistic. We derive the sampling distribution of the sample mean to make inferences, assuming repeated sampling. Park (Wash U.) U25 PS323 Intro to Quantitative Methods February 15, / 23
18 Sampling distribution of the sample mean The sample mean of sample data y = {y 1,y 2,...,y n }: y. Assuming repeated sampling: Mean of y y1 = {y11,y 12,...,y 1n } y2 = {y21,y 22,...,y 2n }... yk = {yk1,y k2,...,y kn } The mean of sampling distribution of y equals the population mean given repeated sampling. Standard error of y The standard deviation of sampling distribution of y, denoted by σ y. σ y describes how y varies from sample to sample. Park (Wash U.) U25 PS323 Intro to Quantitative Methods February 15, / 23
19 Sampling distribution of the sample mean In practice, we don t need to take samples repeatedly to find σ y. Instead, use the following formular: σ y = σ n, where σ is the population standard deviation and n is the sample size. Suppose a population having σ = 10 and a sample size of 100. σ y = = 1. Individual observations tend to vary much more than sample means vary from sample to sample. Park (Wash U.) U25 PS323 Intro to Quantitative Methods February 15, / 23
20 Central limit theorem Central limit theorem For random sampling with a large sample size n, the sampling distribution of the sample mean y is approximately normal. The mean of the distribution is equal to population mean µ. The standard deviation of the distribution is equal to σ n. ) y N (µ, σ n Park (Wash U.) U25 PS323 Intro to Quantitative Methods February 15, / 23
21 Central limit theorem: Some notes The approximate normality of the sampling distribution applies no matter what the shape of the population distribution. Remarkable! Even if the population distribution is U-shaped, highly discrete, or highly skewed. Park (Wash U.) U25 PS323 Intro to Quantitative Methods February 15, / 23
22 It works for EVERYTHING Park (Wash U.) U25 PS323 Intro to Quantitative Methods February 15, / 23
23 Central limit theorem: Some notes If Y is normal, the CLT applies for all n. Otherwise, you need a large enough sample. Usually n=30 is good enough, but it will depend on the distribution. As n, the standard error is going to get smaller and smaller. Knowing that the sampling distribution of y is approximately normal helps us find probabiliteis for possible values of y. For instance, y almost certainly falls within 3σ y = 3σ n of µ. Park (Wash U.) U25 PS323 Intro to Quantitative Methods February 15, / 23
Lecture 6: Confidence Intervals
Lecture 6: Confidence Intervals Taeyong Park Washington University in St. Louis February 22, 2017 Park (Wash U.) U25 PS323 Intro to Quantitative Methods February 22, 2017 1 / 29 Today... Review of sampling
More informationChapter 9: Sampling Distributions
Chapter 9: Sampling Distributions 9. Introduction This chapter connects the material in Chapters 4 through 8 (numerical descriptive statistics, sampling, and probability distributions, in particular) with
More informationSAMPLING DISTRIBUTIONS. Chapter 7
SAMPLING DISTRIBUTIONS Chapter 7 7.1 How Likely Are the Possible Values of a Statistic? The Sampling Distribution Statistic and Parameter Statistic numerical summary of sample data: p-hat or xbar Parameter
More informationCentral Limit Theorem
Central Limit Theorem Lots of Samples 1 Homework Read Sec 6-5. Discussion Question pg 329 Do Ex 6-5 8-15 2 Objective Use the Central Limit Theorem to solve problems involving sample means 3 Sample Means
More informationChapter 7. Sampling Distributions and the Central Limit Theorem
Chapter 7. Sampling Distributions and the Central Limit Theorem 1 Introduction 2 Sampling Distributions related to the normal distribution 3 The central limit theorem 4 The normal approximation to binomial
More informationChapter 7: Point Estimation and Sampling Distributions
Chapter 7: Point Estimation and Sampling Distributions Seungchul Baek Department of Statistics, University of South Carolina STAT 509: Statistics for Engineers 1 / 20 Motivation In chapter 3, we learned
More informationChapter 7 Sampling Distributions and Point Estimation of Parameters
Chapter 7 Sampling Distributions and Point Estimation of Parameters Part 1: Sampling Distributions, the Central Limit Theorem, Point Estimation & Estimators Sections 7-1 to 7-2 1 / 25 Statistical Inferences
More informationDistribution of the Sample Mean
Distribution of the Sample Mean MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2018 Experiment (1 of 3) Suppose we have the following population : 4 8 1 2 3 4 9 1
More informationMaking Sense of Cents
Name: Date: Making Sense of Cents Exploring the Central Limit Theorem Many of the variables that you have studied so far in this class have had a normal distribution. You have used a table of the normal
More information1. Variability in estimates and CLT
Unit3: Foundationsforinference 1. Variability in estimates and CLT Sta 101 - Fall 2015 Duke University, Department of Statistical Science Dr. Çetinkaya-Rundel Slides posted at http://bit.ly/sta101_f15
More informationMath 140 Introductory Statistics
Math 140 Introductory Statistics Let s make our own sampling! If we use a random sample (a survey) or if we randomly assign treatments to subjects (an experiment) we can come up with proper, unbiased conclusions
More informationIntroduction to Statistics I
Introduction to Statistics I Keio University, Faculty of Economics Continuous random variables Simon Clinet (Keio University) Intro to Stats November 1, 2018 1 / 18 Definition (Continuous random variable)
More information1 Sampling Distributions
1 Sampling Distributions 1.1 Statistics and Sampling Distributions When a random sample is selected the numerical descriptive measures calculated from such a sample are called statistics. These statistics
More informationChapter 7. Sampling Distributions and the Central Limit Theorem
Chapter 7. Sampling Distributions and the Central Limit Theorem 1 Introduction 2 Sampling Distributions related to the normal distribution 3 The central limit theorem 4 The normal approximation to binomial
More information6 Central Limit Theorem. (Chs 6.4, 6.5)
6 Central Limit Theorem (Chs 6.4, 6.5) Motivating Example In the next few weeks, we will be focusing on making statistical inference about the true mean of a population by using sample datasets. Examples?
More informationChapter 5. Sampling Distributions
Lecture notes, Lang Wu, UBC 1 Chapter 5. Sampling Distributions 5.1. Introduction In statistical inference, we attempt to estimate an unknown population characteristic, such as the population mean, µ,
More informationModule 4: Probability
Module 4: Probability 1 / 22 Probability concepts in statistical inference Probability is a way of quantifying uncertainty associated with random events and is the basis for statistical inference. Inference
More informationAs you draw random samples of size n, as n increases, the sample means tend to be normally distributed.
The Central Limit Theorem The central limit theorem (clt for short) is one of the most powerful and useful ideas in all of statistics. The clt says that if we collect samples of size n with a "large enough
More informationThe Binomial Distribution
The Binomial Distribution January 31, 2019 Contents The Binomial Distribution The Normal Approximation to the Binomial The Binomial Hypothesis Test Computing Binomial Probabilities in R 30 Problems The
More informationCHAPTER 4 DISCRETE PROBABILITY DISTRIBUTIONS
CHAPTER 4 DISCRETE PROBABILITY DISTRIBUTIONS A random variable is the description of the outcome of an experiment in words. The verbal description of a random variable tells you how to find or calculate
More informationFigure 1: 2πσ is said to have a normal distribution with mean µ and standard deviation σ. This is also denoted
Figure 1: Math 223 Lecture Notes 4/1/04 Section 4.10 The normal distribution Recall that a continuous random variable X with probability distribution function f(x) = 1 µ)2 (x e 2σ 2πσ is said to have a
More informationChapter 15: Sampling distributions
=true true Chapter 15: Sampling distributions Objective (1) Get "big picture" view on drawing inferences from statistical studies. (2) Understand the concept of sampling distributions & sampling variability.
More informationReview: Population, sample, and sampling distributions
Review: Population, sample, and sampling distributions A population with mean µ and standard deviation σ For instance, µ = 0, σ = 1 0 1 Sample 1, N=30 Sample 2, N=30 Sample 100000000000 InterquartileRange
More informationConfidence Intervals Introduction
Confidence Intervals Introduction A point estimate provides no information about the precision and reliability of estimation. For example, the sample mean X is a point estimate of the population mean μ
More informationBusiness Statistics 41000: Probability 4
Business Statistics 41000: Probability 4 Drew D. Creal University of Chicago, Booth School of Business February 14 and 15, 2014 1 Class information Drew D. Creal Email: dcreal@chicagobooth.edu Office:
More informationIntroduction to Statistical Data Analysis II
Introduction to Statistical Data Analysis II JULY 2011 Afsaneh Yazdani Preface Major branches of Statistics: - Descriptive Statistics - Inferential Statistics Preface What is Inferential Statistics? Preface
More informationConfidence Intervals and Sample Size
Confidence Intervals and Sample Size Chapter 6 shows us how we can use the Central Limit Theorem (CLT) to 1. estimate a population parameter (such as the mean or proportion) using a sample, and. determine
More informationECO220Y Estimation: Confidence Interval Estimator for Sample Proportions Readings: Chapter 11 (skip 11.5)
ECO220Y Estimation: Confidence Interval Estimator for Sample Proportions Readings: Chapter 11 (skip 11.5) Fall 2011 Lecture 10 (Fall 2011) Estimation Lecture 10 1 / 23 Review: Sampling Distributions Sample
More informationBIOL The Normal Distribution and the Central Limit Theorem
BIOL 300 - The Normal Distribution and the Central Limit Theorem In the first week of the course, we introduced a few measures of center and spread, and discussed how the mean and standard deviation are
More informationSTA 320 Fall Thursday, Dec 5. Sampling Distribution. STA Fall
STA 320 Fall 2013 Thursday, Dec 5 Sampling Distribution STA 320 - Fall 2013-1 Review We cannot tell what will happen in any given individual sample (just as we can not predict a single coin flip in advance).
More informationStatistics for Managers Using Microsoft Excel 7 th Edition
Statistics for Managers Using Microsoft Excel 7 th Edition Chapter 7 Sampling Distributions Statistics for Managers Using Microsoft Excel 7e Copyright 2014 Pearson Education, Inc. Chap 7-1 Learning Objectives
More informationThe Binomial Distribution
The Binomial Distribution January 31, 2018 Contents The Binomial Distribution The Normal Approximation to the Binomial The Binomial Hypothesis Test Computing Binomial Probabilities in R 30 Problems The
More informationChapter 7 Study Guide: The Central Limit Theorem
Chapter 7 Study Guide: The Central Limit Theorem Introduction Why are we so concerned with means? Two reasons are that they give us a middle ground for comparison and they are easy to calculate. In this
More informationProbability is the tool used for anticipating what the distribution of data should look like under a given model.
AP Statistics NAME: Exam Review: Strand 3: Anticipating Patterns Date: Block: III. Anticipating Patterns: Exploring random phenomena using probability and simulation (20%-30%) Probability is the tool used
More informationMLLunsford 1. Activity: Central Limit Theorem Theory and Computations
MLLunsford 1 Activity: Central Limit Theorem Theory and Computations Concepts: The Central Limit Theorem; computations using the Central Limit Theorem. Prerequisites: The student should be familiar with
More informationSTAT Chapter 6: Sampling Distributions
STAT 515 -- Chapter 6: Sampling Distributions Definition: Parameter = a number that characterizes a population (example: population mean ) it s typically unknown. Statistic = a number that characterizes
More informationStatistics 431 Spring 2007 P. Shaman. Preliminaries
Statistics 4 Spring 007 P. Shaman The Binomial Distribution Preliminaries A binomial experiment is defined by the following conditions: A sequence of n trials is conducted, with each trial having two possible
More informationStatistics 13 Elementary Statistics
Statistics 13 Elementary Statistics Summer Session I 2012 Lecture Notes 5: Estimation with Confidence intervals 1 Our goal is to estimate the value of an unknown population parameter, such as a population
More informationStatistics Class 15 3/21/2012
Statistics Class 15 3/21/2012 Quiz 1. Cans of regular Pepsi are labeled to indicate that they contain 12 oz. Data Set 17 in Appendix B lists measured amounts for a sample of Pepsi cans. The same statistics
More informationSampling Distribution
MAT 2379 (Spring 2012) Sampling Distribution Definition : Let X 1,..., X n be a collection of random variables. We say that they are identically distributed if they have a common distribution. Definition
More informationChapter 5: Statistical Inference (in General)
Chapter 5: Statistical Inference (in General) Shiwen Shen University of South Carolina 2016 Fall Section 003 1 / 17 Motivation In chapter 3, we learn the discrete probability distributions, including Bernoulli,
More informationMidterm Exam III Review
Midterm Exam III Review Dr. Joseph Brennan Math 148, BU Dr. Joseph Brennan (Math 148, BU) Midterm Exam III Review 1 / 25 Permutations and Combinations ORDER In order to count the number of possible ways
More informationFEEG6017 lecture: The normal distribution, estimation, confidence intervals. Markus Brede,
FEEG6017 lecture: The normal distribution, estimation, confidence intervals. Markus Brede, mb8@ecs.soton.ac.uk The normal distribution The normal distribution is the classic "bell curve". We've seen that
More information10/1/2012. PSY 511: Advanced Statistics for Psychological and Behavioral Research 1
PSY 511: Advanced Statistics for Psychological and Behavioral Research 1 Pivotal subject: distributions of statistics. Foundation linchpin important crucial You need sampling distributions to make inferences:
More informationProbability and Sampling Distributions Random variables. Section 4.3 (Continued)
Probability and Sampling Distributions Random variables Section 4.3 (Continued) The mean of a random variable The mean (or expected value) of a random variable, X, is an idealization of the mean,, of quantitative
More informationElementary Statistics Lecture 5
Elementary Statistics Lecture 5 Sampling Distributions Chong Ma Department of Statistics University of South Carolina Chong Ma (Statistics, USC) STAT 201 Elementary Statistics 1 / 24 Outline 1 Introduction
More informationWeek 1 Quantitative Analysis of Financial Markets Distributions B
Week 1 Quantitative Analysis of Financial Markets Distributions B Christopher Ting http://www.mysmu.edu/faculty/christophert/ Christopher Ting : christopherting@smu.edu.sg : 6828 0364 : LKCSB 5036 October
More informationChapter 9 & 10. Multiple Choice.
Chapter 9 & 10 Review Name Multiple Choice. 1. An agricultural researcher plants 25 plots with a new variety of corn. The average yield for these plots is X = 150 bushels per acre. Assume that the yield
More informationChapter 7: SAMPLING DISTRIBUTIONS & POINT ESTIMATION OF PARAMETERS
Chapter 7: SAMPLING DISTRIBUTIONS & POINT ESTIMATION OF PARAMETERS Part 1: Introduction Sampling Distributions & the Central Limit Theorem Point Estimation & Estimators Sections 7-1 to 7-2 Sample data
More informationCHAPTER 5 SAMPLING DISTRIBUTIONS
CHAPTER 5 SAMPLING DISTRIBUTIONS Sampling Variability. We will visualize our data as a random sample from the population with unknown parameter μ. Our sample mean Ȳ is intended to estimate population mean
More informationQuantitative Methods for Economics, Finance and Management (A86050 F86050)
Quantitative Methods for Economics, Finance and Management (A86050 F86050) Matteo Manera matteo.manera@unimib.it Marzio Galeotti marzio.galeotti@unimi.it 1 This material is taken and adapted from Guy Judge
More informationUsing the Central Limit Theorem It is important for you to understand when to use the CLT. If you are being asked to find the probability of the
Using the Central Limit Theorem It is important for you to understand when to use the CLT. If you are being asked to find the probability of the mean, use the CLT for the mean. If you are being asked to
More informationData Analysis and Statistical Methods Statistics 651
Review of previous lecture: Why confidence intervals? Data Analysis and Statistical Methods Statistics 651 http://www.stat.tamu.edu/~suhasini/teaching.html Suhasini Subba Rao Suppose you want to know the
More informationDetermining Sample Size. Slide 1 ˆ ˆ. p q n E = z α / 2. (solve for n by algebra) n = E 2
Determining Sample Size Slide 1 E = z α / 2 ˆ ˆ p q n (solve for n by algebra) n = ( zα α / 2) 2 p ˆ qˆ E 2 Sample Size for Estimating Proportion p When an estimate of ˆp is known: Slide 2 n = ˆ ˆ ( )
More informationTutorial 6. Sampling Distribution. ENGG2450A Tutors. 27 February The Chinese University of Hong Kong 1/6
Tutorial 6 Sampling Distribution ENGG2450A Tutors The Chinese University of Hong Kong 27 February 2017 1/6 Random Sample and Sampling Distribution 2/6 Random sample Consider a random variable X with distribution
More informationStat 213: Intro to Statistics 9 Central Limit Theorem
1 Stat 213: Intro to Statistics 9 Central Limit Theorem H. Kim Fall 2007 2 unknown parameters Example: A pollster is sure that the responses to his agree/disagree questions will follow a binomial distribution,
More informationPoint Estimation. Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage
6 Point Estimation Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage Point Estimation Statistical inference: directed toward conclusions about one or more parameters. We will use the generic
More informationSTAT 509: Statistics for Engineers Dr. Dewei Wang. Copyright 2014 John Wiley & Sons, Inc. All rights reserved.
STAT 509: Statistics for Engineers Dr. Dewei Wang Applied Statistics and Probability for Engineers Sixth Edition Douglas C. Montgomery George C. Runger 7 Point CHAPTER OUTLINE 7-1 Point Estimation 7-2
More information*****CENTRAL LIMIT THEOREM (CLT)*****
Sampling Distributions and CLT Day 5 *****CENTRAL LIMIT THEOREM (CLT)***** (One of the MOST important theorems in Statistics - KNOW AND UNDERSTAND THIS!!!!!!) Draw an SRS of size n from ANY population
More informationLecture 6: Chapter 6
Lecture 6: Chapter 6 C C Moxley UAB Mathematics 3 October 16 6.1 Continuous Probability Distributions Last week, we discussed the binomial probability distribution, which was discrete. 6.1 Continuous Probability
More informationNormal Curves & Sampling Distributions
Chapter 7 Name Normal Curves & Sampling Distributions Section 7.1 Graphs of Normal Probability Distributions Objective: In this lesson you learned how to graph a normal curve and apply the empirical rule
More informationThe Normal Probability Distribution
1 The Normal Probability Distribution Key Definitions Probability Density Function: An equation used to compute probabilities for continuous random variables where the output value is greater than zero
More informationValue (x) probability Example A-2: Construct a histogram for population Ψ.
Calculus 111, section 08.x The Central Limit Theorem notes by Tim Pilachowski If you haven t done it yet, go to the Math 111 page and download the handout: Central Limit Theorem supplement. Today s lecture
More informationAMS 7 Sampling Distributions, Central limit theorem, Confidence Intervals Lecture 4
AMS 7 Sampling Distributions, Central limit theorem, Confidence Intervals Lecture 4 Department of Applied Mathematics and Statistics, University of California, Santa Cruz Summer 2014 1 / 26 Sampling Distributions!!!!!!
More informationchapter 13: Binomial Distribution Exercises (binomial)13.6, 13.12, 13.22, 13.43
chapter 13: Binomial Distribution ch13-links binom-tossing-4-coins binom-coin-example ch13 image Exercises (binomial)13.6, 13.12, 13.22, 13.43 CHAPTER 13: Binomial Distributions The Basic Practice of Statistics
More informationKey Objectives. Module 2: The Logic of Statistical Inference. Z-scores. SGSB Workshop: Using Statistical Data to Make Decisions
SGSB Workshop: Using Statistical Data to Make Decisions Module 2: The Logic of Statistical Inference Dr. Tom Ilvento January 2006 Dr. Mugdim Pašić Key Objectives Understand the logic of statistical inference
More informationChapter 14 : Statistical Inference 1. Note : Here the 4-th and 5-th editions of the text have different chapters, but the material is the same.
Chapter 14 : Statistical Inference 1 Chapter 14 : Introduction to Statistical Inference Note : Here the 4-th and 5-th editions of the text have different chapters, but the material is the same. Data x
More information8.1 Estimation of the Mean and Proportion
8.1 Estimation of the Mean and Proportion Statistical inference enables us to make judgments about a population on the basis of sample information. The mean, standard deviation, and proportions of a population
More informationThe Central Limit Theorem
Section 6-5 The Central Limit Theorem I. Sampling Distribution of Sample Mean ( ) Eample 1: Population Distribution Table 2 4 6 8 P() 1/4 1/4 1/4 1/4 μ (a) Find the population mean and population standard
More informationSection The Sampling Distribution of a Sample Mean
Section 5.2 - The Sampling Distribution of a Sample Mean Statistics 104 Autumn 2004 Copyright c 2004 by Mark E. Irwin The Sampling Distribution of a Sample Mean Example: Quality control check of light
More informationThe Mathematics of Normality
MATH 110 Week 9 Chapter 17 Worksheet The Mathematics of Normality NAME Normal (bell-shaped) distributions play an important role in the world of statistics. One reason the normal distribution is important
More informationNormal Model (Part 1)
Normal Model (Part 1) Formulas New Vocabulary The Standard Deviation as a Ruler The trick in comparing very different-looking values is to use standard deviations as our rulers. The standard deviation
More informationSampling Distributions
Section 8.1 119 Sampling Distributions Section 8.1 C H A P T E R 8 4Example 2 (pg. 378) Sampling Distribution of the Sample Mean The heights of 3-year-old girls are normally distributed with μ=38.72 and
More informationActivity #17b: Central Limit Theorem #2. 1) Explain the Central Limit Theorem in your own words.
Activity #17b: Central Limit Theorem #2 1) Explain the Central Limit Theorem in your own words. Importance of the CLT: You can standardize and use normal distribution tables to calculate probabilities
More informationand µ Asian male > " men
A.P. Statistics Sampling Distributions and the Central Limit Theorem Definitions A parameter is a number that describes the population. A parameter always exists but in practice we rarely know its value
More informationOne sample z-test and t-test
One sample z-test and t-test January 30, 2017 psych10.stanford.edu Announcements / Action Items Install ISI package (instructions in Getting Started with R) Assessment Problem Set #3 due Tu 1/31 at 7 PM
More informationNormal Probability Distributions
Normal Probability Distributions Properties of Normal Distributions The most important probability distribution in statistics is the normal distribution. Normal curve A normal distribution is a continuous
More informationEstimation Y 3. Confidence intervals I, Feb 11,
Estimation Example: Cholesterol levels of heart-attack patients Data: Observational study at a Pennsylvania medical center blood cholesterol levels patients treated for heart attacks measurements 2, 4,
More information19. CONFIDENCE INTERVALS FOR THE MEAN; KNOWN VARIANCE
19. CONFIDENCE INTERVALS FOR THE MEAN; KNOWN VARIANCE We assume here that the population variance σ 2 is known. This is an unrealistic assumption, but it allows us to give a simplified presentation which
More informationName PID Section # (enrolled)
STT 315 - Lecture 3 Instructor: Aylin ALIN 04/02/2014 Midterm # 2 A Name PID Section # (enrolled) * The exam is closed book and 80 minutes. * You may use a calculator and the formula sheet that you brought
More informationSTATISTICS - CLUTCH CH.9: SAMPLING DISTRIBUTIONS: MEAN.
!! www.clutchprep.com SAMPLING DISTRIBUTIONS (MEANS) As of now, the normal distributions we have worked with only deal with the population of observations Example: What is the probability of randomly selecting
More informationPart V - Chance Variability
Part V - Chance Variability Dr. Joseph Brennan Math 148, BU Dr. Joseph Brennan (Math 148, BU) Part V - Chance Variability 1 / 78 Law of Averages In Chapter 13 we discussed the Kerrich coin-tossing experiment.
More informationToday s plan: Section 4.4.2: Capture-Recapture method revisited and Section 4.4.3: Public Opinion Polls
1 Today s plan: Section 4.4.2: Capture-Recapture method revisited and Section 4.4.3: Public Opinion Polls 2 Section 4.4.2: Capture-Recapture method revisited 3 Let s use statistical inference to get a
More informationSampling Distributions
Al Nosedal. University of Toronto. Fall 2017 October 26, 2017 1 What is a Sampling Distribution? 2 3 Sampling Distribution The sampling distribution of a statistic is the distribution of values taken by
More informationMAS1403. Quantitative Methods for Business Management. Semester 1, Module leader: Dr. David Walshaw
MAS1403 Quantitative Methods for Business Management Semester 1, 2018 2019 Module leader: Dr. David Walshaw Additional lecturers: Dr. James Waldren and Dr. Stuart Hall Announcements: Written assignment
More informationECON 214 Elements of Statistics for Economists 2016/2017
ECON 214 Elements of Statistics for Economists 2016/2017 Topic The Normal Distribution Lecturer: Dr. Bernardin Senadza, Dept. of Economics bsenadza@ug.edu.gh College of Education School of Continuing and
More informationTerms & Characteristics
NORMAL CURVE Knowledge that a variable is distributed normally can be helpful in drawing inferences as to how frequently certain observations are likely to occur. NORMAL CURVE A Normal distribution: Distribution
More informationThe Normal Approximation to the Binomial
Lecture 16 The Normal Approximation to the Binomial We can calculate l binomial i probabilities bbilii using The binomial formula The cumulative binomial tables When n is large, and p is not too close
More informationThe normal distribution is a theoretical model derived mathematically and not empirically.
Sociology 541 The Normal Distribution Probability and An Introduction to Inferential Statistics Normal Approximation The normal distribution is a theoretical model derived mathematically and not empirically.
More informationSampling Distributions and the Central Limit Theorem
Sampling Distributions and the Central Limit Theorem February 18 Data distributions and sampling distributions So far, we have discussed the distribution of data (i.e. of random variables in our sample,
More informationUnit 5: Sampling Distributions of Statistics
Unit 5: Sampling Distributions of Statistics Statistics 571: Statistical Methods Ramón V. León 6/12/2004 Unit 5 - Stat 571 - Ramon V. Leon 1 Definitions and Key Concepts A sample statistic used to estimate
More informationUnit 5: Sampling Distributions of Statistics
Unit 5: Sampling Distributions of Statistics Statistics 571: Statistical Methods Ramón V. León 6/12/2004 Unit 5 - Stat 571 - Ramon V. Leon 1 Definitions and Key Concepts A sample statistic used to estimate
More informationStatistics 251: Statistical Methods Sampling Distributions Module
Statistics 251: Statistical Methods Sampling Distributions Module 7 2018 Three Types of Distributions data distribution the distribution of a variable in a sample population distribution the probability
More informationStatistics, Measures of Central Tendency I
Statistics, Measures of Central Tendency I We are considering a random variable X with a probability distribution which has some parameters. We want to get an idea what these parameters are. We perfom
More informationSTAT Chapter 6 The Standard Deviation (SD) as a Ruler and The Normal Model
STAT 203 - Chapter 6 The Standard Deviation (SD) as a Ruler and The Normal Model In Chapter 5, we introduced a few measures of center and spread, and discussed how the mean and standard deviation are good
More informationStatistical Intervals (One sample) (Chs )
7 Statistical Intervals (One sample) (Chs 8.1-8.3) Confidence Intervals The CLT tells us that as the sample size n increases, the sample mean X is close to normally distributed with expected value µ and
More informationSTAT Chapter 6 The Standard Deviation (SD) as a Ruler and The Normal Model
STAT 203 - Chapter 6 The Standard Deviation (SD) as a Ruler and The Normal Model In Chapter 5, we introduced a few measures of center and spread, and discussed how the mean and standard deviation are good
More informationECE 295: Lecture 03 Estimation and Confidence Interval
ECE 295: Lecture 03 Estimation and Confidence Interval Spring 2018 Prof Stanley Chan School of Electrical and Computer Engineering Purdue University 1 / 23 Theme of this Lecture What is Estimation? You
More informationMath 140 Introductory Statistics. First midterm September
Math 140 Introductory Statistics First midterm September 23 2010 Box Plots Graphical display of 5 number summary Q1, Q2 (median), Q3, max, min Outliers If a value is more than 1.5 times the IQR from the
More informationSTAT:2010 Statistical Methods and Computing. Using density curves to describe the distribution of values of a quantitative
STAT:10 Statistical Methods and Computing Normal Distributions Lecture 4 Feb. 6, 17 Kate Cowles 374 SH, 335-0727 kate-cowles@uiowa.edu 1 2 Using density curves to describe the distribution of values of
More information