STATISTICAL INFERENCE. Statistical Inference. Statistical Inference. Sampling Sampling distributions
|
|
- Barbara Harvey
- 7 years ago
- Views:
Transcription
1 STATISTICAL INFERENCE Sampling Sampling distributions QMS 204: STATISTICS FOR MANAGERS Instructor: Moez Hababou page 1 Statistical Inference The use of random samples from a population to make inferences about a population page 2 Statistical Inference Two types of inference procedures Estimation Hypothesis Testing page 3 1
2 Random Sample All individuals or objects in the population have a chance of being selected The selection of any one individual or object does not affect the selection probability for any other individual or object page 4 Simple Random Sample A random sample selected in such a way that all possible samples of the same size have the same chance of being selected Implies that all individuals have the same chance of being selected for the sample page 5 Finite Populations and Sampling With Replacement We assume that samples are drawn without replacement and that the population is large enough that the number of possible samples of size n is still large If the sampling fraction n/n ( where N is the population size) is less than 5% then we ignore the finite population effect page 6 2
3 Finite Populations and Sampling With Replacement If the population is small and finite corrections have to be made(not to be discussed in this course) page 7 Population Distribution This is the distribution of all items in the population of interest It is characterized by the values of certain parameters which are usually unknown page 8 Population Distribution X the random variable µ population mean σ 2 population variance draw samples µ σ X Sample 1 of size n Sample 2 of size n page 9 3
4 Sample Distribution This is the distribution of the items that appeared in the sample This information is used to derive the values of sample statistics which are used to estimate the unknown parameters page 10 Population Distribution X the random variable µ population mean σ 2 population variance Sample of size n x 1, x 2, x 3., x n, x S 2 Sample distribution µ σ x s X X page 11 Sampling Distribution This is the distribution of a sample statistic over all possible samples of the same size taken from the same population page 12 4
5 x Sampling distribution of Sampling distribution of S 2 µ x σ 2 S 2 page 13 Statistical Inference For The Population Mean Requires the sampling distribution of the mean page 14 Sampling Distribution of The Sample Mean This is the distribution of the sample mean over all possible simple random samples of the same size taken from the same population page 15 5
6 Normal Population Variance Known The sampling distribution of the mean is a Normal distribution with mean µ and standard deviation σ/ n The standard deviation of the mean σ/ n is called the standard error of the mean As the sample size n increases the variation of the sample means around the true mean decreases page 16 Small Sample more variation in x Distribution of x Large Sample less variation in x µ x µ x σ/ n σ/ n page 17 It has been determined that heating expenses are normally distributed with a mean of $ and a standard deviation of $ a) What is the probability that in a random sample of 25 accounts the average balance could exceed $550? page 18 6
7 Z = = 2 125/ P[ Z > 2 ] = page 19 b) What is the probability that in a random sample of 25 accounts the average would lie between $ and $540.00? Z = = / 25 Z = = / 25 P[ -1 Z 1.60 ] = page 20 Nonnormal Population Central Limit Theorem page 21 7
8 Central Limit Theorem In large samples the central limit theorem states that the sample mean is normally distributed with mean µ and standard deviation σ/ n Thus in large samples we need not be concerned about whether the underlying population is normal page 22 A random sample of 64 residents of a particular section of the city were asked to indicate the number of hours per week they watch television. The population standard deviation is 16. a) What is the probability that the sample mean will exceed the population mean by 2 in absolute value? page 23 µ-2 µ µ+2 (µ+2) - µ Z = = 1 16/ 64 (µ-2) - µ Z = = -1 16/ 64 P[ Z > 1 ] = 2(0.1587) = page 24 8
9 b) If the sample size were 144 what would be the probability in a)? µ-2 µ µ+2 (µ+2) - µ Z = 16/ 144 = 1.5 (µ-2) - µ Z = 16/ 144 = -1.5 P[ Z > 1.5 ] = 2(0.0668) = page 25 Sampling Distribution for the Mean When Variance Unknown Population variance unknown in large samples Population variance unknown in small samples page 26 Population Variance Unknown in Large Samples By the central limit theorem in large samples the statistic below has a standard normal distribution x µ s/ n This distribution is exact if the population being sampled is normal page 27 9
10 Population Variance Unknown in Small Samples In small samples if the population variance is unknown and the population is normal the statistic below has a t distribution with (n-1) degrees of freedom x µ s/ n If the population is not normal we do not know the distribution of this statistic page 28 t-distribution the t distribution is bell shaped and symmetrical and looks very much like a standard normal distribution Normal distribution t distribution page 29 t-distribution the mean of the distribution is zero and the variance is larger than the standard normal variance of 1 as the number of degrees of freedom increase, the variance of the t distribution approaches 1 page 30 10
11 t-distribution the critical values of the t distribution are larger than the corresponding critical values for the standard normal but approach the standard normal values as the number of degrees of freedom rise by the time d.f. = (n-1) = 30 the t distribution and standard normal are very close examine the critical values of t for α = as the d.f. change page 31 Sampling For Proportions In the special case of a binomial population we are interested in making inferences about the binomial parameter π This is a special case of the central limit theorem In large samples the sampling distribution of the sample proportion p has a mean of π and a standard deviation of π (1-π)/n page 32 Sampling For Proportions The sample proportion p is equivalent to a sample mean for a variable whose underlying values are either 0 or 1 In the case of a sample proportion the sample size needs to be large if the true proportion π is close to zero or 1 A useful rule of thumb is that both n π and n(1-π) should exceed 5 page 33 11
12 If π = 0.10 nπ > 5 n > 50 If π = 0.90 n(1-π) > 5 n > 50 If π = 0.01 nπ > 5 n > 500 If π = 0.99 n(1-π) > 5 n > 500 page 34 Suppose that 60% of the population of Alberta voters wish to eliminate video lottery terminals. Suppose a sample of 100 voters was selected and asked for their opinion. What is the probability that less than 50% of the sample will be in favor of eliminating the terminals? page 35 Z = = (0.60)(0.40) 100 P[ Z < ] = page 36 12
5.1 Identifying the Target Parameter
University of California, Davis Department of Statistics Summer Session II Statistics 13 August 20, 2012 Date of latest update: August 20 Lecture 5: Estimation with Confidence intervals 5.1 Identifying
More informationPr(X = x) = f(x) = λe λx
Old Business - variance/std. dev. of binomial distribution - mid-term (day, policies) - class strategies (problems, etc.) - exponential distributions New Business - Central Limit Theorem, standard error
More informationNeed for Sampling. Very large populations Destructive testing Continuous production process
Chapter 4 Sampling and Estimation Need for Sampling Very large populations Destructive testing Continuous production process The objective of sampling is to draw a valid inference about a population. 4-
More information9. Sampling Distributions
9. Sampling Distributions Prerequisites none A. Introduction B. Sampling Distribution of the Mean C. Sampling Distribution of Difference Between Means D. Sampling Distribution of Pearson's r E. Sampling
More informationSampling Distributions
Sampling Distributions You have seen probability distributions of various types. The normal distribution is an example of a continuous distribution that is often used for quantitative measures such as
More informationWeek 4: Standard Error and Confidence Intervals
Health Sciences M.Sc. Programme Applied Biostatistics Week 4: Standard Error and Confidence Intervals Sampling Most research data come from subjects we think of as samples drawn from a larger population.
More informationPart 2: Analysis of Relationship Between Two Variables
Part 2: Analysis of Relationship Between Two Variables Linear Regression Linear correlation Significance Tests Multiple regression Linear Regression Y = a X + b Dependent Variable Independent Variable
More informationConfidence Intervals for One Standard Deviation Using Standard Deviation
Chapter 640 Confidence Intervals for One Standard Deviation Using Standard Deviation Introduction This routine calculates the sample size necessary to achieve a specified interval width or distance from
More informationSimple Regression Theory II 2010 Samuel L. Baker
SIMPLE REGRESSION THEORY II 1 Simple Regression Theory II 2010 Samuel L. Baker Assessing how good the regression equation is likely to be Assignment 1A gets into drawing inferences about how close the
More information4. Continuous Random Variables, the Pareto and Normal Distributions
4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random
More informationConfidence Intervals for the Difference Between Two Means
Chapter 47 Confidence Intervals for the Difference Between Two Means Introduction This procedure calculates the sample size necessary to achieve a specified distance from the difference in sample means
More informationChapter 4. Probability and Probability Distributions
Chapter 4. robability and robability Distributions Importance of Knowing robability To know whether a sample is not identical to the population from which it was selected, it is necessary to assess the
More informationStatistics 104: Section 6!
Page 1 Statistics 104: Section 6! TF: Deirdre (say: Dear-dra) Bloome Email: dbloome@fas.harvard.edu Section Times Thursday 2pm-3pm in SC 109, Thursday 5pm-6pm in SC 705 Office Hours: Thursday 6pm-7pm SC
More information8. THE NORMAL DISTRIBUTION
8. THE NORMAL DISTRIBUTION The normal distribution with mean μ and variance σ 2 has the following density function: The normal distribution is sometimes called a Gaussian Distribution, after its inventor,
More informationNon-Inferiority Tests for One Mean
Chapter 45 Non-Inferiority ests for One Mean Introduction his module computes power and sample size for non-inferiority tests in one-sample designs in which the outcome is distributed as a normal random
More informationUniversity of Chicago Graduate School of Business. Business 41000: Business Statistics Solution Key
Name: OUTLINE SOLUTIONS University of Chicago Graduate School of Business Business 41000: Business Statistics Solution Key Special Notes: 1. This is a closed-book exam. You may use an 8 11 piece of paper
More informationMEASURES OF VARIATION
NORMAL DISTRIBTIONS MEASURES OF VARIATION In statistics, it is important to measure the spread of data. A simple way to measure spread is to find the range. But statisticians want to know if the data are
More informationLecture Notes Module 1
Lecture Notes Module 1 Study Populations A study population is a clearly defined collection of people, animals, plants, or objects. In psychological research, a study population usually consists of a specific
More informationFairfield Public Schools
Mathematics Fairfield Public Schools AP Statistics AP Statistics BOE Approved 04/08/2014 1 AP STATISTICS Critical Areas of Focus AP Statistics is a rigorous course that offers advanced students an opportunity
More informationNormal distribution. ) 2 /2σ. 2π σ
Normal distribution The normal distribution is the most widely known and used of all distributions. Because the normal distribution approximates many natural phenomena so well, it has developed into a
More informationSummary of Formulas and Concepts. Descriptive Statistics (Ch. 1-4)
Summary of Formulas and Concepts Descriptive Statistics (Ch. 1-4) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume
More informationBetting rules and information theory
Betting rules and information theory Giulio Bottazzi LEM and CAFED Scuola Superiore Sant Anna September, 2013 Outline Simple betting in favorable games The Central Limit Theorem Optimal rules The Game
More information1.5 Oneway Analysis of Variance
Statistics: Rosie Cornish. 200. 1.5 Oneway Analysis of Variance 1 Introduction Oneway analysis of variance (ANOVA) is used to compare several means. This method is often used in scientific or medical experiments
More informationWeek 3&4: Z tables and the Sampling Distribution of X
Week 3&4: Z tables and the Sampling Distribution of X 2 / 36 The Standard Normal Distribution, or Z Distribution, is the distribution of a random variable, Z N(0, 1 2 ). The distribution of any other normal
More informationClass 19: Two Way Tables, Conditional Distributions, Chi-Square (Text: Sections 2.5; 9.1)
Spring 204 Class 9: Two Way Tables, Conditional Distributions, Chi-Square (Text: Sections 2.5; 9.) Big Picture: More than Two Samples In Chapter 7: We looked at quantitative variables and compared the
More informationName: Date: Use the following to answer questions 3-4:
Name: Date: 1. Determine whether each of the following statements is true or false. A) The margin of error for a 95% confidence interval for the mean increases as the sample size increases. B) The margin
More informationPermutation Tests for Comparing Two Populations
Permutation Tests for Comparing Two Populations Ferry Butar Butar, Ph.D. Jae-Wan Park Abstract Permutation tests for comparing two populations could be widely used in practice because of flexibility of
More information1. How different is the t distribution from the normal?
Statistics 101 106 Lecture 7 (20 October 98) c David Pollard Page 1 Read M&M 7.1 and 7.2, ignoring starred parts. Reread M&M 3.2. The effects of estimated variances on normal approximations. t-distributions.
More information1 Sufficient statistics
1 Sufficient statistics A statistic is a function T = rx 1, X 2,, X n of the random sample X 1, X 2,, X n. Examples are X n = 1 n s 2 = = X i, 1 n 1 the sample mean X i X n 2, the sample variance T 1 =
More informationCurriculum Map Statistics and Probability Honors (348) Saugus High School Saugus Public Schools 2009-2010
Curriculum Map Statistics and Probability Honors (348) Saugus High School Saugus Public Schools 2009-2010 Week 1 Week 2 14.0 Students organize and describe distributions of data by using a number of different
More information6.4 Normal Distribution
Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under
More informationSimple Random Sampling
Source: Frerichs, R.R. Rapid Surveys (unpublished), 2008. NOT FOR COMMERCIAL DISTRIBUTION 3 Simple Random Sampling 3.1 INTRODUCTION Everyone mentions simple random sampling, but few use this method for
More informationCharacteristics of Binomial Distributions
Lesson2 Characteristics of Binomial Distributions In the last lesson, you constructed several binomial distributions, observed their shapes, and estimated their means and standard deviations. In Investigation
More informationQUANTITATIVE METHODS BIOLOGY FINAL HONOUR SCHOOL NON-PARAMETRIC TESTS
QUANTITATIVE METHODS BIOLOGY FINAL HONOUR SCHOOL NON-PARAMETRIC TESTS This booklet contains lecture notes for the nonparametric work in the QM course. This booklet may be online at http://users.ox.ac.uk/~grafen/qmnotes/index.html.
More informationPoint and Interval Estimates
Point and Interval Estimates Suppose we want to estimate a parameter, such as p or µ, based on a finite sample of data. There are two main methods: 1. Point estimate: Summarize the sample by a single number
More informationSimple Linear Regression Inference
Simple Linear Regression Inference 1 Inference requirements The Normality assumption of the stochastic term e is needed for inference even if it is not a OLS requirement. Therefore we have: Interpretation
More informationAn Introduction to Basic Statistics and Probability
An Introduction to Basic Statistics and Probability Shenek Heyward NCSU An Introduction to Basic Statistics and Probability p. 1/4 Outline Basic probability concepts Conditional probability Discrete Random
More informationStat 411/511 THE RANDOMIZATION TEST. Charlotte Wickham. stat511.cwick.co.nz. Oct 16 2015
Stat 411/511 THE RANDOMIZATION TEST Oct 16 2015 Charlotte Wickham stat511.cwick.co.nz Today Review randomization model Conduct randomization test What about CIs? Using a t-distribution as an approximation
More informationThe Normal distribution
The Normal distribution The normal probability distribution is the most common model for relative frequencies of a quantitative variable. Bell-shaped and described by the function f(y) = 1 2σ π e{ 1 2σ
More informationHYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1. used confidence intervals to answer questions such as...
HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1 PREVIOUSLY used confidence intervals to answer questions such as... You know that 0.25% of women have red/green color blindness. You conduct a study of men
More informationChi Square Tests. Chapter 10. 10.1 Introduction
Contents 10 Chi Square Tests 703 10.1 Introduction............................ 703 10.2 The Chi Square Distribution.................. 704 10.3 Goodness of Fit Test....................... 709 10.4 Chi Square
More informationSENSITIVITY ANALYSIS AND INFERENCE. Lecture 12
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. Your use of this material constitutes acceptance of that license and the conditions of use of materials on this
More informationSOLUTIONS: 4.1 Probability Distributions and 4.2 Binomial Distributions
SOLUTIONS: 4.1 Probability Distributions and 4.2 Binomial Distributions 1. The following table contains a probability distribution for a random variable X. a. Find the expected value (mean) of X. x 1 2
More informationUnit 26 Estimation with Confidence Intervals
Unit 26 Estimation with Confidence Intervals Objectives: To see how confidence intervals are used to estimate a population proportion, a population mean, a difference in population proportions, or a difference
More informationConfidence Intervals for Cp
Chapter 296 Confidence Intervals for Cp Introduction This routine calculates the sample size needed to obtain a specified width of a Cp confidence interval at a stated confidence level. Cp is a process
More informationLesson 17: Margin of Error When Estimating a Population Proportion
Margin of Error When Estimating a Population Proportion Classwork In this lesson, you will find and interpret the standard deviation of a simulated distribution for a sample proportion and use this information
More informationChicago Booth BUSINESS STATISTICS 41000 Final Exam Fall 2011
Chicago Booth BUSINESS STATISTICS 41000 Final Exam Fall 2011 Name: Section: I pledge my honor that I have not violated the Honor Code Signature: This exam has 34 pages. You have 3 hours to complete this
More information3.2 Measures of Spread
3.2 Measures of Spread In some data sets the observations are close together, while in others they are more spread out. In addition to measures of the center, it's often important to measure the spread
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. A) ±1.88 B) ±1.645 C) ±1.96 D) ±2.
Ch. 6 Confidence Intervals 6.1 Confidence Intervals for the Mean (Large Samples) 1 Find a Critical Value 1) Find the critical value zc that corresponds to a 94% confidence level. A) ±1.88 B) ±1.645 C)
More information99.37, 99.38, 99.38, 99.39, 99.39, 99.39, 99.39, 99.40, 99.41, 99.42 cm
Error Analysis and the Gaussian Distribution In experimental science theory lives or dies based on the results of experimental evidence and thus the analysis of this evidence is a critical part of the
More informationPROBABILITY AND SAMPLING DISTRIBUTIONS
PROBABILITY AND SAMPLING DISTRIBUTIONS SEEMA JAGGI AND P.K. BATRA Indian Agricultural Statistics Research Institute Library Avenue, New Delhi - 0 0 seema@iasri.res.in. Introduction The concept of probability
More informationConfidence Intervals in Public Health
Confidence Intervals in Public Health When public health practitioners use health statistics, sometimes they are interested in the actual number of health events, but more often they use the statistics
More informationIntroduction to General and Generalized Linear Models
Introduction to General and Generalized Linear Models General Linear Models - part I Henrik Madsen Poul Thyregod Informatics and Mathematical Modelling Technical University of Denmark DK-2800 Kgs. Lyngby
More informationAMS 5 CHANCE VARIABILITY
AMS 5 CHANCE VARIABILITY The Law of Averages When tossing a fair coin the chances of tails and heads are the same: 50% and 50%. So if the coin is tossed a large number of times, the number of heads and
More informationCHAPTER 6: Continuous Uniform Distribution: 6.1. Definition: The density function of the continuous random variable X on the interval [A, B] is.
Some Continuous Probability Distributions CHAPTER 6: Continuous Uniform Distribution: 6. Definition: The density function of the continuous random variable X on the interval [A, B] is B A A x B f(x; A,
More informationMath 58. Rumbos Fall 2008 1. Solutions to Review Problems for Exam 2
Math 58. Rumbos Fall 2008 1 Solutions to Review Problems for Exam 2 1. For each of the following scenarios, determine whether the binomial distribution is the appropriate distribution for the random variable
More informationWHERE DOES THE 10% CONDITION COME FROM?
1 WHERE DOES THE 10% CONDITION COME FROM? The text has mentioned The 10% Condition (at least) twice so far: p. 407 Bernoulli trials must be independent. If that assumption is violated, it is still okay
More informationStatistical tests for SPSS
Statistical tests for SPSS Paolo Coletti A.Y. 2010/11 Free University of Bolzano Bozen Premise This book is a very quick, rough and fast description of statistical tests and their usage. It is explicitly
More informationNon Parametric Inference
Maura Department of Economics and Finance Università Tor Vergata Outline 1 2 3 Inverse distribution function Theorem: Let U be a uniform random variable on (0, 1). Let X be a continuous random variable
More informationStudy Guide for the Final Exam
Study Guide for the Final Exam When studying, remember that the computational portion of the exam will only involve new material (covered after the second midterm), that material from Exam 1 will make
More informationChapter 7 Section 7.1: Inference for the Mean of a Population
Chapter 7 Section 7.1: Inference for the Mean of a Population Now let s look at a similar situation Take an SRS of size n Normal Population : N(, ). Both and are unknown parameters. Unlike what we used
More informationThe Normal Approximation to Probability Histograms. Dice: Throw a single die twice. The Probability Histogram: Area = Probability. Where are we going?
The Normal Approximation to Probability Histograms Where are we going? Probability histograms The normal approximation to binomial histograms The normal approximation to probability histograms of sums
More informationNovember 08, 2010. 155S8.6_3 Testing a Claim About a Standard Deviation or Variance
Chapter 8 Hypothesis Testing 8 1 Review and Preview 8 2 Basics of Hypothesis Testing 8 3 Testing a Claim about a Proportion 8 4 Testing a Claim About a Mean: σ Known 8 5 Testing a Claim About a Mean: σ
More informationCA200 Quantitative Analysis for Business Decisions. File name: CA200_Section_04A_StatisticsIntroduction
CA200 Quantitative Analysis for Business Decisions File name: CA200_Section_04A_StatisticsIntroduction Table of Contents 4. Introduction to Statistics... 1 4.1 Overview... 3 4.2 Discrete or continuous
More informationLAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING
LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING In this lab you will explore the concept of a confidence interval and hypothesis testing through a simulation problem in engineering setting.
More informationt Tests in Excel The Excel Statistical Master By Mark Harmon Copyright 2011 Mark Harmon
t-tests in Excel By Mark Harmon Copyright 2011 Mark Harmon No part of this publication may be reproduced or distributed without the express permission of the author. mark@excelmasterseries.com www.excelmasterseries.com
More informationCHAPTER 7 SECTION 5: RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTIONS
CHAPTER 7 SECTION 5: RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTIONS TRUE/FALSE 235. The Poisson probability distribution is a continuous probability distribution. F 236. In a Poisson distribution,
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
STT315 Practice Ch 5-7 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) The length of time a traffic signal stays green (nicknamed
More informationThe Math. P (x) = 5! = 1 2 3 4 5 = 120.
The Math Suppose there are n experiments, and the probability that someone gets the right answer on any given experiment is p. So in the first example above, n = 5 and p = 0.2. Let X be the number of correct
More informationRecall this chart that showed how most of our course would be organized:
Chapter 4 One-Way ANOVA Recall this chart that showed how most of our course would be organized: Explanatory Variable(s) Response Variable Methods Categorical Categorical Contingency Tables Categorical
More information5/31/2013. 6.1 Normal Distributions. Normal Distributions. Chapter 6. Distribution. The Normal Distribution. Outline. Objectives.
The Normal Distribution C H 6A P T E R The Normal Distribution Outline 6 1 6 2 Applications of the Normal Distribution 6 3 The Central Limit Theorem 6 4 The Normal Approximation to the Binomial Distribution
More informationA POPULATION MEAN, CONFIDENCE INTERVALS AND HYPOTHESIS TESTING
CHAPTER 5. A POPULATION MEAN, CONFIDENCE INTERVALS AND HYPOTHESIS TESTING 5.1 Concepts When a number of animals or plots are exposed to a certain treatment, we usually estimate the effect of the treatment
More informationNonparametric tests these test hypotheses that are not statements about population parameters (e.g.,
CHAPTER 13 Nonparametric and Distribution-Free Statistics Nonparametric tests these test hypotheses that are not statements about population parameters (e.g., 2 tests for goodness of fit and independence).
More informationE3: PROBABILITY AND STATISTICS lecture notes
E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................
More informationDensity Curve. A density curve is the graph of a continuous probability distribution. It must satisfy the following properties:
Density Curve A density curve is the graph of a continuous probability distribution. It must satisfy the following properties: 1. The total area under the curve must equal 1. 2. Every point on the curve
More informationComparing the Means of Two Populations: Independent Samples
CHAPTER 14 Comparing the Means of Two Populations: Independent Samples 14.1 From One Mu to Two Do children in phonics-based reading programs become better readers than children in whole language programs?
More informationTests for One Proportion
Chapter 100 Tests for One Proportion Introduction The One-Sample Proportion Test is used to assess whether a population proportion (P1) is significantly different from a hypothesized value (P0). This is
More informationThe Binomial Distribution
The Binomial Distribution James H. Steiger November 10, 00 1 Topics for this Module 1. The Binomial Process. The Binomial Random Variable. The Binomial Distribution (a) Computing the Binomial pdf (b) Computing
More informationIEOR 6711: Stochastic Models I Fall 2012, Professor Whitt, Tuesday, September 11 Normal Approximations and the Central Limit Theorem
IEOR 6711: Stochastic Models I Fall 2012, Professor Whitt, Tuesday, September 11 Normal Approximations and the Central Limit Theorem Time on my hands: Coin tosses. Problem Formulation: Suppose that I have
More informationTEACHER NOTES MATH NSPIRED
Math Objectives Students will understand that normal distributions can be used to approximate binomial distributions whenever both np and n(1 p) are sufficiently large. Students will understand that when
More informationChapter 7 - Practice Problems 1
Chapter 7 - Practice Problems 1 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) Define a point estimate. What is the
More informationMBA 611 STATISTICS AND QUANTITATIVE METHODS
MBA 611 STATISTICS AND QUANTITATIVE METHODS Part I. Review of Basic Statistics (Chapters 1-11) A. Introduction (Chapter 1) Uncertainty: Decisions are often based on incomplete information from uncertain
More informationNCSS Statistical Software. One-Sample T-Test
Chapter 205 Introduction This procedure provides several reports for making inference about a population mean based on a single sample. These reports include confidence intervals of the mean or median,
More informationCHAPTER 14 NONPARAMETRIC TESTS
CHAPTER 14 NONPARAMETRIC TESTS Everything that we have done up until now in statistics has relied heavily on one major fact: that our data is normally distributed. We have been able to make inferences
More informationOne-Way Analysis of Variance (ANOVA) Example Problem
One-Way Analysis of Variance (ANOVA) Example Problem Introduction Analysis of Variance (ANOVA) is a hypothesis-testing technique used to test the equality of two or more population (or treatment) means
More informationCONTENTS OF DAY 2. II. Why Random Sampling is Important 9 A myth, an urban legend, and the real reason NOTES FOR SUMMER STATISTICS INSTITUTE COURSE
1 2 CONTENTS OF DAY 2 I. More Precise Definition of Simple Random Sample 3 Connection with independent random variables 3 Problems with small populations 8 II. Why Random Sampling is Important 9 A myth,
More informationNon-Inferiority Tests for Two Means using Differences
Chapter 450 on-inferiority Tests for Two Means using Differences Introduction This procedure computes power and sample size for non-inferiority tests in two-sample designs in which the outcome is a continuous
More informationExperimental Design. Power and Sample Size Determination. Proportions. Proportions. Confidence Interval for p. The Binomial Test
Experimental Design Power and Sample Size Determination Bret Hanlon and Bret Larget Department of Statistics University of Wisconsin Madison November 3 8, 2011 To this point in the semester, we have largely
More informationTwo-sample inference: Continuous data
Two-sample inference: Continuous data Patrick Breheny April 5 Patrick Breheny STA 580: Biostatistics I 1/32 Introduction Our next two lectures will deal with two-sample inference for continuous data As
More information7. Normal Distributions
7. Normal Distributions A. Introduction B. History C. Areas of Normal Distributions D. Standard Normal E. Exercises Most of the statistical analyses presented in this book are based on the bell-shaped
More informationLesson 1: Comparison of Population Means Part c: Comparison of Two- Means
Lesson : Comparison of Population Means Part c: Comparison of Two- Means Welcome to lesson c. This third lesson of lesson will discuss hypothesis testing for two independent means. Steps in Hypothesis
More informationT test as a parametric statistic
KJA Statistical Round pissn 2005-619 eissn 2005-7563 T test as a parametric statistic Korean Journal of Anesthesiology Department of Anesthesia and Pain Medicine, Pusan National University School of Medicine,
More informationHypothesis Testing for Beginners
Hypothesis Testing for Beginners Michele Piffer LSE August, 2011 Michele Piffer (LSE) Hypothesis Testing for Beginners August, 2011 1 / 53 One year ago a friend asked me to put down some easy-to-read notes
More informationBasic Data Analysis. Stephen Turnbull Business Administration and Public Policy Lecture 12: June 22, 2012. Abstract. Review session.
June 23, 2012 1 review session Basic Data Analysis Stephen Turnbull Business Administration and Public Policy Lecture 12: June 22, 2012 Review session. Abstract Quantitative methods in business Accounting
More informationTutorial 5: Hypothesis Testing
Tutorial 5: Hypothesis Testing Rob Nicholls nicholls@mrc-lmb.cam.ac.uk MRC LMB Statistics Course 2014 Contents 1 Introduction................................ 1 2 Testing distributional assumptions....................
More informationIn the past, the increase in the price of gasoline could be attributed to major national or global
Chapter 7 Testing Hypotheses Chapter Learning Objectives Understanding the assumptions of statistical hypothesis testing Defining and applying the components in hypothesis testing: the research and null
More information2013 MBA Jump Start Program. Statistics Module Part 3
2013 MBA Jump Start Program Module 1: Statistics Thomas Gilbert Part 3 Statistics Module Part 3 Hypothesis Testing (Inference) Regressions 2 1 Making an Investment Decision A researcher in your firm just
More informationQuestion: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?
ECS20 Discrete Mathematics Quarter: Spring 2007 Instructor: John Steinberger Assistant: Sophie Engle (prepared by Sophie Engle) Homework 8 Hints Due Wednesday June 6 th 2007 Section 6.1 #16 What is the
More informationUNDERSTANDING THE TWO-WAY ANOVA
UNDERSTANDING THE e have seen how the one-way ANOVA can be used to compare two or more sample means in studies involving a single independent variable. This can be extended to two independent variables
More informationChapter 7 - Practice Problems 2
Chapter 7 - Practice Problems 2 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the requested value. 1) A researcher for a car insurance company
More information