ChiSquare Test. Contingency Tables. Contingency Tables. ChiSquare Test for Independence. ChiSquare Tests for GoodnessofFit


 Milo Gregory
 2 years ago
 Views:
Transcription
1 ChiSquare Tests 15 Chapter ChiSquare Test for Independence ChiSquare Tests for Goodness Uniform Goodness Poisson Goodness Goodness Test ECDF Tests (Optional) McGrawHill/Irwin Copyright 2009 by The McGrawHill Companies, Inc Contingency Tables A contingency table is a crosstabulation of n paired observations into categories. Each cell shows the count of observations that fall into the B category A defined by its row (r)( ) and column (c)( heading Contingency Tables For example: Table ChiSquare Test In a test of independence for an r x c contingency table, the hypotheses are H 0 : Variable A is independent of variable B H 1 : Variable A is not independent of variable B Use the chisquare test for independence to test these hypotheses. This nonparametric test is based on frequencies. The n data pairs are classified into c columns and r rows and then the observed frequency f jk is compared with the expected frequency e jk.
2 ChiSquare Distribution The critical value comes from the chisquare probability distribution with ν degrees of freedom. ν = degrees of freedom = (r( 1)(c 1) where r = number of rows in the table c = number of columns in the table Appendix E contains critical values for right tail areas of the chisquare distribution. The mean of a chisquare distribution is ν with variance 2ν. 2 ChiSquare Distribution Consider the shape of the chisquare distribution: Figure Expected Frequencies Assuming that H 0 is true, the expected frequency of row j and column k is: e jk = R j C k /n where R j = total for row j (j = 1, 2,, r) C k = total for column k (k = 1, 2,, c) n = sample size Steps in Testing the Hypotheses Step 1: State the Hypotheses H 0 : Variable A is independent of variable B H 1 : Variable A is not independent of variable B Step 2: Specify the Decision Rule Calculate ν = (r( 1)(c 1) For a given α,, look up the righttail tail critical value (χ( 2 R) ) from Appendix E or by using Excel. Reject H 0 if χ 2 R > test statistic
3 Steps in Testing the Hypotheses For example, for ν = 6 and α =.05, χ 2.05 = Steps in Testing the Hypotheses Here is the rejection region Figure Figure 15.3 Steps in Testing the Hypotheses Step 3: Calculate the Expected Frequencies e jk = R j C k /n For example, Steps in Testing the Hypotheses Step 4: Calculate the Test Statistic The chisquare test statistic is calc Step 5: Make the Decision Reject H 0 if χ 2 R > test statistic or if the pvalue < α
4 Small Expected Frequencies The chisquare test is unreliable if the expected frequencies are too small. Rules of thumb: Cochran s s Rule requires that e jk > 5 for all cells. Up to 20% of the cells may have e jk < 5 Most agree that a chisquare test is infeasible if e jk < 1 in any cell. If this happens, try combining adjacent rows or columns to enlarge the expected frequencies. CrossTabulating Raw Data Chisquare tests for independence can also be used to analyze quantitative variables by coding them into categories. For example, the variables Infant Deaths per 1,000 and Doctors per 100,000 can each be coded into various categories: Figure ChiSquare Test for Why Do a ChiSquare Test on Numerical Data? The researcher may believe there s s a relationship between X and Y, but doesn t want to use regression. There are outliers or anomalies that prevent us from assuming that the data came from a normal population. The researcher has numerical data for one variable but not the other. 3Way Tables and Higher More than two variables can be compared using contingency tables. However, it is difficult to visualize a higher order table. For example, you could visualize a cube as a stack of tiled 2way 2 contingency tables. Major computer packages permit 3way 3 tables
5 Purpose of the Test The goodnessof fitfit (GOF)) test helps you decide whether your sample resembles a particular kind of population. The chisquare test will be used because it is versatile and easy to understand. Multinomial GOF Test A multinomial distribution is defined by any k probabilities π 1, π 2,, π k that sum to unity. For example, consider the following official proportions of M&M colors. calc Multinomial GOF Test The hypotheses are H 0 : π 1 =.30, π 2 =.20, π 3 =.10, π 4 =.10, π 5 =.10, π 6 =.20 H 1 : At least one of the π j differs from the hypothesized value No parameters are estimated (m( = 0) and there are c = 6 classes, so the degrees of freedom are ν = c m 1 = ChiSquare Test for ChiSquare Test for ChiSquare Test for ChiSquare Test for Hypotheses for GOF The hypotheses are: H 0 : The population follows a distribution H 1 : The population does not follow a distribution The blank may contain the name of any theoretical distribution (e.g., uniform, Poisson, normal)
6 1521 Test Statistic and Degrees of Freedom for GOF ChiSquare Test for Assuming n observations, the observations are grouped into c classes and then the chi square test statistic is found using: where calc f j = the observed frequency of observations in class j e j = the expected frequency in class j if H 0 were true Test Statistic and Degrees of Freedom for GOF If the proposed distribution gives a good fit to the sample, the test statistic will be near zero. The test statistic follows the chisquare distribution with degrees of freedom ν = c m 1 where c is the no. of classes used in the test m is the no. of parameters estimated Test Statistic and Degrees of Freedom for GOF v = c m = c 0 1 = c 1 v = c m = c 1 1 = c 2 v = c m = c 2 1 = c 3 ChiSquare Test for ChiSquare Test for ChiSquare Test for DataGenerating Situations Instead of fishing for a goodfitting model, visualize a priori the characteristics of the underlying datagenerating process. Mixtures: A Problem Mixtures occur when more than one data generating process is superimposed on top of one another
7 ChiSquare Test for Eyeball Tests A simple eyeball inspection of the histogram or dot plot may suffice to rule out a hypothesized population. Small Expected Frequencies fit fit tests may lack power in small samples. As a guideline, a chi square goodnessof fit fit test should be avoided if n < 25. Uniform Uniform Distribution The uniform goodnessof fitfit test is a special case of the multinomial in which every value has the same chance of occurrence. The chisquare test for a uniform distribution compares all c groups simultaneously. The hypotheses are: H 0 : π 1 = π 2 =, π c = 1/c H 1 : Not all π j are equal Uniform Uniform Uniform GOF Test: Grouped Data The test can be performed on data that are already tabulated into groups. Calculate the expected frequency e j for each cell. The degrees of freedom are ν = c 1 since there are no parameters for the uniform distribution. Obtain the critical value χ 2 α from Appendix E for the desired level of significance α. The pvalue can be obtained from Excel. Reject H 0 if pvalue < α Uniform GOF Test: Raw Data First form c bins of equal width and create a frequency distribution. Calculate the observed frequency f j for each bin. Define e j = n/c. Perform the chisquare calculations. The degrees of freedom are ν = c 1 since there are no parameters for the uniform distribution. Obtain the critical value from Appendix E for a given significance level α and make the decision.
8 Uniform Uniform Uniform GOF Test: Raw Data Maximize the test s s power by defining bin width as As a result, the expected frequencies will be as large as possible. Uniform GOF Test: Raw Data Calculate the mean and standard deviation of the uniform distribution as: µ = (a + b)/2 σ = [(b a + 1)2 1)/12 If the data are not skewed and the sample size is large (n( > 30), then the mean is approximately normally distributed. So, test the hypothesized uniform mean using Poisson Poisson Poisson DataGenerating Situations In a Poisson distribution model, X represents the number of events per unit of time or space. X is a discrete nonnegative integer (X( = 0, 1, 2, ) Event arrivals must be independent of each other. Sometimes called a model of rare events because X typically has a small mean. Poisson The mean λ is the only parameter. Assuming that λ is unknown and must be estimated from the sample, the steps are: Step 1: Tally the observed frequency f j of each Xvalue. Step 2: Estimate the mean λ from the sample. Step 3: Use the estimated λ to find the Poisson probability P(X) ) for each value of X
9 Poisson Poisson Poisson Step 4: Multiply P(X) ) by the sample size n to get expected Poisson frequencies e j. Step 5: Perform the chisquare calculations. Step 6: Make the decision. You may need to combine classes until expected frequencies become large enough for the test (at least until e j > 2). Poisson GOF Test: Tabulated Data Calculate the sample mean as: ^λ = c Σ x j f j =1 j n Using this estimate mean, calculate the Poisson probabilities either by using the Poisson formula P(x) ) = (λ( x e λ )/x!! or Excel Poisson Poisson GOF Test: Tabulated Data For c classes with m = 1 parameter estimated, the degrees of freedom are ν = c m 1 Obtain the critical value for a given α from Appendix E. Make the decision. Normal Data Generating Situations Two parameters, µ and σ,, fully describe the normal distribution. Unless µ and σ are know a priori,, they must be estimated from a sample by using x and s. Using these statistics, the chisquare goodnessof fit fit test can be used
10 Method 1: Standardizing the Data Transform the sample observations x 1, x 2,, x n into standardized values. Count the sample observations f j within intervals of the form x + ks and compare them with the known frequencies e j based on the normal distribution. Method 1: Standardizing the Data Advantage is a standardized scale. Disadvantage is that data are no longer in the original units Figure Method 2: Equal Bin Widths To obtain equalwidth bins, divide the exact data range into c groups of equal width. Step 1: Count the sample observations in each bin to get observed frequencies f j. Step 2: Convert the bin limits into standardized zvalues z by using the formula. Method 2: Equal Bin Widths Step 3: Find the normal area within each bin assuming a normal distribution. Step 4: Find expected frequencies e j by multiplying each normal area by the sample size n. Classes may need to be collapsed from the ends inward to enlarge expected frequencies
11 1541 Method 3: Equal Expected Frequencies Define histogram bins in such a way that an equal number of observations would be expected within each bin under the null hypothesis. Define bin limits so that e j = n/c A normal area of 1/c in each of the c bins is desired. The first and last classes must be openended ended for a normal distribution, so to define c bins, we need c 1 cutpoints Method 3: Equal Expected Frequencies The upper limit of bin j can be found directly by using Excel. Alternatively, find z j for bin j using Excel and then calculate the upper limit for bin j as x + z j s Once the bins are defined, count the observations f j within each bin and compare them with the expected frequencies e j = n/c. Method 3: Equal Expected Frequencies Standard normal cutpoints for equal area bins. Histograms The fitted normal histogram gives visual clues as to the likely outcome of the GOF test. Histograms reveal any outliers or other non normality issues. Further tests are needed since histograms vary Table Figure 15.15
12 1545 Critical Values for Normal GOF Test Since two parameters, m and s, are estimated from the sample, the degrees of freedom are ν = c m 1 Table At least 4 bins are needed to ensure 1 df ECDF Tests KolmogorovSmirnov and Lilliefors Tests There are many alternatives to the chisquare test based on the Empirical Cumulative Distribution Function (ECDF). The KolmogorovSmirnov (KS) test statistic D is the largest absolute difference between the actual and expected cumulative relative frequency of the n data values: D = Max F a F e The KS K S test is not recommended for grouped data. ECDF Tests KolmogorovSmirnov and Lilliefors Tests F a is the actual cumulative frequency at observation i. F e is the expected cumulative frequency at observation i under the assumption that the data came from the hypothesized distribution. The KS K S test assumes that no parameters are estimated. If parameters are estimated, use a Lilliefors test. Both of these tests are done by computer. ECDF Tests KolmogorovSmirnov and Lilliefors Tests KS test for uniformity. Figure
13 1549 ECDF Tests KolmogorovSmirnov and Lilliefors Tests KS test for normality. Figure ECDF Tests AndersonDarling Tests The AndersonDarling (AD) test is widely used for nonnormality normality because of its power. The AD A D test is based on a probability plot. When the data fit the hypothesized distribution closely, the probability plot will be close to a straight line. The AD A D test statistic measures the overall distance between the actual and the hypothesized distributions, using a weighted squared distance. ECDF Tests AndersonDarling Tests with MINITAB Applied Statistics in Business & Economics End of Chapter 15 Figure McGrawHill/Irwin Copyright 2009 by The McGrawHill Companies, Inc.
12.5: CHISQUARE GOODNESS OF FIT TESTS
125: ChiSquare Goodness of Fit Tests CD121 125: CHISQUARE GOODNESS OF FIT TESTS In this section, the χ 2 distribution is used for testing the goodness of fit of a set of data to a specific probability
More informationNormality Testing in Excel
Normality Testing 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
More informationMinitab Guide. This packet contains: A Friendly Guide to Minitab. Minitab StepByStep
Minitab Guide This packet contains: A Friendly Guide to Minitab An introduction to Minitab; including basic Minitab functions, how to create sets of data, and how to create and edit graphs of different
More informationComputer Lab 3 Thursday, 24 February, 2011 DMS 106 4:00 5:15PM
Statistics: Continuous Methods STAT452/652, Spring 2011 Computer Lab 3 Thursday, 24 February, 2011 DMS 106 4:00 5:15PM Goodness of Fit tests: Chisquare, KolmogorovSmirnov, AndersonDarling, ShapiroWilk
More informationTechnology StepbyStep Using StatCrunch
Technology StepbyStep Using StatCrunch Section 1.3 Simple Random Sampling 1. Select Data, highlight Simulate Data, then highlight Discrete Uniform. 2. Fill in the following window with the appropriate
More informationChapter Additional: Standard Deviation and Chi Square
Chapter Additional: Standard Deviation and Chi Square Chapter Outline: 6.4 Confidence Intervals for the Standard Deviation 7.5 Hypothesis testing for Standard Deviation Section 6.4 Objectives Interpret
More informationThe GoodnessofFit Test
on the Lecture 49 Section 14.3 HampdenSydney College Tue, Apr 21, 2009 Outline 1 on the 2 3 on the 4 5 Hypotheses on the (Steps 1 and 2) (1) H 0 : H 1 : H 0 is false. (2) α = 0.05. p 1 = 0.24 p 2 = 0.20
More informationComparing Multiple Proportions, Test of Independence and Goodness of Fit
Comparing Multiple Proportions, Test of Independence and Goodness of Fit Content Testing the Equality of Population Proportions for Three or More Populations Test of Independence Goodness of Fit Test 2
More informationHypothesis Testing COMP 245 STATISTICS. Dr N A Heard. 1 Hypothesis Testing 2 1.1 Introduction... 2 1.2 Error Rates and Power of a Test...
Hypothesis Testing COMP 45 STATISTICS Dr N A Heard Contents 1 Hypothesis Testing 1.1 Introduction........................................ 1. Error Rates and Power of a Test.............................
More informationCalculating PValues. Parkland College. Isela Guerra Parkland College. Recommended Citation
Parkland College A with Honors Projects Honors Program 2014 Calculating PValues Isela Guerra Parkland College Recommended Citation Guerra, Isela, "Calculating PValues" (2014). A with Honors Projects.
More informationt Tests in Excel The Excel Statistical Master By Mark Harmon Copyright 2011 Mark Harmon
ttests 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 information93.4 Likelihood ratio test. NeymanPearson lemma
93.4 Likelihood ratio test NeymanPearson lemma 91 Hypothesis Testing 91.1 Statistical Hypotheses Statistical hypothesis testing and confidence interval estimation of parameters are the fundamental
More informationAdditional sources Compilation of sources: http://lrs.ed.uiuc.edu/tseportal/datacollectionmethodologies/jintselink/tselink.htm
Mgt 540 Research Methods Data Analysis 1 Additional sources Compilation of sources: http://lrs.ed.uiuc.edu/tseportal/datacollectionmethodologies/jintselink/tselink.htm http://web.utk.edu/~dap/random/order/start.htm
More informationChapter 3 RANDOM VARIATE GENERATION
Chapter 3 RANDOM VARIATE GENERATION In order to do a Monte Carlo simulation either by hand or by computer, techniques must be developed for generating values of random variables having known distributions.
More informationStatistics 641  EXAM II  1999 through 2003
Statistics 641  EXAM II  1999 through 2003 December 1, 1999 I. (40 points ) Place the letter of the best answer in the blank to the left of each question. (1) In testing H 0 : µ 5 vs H 1 : µ > 5, the
More information3.6: General Hypothesis Tests
3.6: General Hypothesis Tests The χ 2 goodness of fit tests which we introduced in the previous section were an example of a hypothesis test. In this section we now consider hypothesis tests more generally.
More informationINTERPRETING THE ONEWAY ANALYSIS OF VARIANCE (ANOVA)
INTERPRETING THE ONEWAY ANALYSIS OF VARIANCE (ANOVA) As with other parametric statistics, we begin the oneway ANOVA with a test of the underlying assumptions. Our first assumption is the assumption of
More information1 SAMPLE SIGN TEST. NonParametric Univariate Tests: 1 Sample Sign Test 1. A nonparametric equivalent of the 1 SAMPLE TTEST.
NonParametric Univariate Tests: 1 Sample Sign Test 1 1 SAMPLE SIGN TEST A nonparametric equivalent of the 1 SAMPLE TTEST. ASSUMPTIONS: Data is nonnormally distributed, even after log transforming.
More informationChapter 23. Two Categorical Variables: The ChiSquare Test
Chapter 23. Two Categorical Variables: The ChiSquare Test 1 Chapter 23. Two Categorical Variables: The ChiSquare Test TwoWay Tables Note. We quickly review twoway tables with an example. Example. Exercise
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 informationUNDERSTANDING THE INDEPENDENTSAMPLES t TEST
UNDERSTANDING The independentsamples t test evaluates the difference between the means of two independent or unrelated groups. That is, we evaluate whether the means for two independent groups are significantly
More information13.2 The Chi Square Test for Homogeneity of Populations The setting: Used to compare distribution of proportions in two or more populations.
13.2 The Chi Square Test for Homogeneity of Populations The setting: Used to compare distribution of proportions in two or more populations. Data is organized in a two way table Explanatory variable (Treatments)
More informationChapter 9, Part A Hypothesis Tests. Learning objectives
Chapter 9, Part A Hypothesis Tests Slide 1 Learning objectives 1. Understand how to develop Null and Alternative Hypotheses 2. Understand Type I and Type II Errors 3. Able to do hypothesis test about population
More informationClass 19: Two Way Tables, Conditional Distributions, ChiSquare (Text: Sections 2.5; 9.1)
Spring 204 Class 9: Two Way Tables, Conditional Distributions, ChiSquare (Text: Sections 2.5; 9.) Big Picture: More than Two Samples In Chapter 7: We looked at quantitative variables and compared the
More informationChi Square for Contingency Tables
2 x 2 Case Chi Square for Contingency Tables A test for p 1 = p 2 We have learned a confidence interval for p 1 p 2, the difference in the population proportions. We want a hypothesis testing procedure
More informationThe Chi Square Test. Diana Mindrila, Ph.D. Phoebe Balentyne, M.Ed. Based on Chapter 23 of The Basic Practice of Statistics (6 th ed.
The Chi Square Test Diana Mindrila, Ph.D. Phoebe Balentyne, M.Ed. Based on Chapter 23 of The Basic Practice of Statistics (6 th ed.) Concepts: TwoWay Tables The Problem of Multiple Comparisons Expected
More informationFoundation of Quantitative Data Analysis
Foundation of Quantitative Data Analysis Part 1: Data manipulation and descriptive statistics with SPSS/Excel HSRS #10  October 17, 2013 Reference : A. Aczel, Complete Business Statistics. Chapters 1
More informationStatistical Testing of Randomness Masaryk University in Brno Faculty of Informatics
Statistical Testing of Randomness Masaryk University in Brno Faculty of Informatics Jan Krhovják Basic Idea Behind the Statistical Tests Generated random sequences properties as sample drawn from uniform/rectangular
More information13 id id no. of respondents 101300 4 respon 1 responsible for maintenance? 1 = no, 2 = yes, 9 = blank
Basic Data Analysis Graziadio School of Business and Management Data Preparation & Entry Editing: Inspection & Correction Field Edit: Immediate followup (complete? legible? comprehensible? consistent?
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 information2. DATA AND EXERCISES (Geos2911 students please read page 8)
2. DATA AND EXERCISES (Geos2911 students please read page 8) 2.1 Data set The data set available to you is an Excel spreadsheet file called cyclones.xls. The file consists of 3 sheets. Only the third is
More informationBowerman, O'Connell, Aitken Schermer, & Adcock, Business Statistics in Practice, Canadian edition
Bowerman, O'Connell, Aitken Schermer, & Adcock, Business Statistics in Practice, Canadian edition Online Learning Centre Technology StepbyStep  Excel Microsoft Excel is a spreadsheet software application
More informationInferential Statistics
Inferential Statistics Sampling and the normal distribution Zscores Confidence levels and intervals Hypothesis testing Commonly used statistical methods Inferential Statistics Descriptive statistics are
More information4) The goodness of fit test is always a one tail test with the rejection region in the upper tail. Answer: TRUE
Business Statistics, 9e (Groebner/Shannon/Fry) Chapter 13 Goodness of Fit Tests and Contingency Analysis 1) A goodness of fit test can be used to determine whether a set of sample data comes from a specific
More informationChiSquare Tests. In This Chapter BONUS CHAPTER
BONUS CHAPTER ChiSquare Tests In the previous chapters, we explored the wonderful world of hypothesis testing as we compared means and proportions of one, two, three, and more populations, making an educated
More informationFirstyear Statistics for Psychology Students Through Worked Examples
Firstyear Statistics for Psychology Students Through Worked Examples 1. THE CHISQUARE TEST A test of association between categorical variables by Charles McCreery, D.Phil Formerly Lecturer in Experimental
More informationModule 9: Nonparametric Tests. The Applied Research Center
Module 9: Nonparametric Tests The Applied Research Center Module 9 Overview } Nonparametric Tests } Parametric vs. Nonparametric Tests } Restrictions of Nonparametric Tests } OneSample ChiSquare Test
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 informationChisquare test Fisher s Exact test
Lesson 1 Chisquare test Fisher s Exact test McNemar s Test Lesson 1 Overview Lesson 11 covered two inference methods for categorical data from groups Confidence Intervals for the difference of two proportions
More informationCHISQUARE: TESTING FOR GOODNESS OF FIT
CHISQUARE: TESTING FOR GOODNESS OF FIT In the previous chapter we discussed procedures for fitting a hypothesized function to a set of experimental data points. Such procedures involve minimizing a quantity
More informationMeasuring the Power of a Test
Textbook Reference: Chapter 9.5 Measuring the Power of a Test An economic problem motivates the statement of a null and alternative hypothesis. For a numeric data set, a decision rule can lead to the rejection
More informationMINITAB ASSISTANT WHITE PAPER
MINITAB ASSISTANT WHITE PAPER This paper explains the research conducted by Minitab statisticians to develop the methods and data checks used in the Assistant in Minitab 17 Statistical Software. OneWay
More informationCHAPTER 11. GOODNESS OF FIT AND CONTINGENCY TABLES
CHAPTER 11. GOODNESS OF FIT AND CONTINGENCY TABLES The chisquare distribution was discussed in Chapter 4. We now turn to some applications of this distribution. As previously discussed, chisquare is
More information1. What is the critical value for this 95% confidence interval? CV = z.025 = invnorm(0.025) = 1.96
1 Final Review 2 Review 2.1 CI 1propZint Scenario 1 A TV manufacturer claims in its warranty brochure that in the past not more than 10 percent of its TV sets needed any repair during the first two years
More informationUnit 29 ChiSquare GoodnessofFit Test
Unit 29 ChiSquare GoodnessofFit Test Objectives: To perform the chisquare hypothesis test concerning proportions corresponding to more than two categories of a qualitative variable To perform the Bonferroni
More informationRecall this chart that showed how most of our course would be organized:
Chapter 4 OneWay 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 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 informationVariables Control Charts
MINITAB ASSISTANT WHITE PAPER This paper explains the research conducted by Minitab statisticians to develop the methods and data checks used in the Assistant in Minitab 17 Statistical Software. Variables
More informationNCSS Statistical Software. OneSample TTest
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 informationRandom Uniform Clumped. 0 1 2 3 4 5 6 7 8 9 Number of Individuals per SubQuadrat. Number of Individuals per SubQuadrat
41 Population ecology Lab 4: Population dispersion patterns I. Introduction to population dispersion patterns The dispersion of individuals in a population describes their spacing relative to each other.
More informationThe ChiSquare Test. STAT E50 Introduction to Statistics
STAT 50 Introduction to Statistics The ChiSquare Test The Chisquare test is a nonparametric test that is used to compare experimental results with theoretical models. That is, we will be comparing observed
More informationBusiness Statistics. Successful completion of Introductory and/or Intermediate Algebra courses is recommended before taking Business Statistics.
Business Course Text Bowerman, Bruce L., Richard T. O'Connell, J. B. Orris, and Dawn C. Porter. Essentials of Business, 2nd edition, McGrawHill/Irwin, 2008, ISBN: 9780073319889. Required Computing
More informationNotes for STA 437/1005 Methods for Multivariate Data
Notes for STA 437/1005 Methods for Multivariate Data Radford M. Neal, 26 November 2010 Random Vectors Notation: Let X be a random vector with p elements, so that X = [X 1,..., X p ], where denotes transpose.
More informationMT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo. 3 MT426 Notebook 3 3. 3.1 Definitions... 3. 3.2 Joint Discrete Distributions...
MT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo c Copyright 20042012 by Jenny A. Baglivo. All Rights Reserved. Contents 3 MT426 Notebook 3 3 3.1 Definitions............................................
More informationDescriptive Statistics
Y520 Robert S Michael Goal: Learn to calculate indicators and construct graphs that summarize and describe a large quantity of values. Using the textbook readings and other resources listed on the web
More informationPASS Sample Size Software
Chapter 250 Introduction The Chisquare test is often used to test whether sets of frequencies or proportions follow certain patterns. The two most common instances are tests of goodness of fit using multinomial
More informationBivariate Statistics Session 2: Measuring Associations ChiSquare Test
Bivariate Statistics Session 2: Measuring Associations ChiSquare Test Features Of The ChiSquare Statistic The chisquare test is nonparametric. That is, it makes no assumptions about the distribution
More informationSTART Selected Topics in Assurance
START Selected Topics in Assurance Related Technologies Table of Contents Introduction Some Statistical Background Fitting a Normal Using the Anderson Darling GoF Test Fitting a Weibull Using the Anderson
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 informationChapter 8 Hypothesis Testing
Chapter 8 Hypothesis Testing Chapter problem: Does the MicroSort method of gender selection increase the likelihood that a baby will be girl? MicroSort: a genderselection method developed by Genetics
More informationData Analysis Tools. Tools for Summarizing Data
Data Analysis Tools This section of the notes is meant to introduce you to many of the tools that are provided by Excel under the Tools/Data Analysis menu item. If your computer does not have that tool
More informationHow to Conduct a Hypothesis Test
How to Conduct a Hypothesis Test The idea of hypothesis testing is relatively straightforward. In various studies we observe certain events. We must ask, is the event due to chance alone, or is there some
More informationChapter Five: Paired Samples Methods 1/38
Chapter Five: Paired Samples Methods 1/38 5.1 Introduction 2/38 Introduction Paired data arise with some frequency in a variety of research contexts. Patients might have a particular type of laser surgery
More informationCalculate Confidence Intervals Using the TI Graphing Calculator
Calculate Confidence Intervals Using the TI Graphing Calculator Confidence Interval for Population Proportion p Confidence Interval for Population μ (σ is known 1 Select: STAT / TESTS / 1PropZInt x: number
More informationbusiness statistics using Excel OXFORD UNIVERSITY PRESS Glyn Davis & Branko Pecar
business statistics using Excel Glyn Davis & Branko Pecar OXFORD UNIVERSITY PRESS Detailed contents Introduction to Microsoft Excel 2003 Overview Learning Objectives 1.1 Introduction to Microsoft Excel
More information3.4 Statistical inference for 2 populations based on two samples
3.4 Statistical inference for 2 populations based on two samples Tests for a difference between two population means The first sample will be denoted as X 1, X 2,..., X m. The second sample will be denoted
More informationCHAPTER 11 CHISQUARE: NONPARAMETRIC COMPARISONS OF FREQUENCY
CHAPTER 11 CHISQUARE: NONPARAMETRIC COMPARISONS OF FREQUENCY The hypothesis testing statistics detailed thus far in this text have all been designed to allow comparison of the means of two or more samples
More informationNPTEL STRUCTURAL RELIABILITY
NPTEL Course On STRUCTURAL RELIABILITY Module # 02 Lecture 6 Course Format: Web Instructor: Dr. Arunasis Chakraborty Department of Civil Engineering Indian Institute of Technology Guwahati 6. Lecture 06:
More information1. Comparing Two Means: Dependent Samples
1. Comparing Two Means: ependent Samples In the preceding lectures we've considered how to test a difference of two means for independent samples. Now we look at how to do the same thing with dependent
More information3. Nonparametric methods
3. Nonparametric methods If the probability distributions of the statistical variables are unknown or are not as required (e.g. normality assumption violated), then we may still apply nonparametric tests
More informationExercise 1.12 (Pg. 2223)
Individuals: The objects that are described by a set of data. They may be people, animals, things, etc. (Also referred to as Cases or Records) Variables: The characteristics recorded about each individual.
More informationIntroduction to Hypothesis Testing. Point estimation and confidence intervals are useful statistical inference procedures.
Introduction to Hypothesis Testing Point estimation and confidence intervals are useful statistical inference procedures. Another type of inference is used frequently used concerns tests of hypotheses.
More informationSPSS for Exploratory Data Analysis Data used in this guide: studentp.sav (http://people.ysu.edu/~gchang/stat/studentp.sav)
Data used in this guide: studentp.sav (http://people.ysu.edu/~gchang/stat/studentp.sav) Organize and Display One Quantitative Variable (Descriptive Statistics, Boxplot & Histogram) 1. Move the mouse pointer
More information7 Hypothesis testing  one sample tests
7 Hypothesis testing  one sample tests 7.1 Introduction Definition 7.1 A hypothesis is a statement about a population parameter. Example A hypothesis might be that the mean age of students taking MAS113X
More informationTHE KRUSKAL WALLLIS TEST
THE KRUSKAL WALLLIS TEST TEODORA H. MEHOTCHEVA Wednesday, 23 rd April 08 THE KRUSKALWALLIS TEST: The nonparametric alternative to ANOVA: testing for difference between several independent groups 2 NON
More informationBasic concepts and introduction to statistical inference
Basic concepts and introduction to statistical inference Anna Helga Jonsdottir Gunnar Stefansson Sigrun Helga Lund University of Iceland (UI) Basic concepts 1 / 19 A review of concepts Basic concepts Confidence
More informationOdds ratio, Odds ratio test for independence, chisquared statistic.
Odds ratio, Odds ratio test for independence, chisquared statistic. Announcements: Assignment 5 is live on webpage. Due Wed Aug 1 at 4:30pm. (9 days, 1 hour, 58.5 minutes ) Final exam is Aug 9. Review
More informationDescriptive Statistics. Understanding Data: Categorical Variables. Descriptive Statistics. Dataset: Shellfish Contamination
Descriptive Statistics Understanding Data: Dataset: Shellfish Contamination Location Year Species Species2 Method Metals Cadmium (mg kg  ) Chromium (mg kg  ) Copper (mg kg  ) Lead (mg kg  ) Mercury
More informationCourse Text. Required Computing Software. Course Description. Course Objectives. StraighterLine. Business Statistics
Course Text Business Statistics Lind, Douglas A., Marchal, William A. and Samuel A. Wathen. Basic Statistics for Business and Economics, 7th edition, McGrawHill/Irwin, 2010, ISBN: 9780077384470 [This
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 informationIntroduction to Quantitative Methods
Introduction to Quantitative Methods October 15, 2009 Contents 1 Definition of Key Terms 2 2 Descriptive Statistics 3 2.1 Frequency Tables......................... 4 2.2 Measures of Central Tendencies.................
More informationBIOSTATISTICS QUIZ ANSWERS
BIOSTATISTICS QUIZ ANSWERS 1. When you read scientific literature, do you know whether the statistical tests that were used were appropriate and why they were used? a. Always b. Mostly c. Rarely d. Never
More informationElementary Statistics Sample Exam #3
Elementary Statistics Sample Exam #3 Instructions. No books or telephones. Only the supplied calculators are allowed. The exam is worth 100 points. 1. A chi square goodness of fit test is considered to
More informationTutorial 5: Hypothesis Testing
Tutorial 5: Hypothesis Testing Rob Nicholls nicholls@mrclmb.cam.ac.uk MRC LMB Statistics Course 2014 Contents 1 Introduction................................ 1 2 Testing distributional assumptions....................
More informationIntroduction to Analysis of Variance (ANOVA) Limitations of the ttest
Introduction to Analysis of Variance (ANOVA) The Structural Model, The Summary Table, and the One Way ANOVA Limitations of the ttest Although the ttest is commonly used, it has limitations Can only
More informationTABLE OF CONTENTS. About Chi Squares... 1. What is a CHI SQUARE?... 1. Chi Squares... 1. Hypothesis Testing with Chi Squares... 2
About Chi Squares TABLE OF CONTENTS About Chi Squares... 1 What is a CHI SQUARE?... 1 Chi Squares... 1 Goodness of fit test (Oneway χ 2 )... 1 Test of Independence (Twoway χ 2 )... 2 Hypothesis Testing
More informationCHAPTER 11 CHISQUARE AND F DISTRIBUTIONS
CHAPTER 11 CHISQUARE AND F DISTRIBUTIONS CHISQUARE TESTS OF INDEPENDENCE (SECTION 11.1 OF UNDERSTANDABLE STATISTICS) In chisquare tests of independence we use the hypotheses. H0: The variables are independent
More informationDescriptive Statistics
Descriptive Statistics Suppose following data have been collected (heights of 99 fiveyearold boys) 117.9 11.2 112.9 115.9 18. 14.6 17.1 117.9 111.8 16.3 111. 1.4 112.1 19.2 11. 15.4 99.4 11.1 13.3 16.9
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 informationMCQ TESTING OF HYPOTHESIS
MCQ TESTING OF HYPOTHESIS MCQ 13.1 A statement about a population developed for the purpose of testing is called: (a) Hypothesis (b) Hypothesis testing (c) Level of significance (d) Teststatistic MCQ
More informationMBA 611 STATISTICS AND QUANTITATIVE METHODS
MBA 611 STATISTICS AND QUANTITATIVE METHODS Part I. Review of Basic Statistics (Chapters 111) A. Introduction (Chapter 1) Uncertainty: Decisions are often based on incomplete information from uncertain
More informationPoisson Models for Count Data
Chapter 4 Poisson Models for Count Data In this chapter we study loglinear models for count data under the assumption of a Poisson error structure. These models have many applications, not only to the
More informationModule 5 Hypotheses Tests: Comparing Two Groups
Module 5 Hypotheses Tests: Comparing Two Groups Objective: In medical research, we often compare the outcomes between two groups of patients, namely exposed and unexposed groups. At the completion of this
More informationtests whether there is an association between the outcome variable and a predictor variable. In the Assistant, you can perform a ChiSquare Test for
This paper explains the research conducted by Minitab statisticians to develop the methods and data checks used in the Assistant in Minitab 17 Statistical Software. In practice, quality professionals sometimes
More informationHow Does My TI84 Do That
How Does My TI84 Do That A guide to using the TI84 for statistics Austin Peay State University Clarksville, Tennessee How Does My TI84 Do That A guide to using the TI84 for statistics Table of Contents
More informationFinal Exam Practice Problem Answers
Final Exam Practice Problem Answers The following data set consists of data gathered from 77 popular breakfast cereals. The variables in the data set are as follows: Brand: The brand name of the cereal
More informationChapter 2  Graphical Summaries of Data
Chapter 2  Graphical Summaries of Data Data recorded in the sequence in which they are collected and before they are processed or ranked are called raw data. Raw data is often difficult to make sense
More informationTwo Correlated Proportions (McNemar Test)
Chapter 50 Two Correlated Proportions (Mcemar Test) Introduction This procedure computes confidence intervals and hypothesis tests for the comparison of the marginal frequencies of two factors (each with
More informationNonparametric TwoSample Tests. Nonparametric Tests. Sign Test
Nonparametric TwoSample Tests Sign test MannWhitney Utest (a.k.a. Wilcoxon twosample test) KolmogorovSmirnov Test Wilcoxon SignedRank Test TukeyDuckworth Test 1 Nonparametric Tests Recall, nonparametric
More information" Y. Notation and Equations for Regression Lecture 11/4. Notation:
Notation: Notation and Equations for Regression Lecture 11/4 m: The number of predictor variables in a regression Xi: One of multiple predictor variables. The subscript i represents any number from 1 through
More information