# Chi-square test Testing for independeny The r x c contingency tables square test

Save this PDF as:

Size: px
Start display at page:

Download "Chi-square test Testing for independeny The r x c contingency tables square test"

## Transcription

1 Chi-square test Testing for independeny The r x c contingency tables square test 1

2 The chi-square distribution HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach

3 Example A study was carried out to investigate the proportion of persons getting influenza vary according to the type of vaccine. Given below is a 3 x table of observed frequencies showing the number of persons who did or did not get influenza after inoculation with one of three vaccines. Does proportion of getting influenza depend on the type of vaccine? Type of vaccine Number getting influenza Number not getting influenza Total Seasonal only 43 (15.35%) (100%) H1N1 only 5 (0.8%) (100%) Combined 5 (9.%) (100%) Totals HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 3

4 Test of independence In biology the most common application for chi-squared is in comparing observed counts of particular cases to the expected counts. A total of n experiments may have been performed whose results are characterized by the values of two random variable X and Y. We assume that the variables are discrete and the values of X and Y are x 1, x,...,x r and y 1, y,...,y c, respectively, which are the outcomes of the events A 1,A,...,A r and B 1, B,...,B c. Let's denote by k ij the number of the outcomes of the event (A i, B j ). These numbers can be grouped into a matrix, called a contingency table. It has the following form: HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 4

5 Contingency table B 1 B... B c Total A 1 k 11 k 1... k 1c k 1 + A k 1 k... k c k Frequency of A i event i1,, r c k i + k ij j 1 A r k r1 k r... k rc k r + Total k + 1 k +... k +c n Frequency of B i event j1,, c k r + j k ij i 1 HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 5

6 Chi-square test (Pearson) H 0 : The two variables are independent. Mathematically: P(A i B j ) P (A i ) P( B j ) Test statistic: χ r s ( k ij k i 1 j 1 + If H 0 is true, then χ has asymptotically χ distribution with (r-1)(c-1) degrees of freedom., namely (number of rows -1)(number of columns - ) Decision: if χ > χ table then we reject the null hypothesis that the two variables are independent, in the k + i + n k n k j + j ) opposite case we do not reject the null hypothesis. HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 6

7 Observed and expected frequencies B 1 B... B j... B c Total A 1 k 11 k 1... k 1j... k 1c k 1 + A k 1 k... k j... k c k A i k i1 k i... k ij... k ic k i A r k r1 k r... k rj... k rc k r + χ r s r i 1 j 1 s ij ij ij ij i + i 1 j 1 i + + j ( O ( k E E k k n ) n k k + j ) Total k + 1 k +... k + j... k + c n Observed (O ij ) k ij Expected (E ij ): Row total*column total/n HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 7 k i + k+ n j

8 Example A study was carried out to investigate the proportion of persons getting influenza vary according to the type of vaccine. Given below is a 3 x table of observed frequencies showing the number of persons who did or did not get influenza after inoculation with one of three vaccines. Type of vaccine Number getting influenza Number not getting influenza Total Seasonal only H1N1 only Combined Totals There are two categorical variables (type of vaccine, getting influenza) H 0 : The two variables are independent proportions getting influenza are the same for each vaccine HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 8

9 Type of vaccine Calculation of the test statistic Observed frequencies Number getting influenza Number not getting influenza Tot al Type of vaccine Expected frequencies Number getting influenza Number not getting influenza Tot al Seasonal only H1N1 only Combined Totals Seasonal only H1N1 only Combined Totals χ ki+ k+ j ( kij ) n (43 4) (37 38) (5 37.5) ( ) (5 40.5) (45 9.5) ki + k j n r s i 1 j 1 + χ² χ Degrees of freedom: {(r 1 )(c 1 )} (-1)*(3-1) Here χ > χ table 5.991; (df;α0.05). We reject the null hypothesis We conclude that the proportions getting influenza are not the same for each type of vaccine HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 9

10 Assumption of the chi-square test Expected frequencies should be big enough The number of cells with expected frequencies < 5 can be maximum 0% of the cells. For example, in case of 6 cells, expected frequencies <5 can be in maximum 1 cell (0% of 6 is 1.) HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 10

11 SPSS results Chi-Square Tests Pearson Chi-Square Likelihood Ratio Linear-by-Linear Association N of Valid Cases a. χ² and p0.001 Asymp. Sig. Value df (-sided) 13,603 a,001 13,941,001 3,878 1, cells (,0%) have expected count less than 5. The minimum expected count is 37,50. Here p0.001 < α0.05 we reject the null hypothesis. We conclude that the proportions getting influenza are not the same for each type of vaccine HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 11

12 The chi-square test for goodness of fit Goodness of fit tests are used to determine whether sample observation fall into categories in the way they "should" according to some ideal model. When they come out as expected, we say that the data fit the model. The chi-square statistic helps us to decide whether the fit of the data to the model is good. H0: the distribution of the variable X is a given distribution HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 1

13 The distribution of the sample depending on the type of the variable Categorical variable. Example. A dice is thrown 10 times. We would like to test whether the dice is fair or biased. Observed frequencies Continuous variable Example. We would like to test whether the sample is drawn from a normally distributed population. Distribution of ages HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 13

14 Suppose we have a sample of n observations. Let's prepare a bar chart or a histogram of the sample depending on the type of the variable. In both cases, we have frequencies of categories or frequencies in the interval. Let's denote the frequency in the i-th category or interval by k i, i1,,,r (r is the number of categories). Let's denote p i the probabilities of falling into a given category or interval in the case of the given distribution. If H0 is true and n is large, then the relative frequencies are approximations of p i -s, or. ki n p i k np The formula of the test statistic has χ distribution with (r-1-s) degrees of freedom. Here s is the number Observed of the parameters frequency of the Expected distribution frequency (if there are). i i X r r ( Oi Ei ) ( ki n pi ) i 1 Ei i 1 n pi HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 14

15 Test for uniform distribution Example. We would like to test whether a dice is fair or biased. The dice is thrown 10 times. H0: the dice is fair, the probability of each category, p i 1/6. Calculation of expected frequencies: n p i 10 1/6 0. If it is fair, every throwing are equally probable so in ideal case we would expect 0 frequencies for each number Observed frequencies Expected frequencies X 6 ( i 1 k i 0) 0 1 [(5 0) + (18 0) + (1 0) 0 1 ( ) ) + (0 0) + (19 0) The degrees of freedom is 5, the critical value in the table is As our test statistic, < we do not reject H0 and claim that the dice is fair. HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 15

16 Test for uniform distribution Example. We would like to test whether a dice is fair or biased. The dice is thrown 10 times. H0: the dice is fair, the probability of each category, p i 1/6. Calculation of expected frequencies: n p i 10 1/6 0. If it is fair, every throwing are equally probable so in ideal case we would expect 0 frequencies for each number Observed frequencies Expected frequencies X 6 ( i 1 k i 0) 0 1 [(5 0) + (18 0) + (1 0) 0 1 ( ) (17 0) + (0 0) + (39 0) The degrees of freedom is 5, the critical value in the table is As our test statistic, 30 > we reject H0 and claim that the dice is not fair. HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 16

17 Test for normality Let's suppose we have a sample and would like to know whether it comes from a normally distributed population. H0: the sample is drawn from a normally distributed population. Let's make a histogram from the sample, so we get the "observed" frequencies. To test the null hypothesis we need the expected frequencies. We have to estimate the parameters of the normal density functions. We use the sample mean and sample standard deviation. The expected frequencies can be computed using the tables of the normal distribution HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 17

18 30 Body height X r ki npi ( ) np i 1 i 0 10 k i Frequency Std. Dev 8.5 Mean N np i Body height HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 18

19 Using Gauss-paper There is a graphical method to check normality. The "Gauss-paper" is a special coordinate system, the tick marks of the y axis are the inverse of the normal distribution and are given in percentages. We simply have to draw the distribution function of the sample into this paper. In the case of normality the points are arranged approximately in a straight line. HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 19

20 SPSS: Q-Q plot (quantile-quantile plot) HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 0

### PASS Sample Size Software

Chapter 250 Introduction The Chi-square 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

### Data Mining Techniques Chapter 5: The Lure of Statistics: Data Mining Using Familiar Tools

Data Mining Techniques Chapter 5: The Lure of Statistics: Data Mining Using Familiar Tools Occam s razor.......................................................... 2 A look at data I.........................................................

### Chapter 23. Two Categorical Variables: The Chi-Square Test

Chapter 23. Two Categorical Variables: The Chi-Square Test 1 Chapter 23. Two Categorical Variables: The Chi-Square Test Two-Way Tables Note. We quickly review two-way tables with an example. Example. Exercise

### The Chi-Square Test. STAT E-50 Introduction to Statistics

STAT -50 Introduction to Statistics The Chi-Square Test The Chi-square test is a nonparametric test that is used to compare experimental results with theoretical models. That is, we will be comparing observed

### Statistical Study: Comparing Fast Food Consumption to Body Mass Index

Statistical Study: Comparing Fast Food Consumption to Body Mass Index Janet Hiestan Biology major Jennifer Sanchez Biology major Jason Majirsky Biology major Introduction: Does fast food consumption influence

### Projects Involving Statistics (& SPSS)

Projects Involving Statistics (& SPSS) Academic Skills Advice Starting a project which involves using statistics can feel confusing as there seems to be many different things you can do (charts, graphs,

### CHAPTER 11 CHI-SQUARE: NON-PARAMETRIC COMPARISONS OF FREQUENCY

CHAPTER 11 CHI-SQUARE: NON-PARAMETRIC 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

### SPSS 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

### Technology Step-by-Step Using StatCrunch

Technology Step-by-Step 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

### Comparing 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

### SPSS: Expected frequencies, chi-squared test. In-depth example: Age groups and radio choices. Dealing with small frequencies.

SPSS: Expected frequencies, chi-squared test. In-depth example: Age groups and radio choices. Dealing with small frequencies. Quick Example: Handedness and Careers Last time we tested whether one nominal

### CHAPTER 7 INTRODUCTION TO SAMPLING DISTRIBUTIONS

CHAPTER 7 INTRODUCTION TO SAMPLING DISTRIBUTIONS CENTRAL LIMIT THEOREM (SECTION 7.2 OF UNDERSTANDABLE STATISTICS) The Central Limit Theorem says that if x is a random variable with any distribution having

### Recall 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

### Chapter 5 Analysis of variance SPSS Analysis of variance

Chapter 5 Analysis of variance SPSS Analysis of variance Data file used: gss.sav How to get there: Analyze Compare Means One-way ANOVA To test the null hypothesis that several population means are equal,

### Data Analysis. Lecture Empirical Model Building and Methods (Empirische Modellbildung und Methoden) SS Analysis of Experiments - Introduction

Data Analysis Lecture Empirical Model Building and Methods (Empirische Modellbildung und Methoden) Prof. Dr. Dr. h.c. Dieter Rombach Dr. Andreas Jedlitschka SS 2014 Analysis of Experiments - Introduction

### Chapter 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.

### NCSS Statistical Software Principal Components Regression. In ordinary least squares, the regression coefficients are estimated using the formula ( )

Chapter 340 Principal Components Regression Introduction is a technique for analyzing multiple regression data that suffer from multicollinearity. When multicollinearity occurs, least squares estimates

### Inferential Statistics

Inferential Statistics Sampling and the normal distribution Z-scores Confidence levels and intervals Hypothesis testing Commonly used statistical methods Inferential Statistics Descriptive statistics are

### CATEGORICAL DATA Chi-Square Tests for Univariate Data

CATEGORICAL DATA Chi-Square Tests For Univariate Data 1 CATEGORICAL DATA Chi-Square Tests for Univariate Data Recall that a categorical variable is one in which the possible values are categories or groupings.

### TABLE 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 (One-way χ 2 )... 1 Test of Independence (Two-way χ 2 )... 2 Hypothesis Testing

### CHAPTER IV FINDINGS AND CONCURRENT DISCUSSIONS

CHAPTER IV FINDINGS AND CONCURRENT DISCUSSIONS Hypothesis 1: People are resistant to the technological change in the security system of the organization. Hypothesis 2: information hacked and misused. Lack

### Recommend Continued CPS Monitoring. 63 (a) 17 (b) 10 (c) 90. 35 (d) 20 (e) 25 (f) 80. Totals/Marginal 98 37 35 170

Work Sheet 2: Calculating a Chi Square Table 1: Substance Abuse Level by ation Total/Marginal 63 (a) 17 (b) 10 (c) 90 35 (d) 20 (e) 25 (f) 80 Totals/Marginal 98 37 35 170 Step 1: Label Your Table. Label

### Analysis of categorical data: Course quiz instructions for SPSS

Analysis of categorical data: Course quiz instructions for SPSS The dataset Please download the Online sales dataset from the Download pod in the Course quiz resources screen. The filename is smr_bus_acd_clo_quiz_online_250.xls.

### Statistics and research

Statistics and research Usaneya Perngparn Chitlada Areesantichai Drug Dependence Research Center (WHOCC for Research and Training in Drug Dependence) College of Public Health Sciences Chulolongkorn University,

### Module 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 } One-Sample Chi-Square Test

### UNDERSTANDING 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

### Elementary Statistics

lementary Statistics Chap10 Dr. Ghamsary Page 1 lementary Statistics M. Ghamsary, Ph.D. Chapter 10 Chi-square Test for Goodness of fit and Contingency tables lementary Statistics Chap10 Dr. Ghamsary Page

### Descriptive Analysis

Research Methods William G. Zikmund Basic Data Analysis: Descriptive Statistics Descriptive Analysis The transformation of raw data into a form that will make them easy to understand and interpret; rearranging,

### Chi-Square Test. Contingency Tables. Contingency Tables. Chi-Square Test for Independence. Chi-Square Tests for Goodnessof-Fit

Chi-Square Tests 15 Chapter Chi-Square Test for Independence Chi-Square Tests for Goodness Uniform Goodness- Poisson Goodness- Goodness Test ECDF Tests (Optional) McGraw-Hill/Irwin Copyright 2009 by The

### 9-3.4 Likelihood ratio test. Neyman-Pearson lemma

9-3.4 Likelihood ratio test Neyman-Pearson lemma 9-1 Hypothesis Testing 9-1.1 Statistical Hypotheses Statistical hypothesis testing and confidence interval estimation of parameters are the fundamental

### CHAPTER 11. GOODNESS OF FIT AND CONTINGENCY TABLES

CHAPTER 11. GOODNESS OF FIT AND CONTINGENCY TABLES The chi-square distribution was discussed in Chapter 4. We now turn to some applications of this distribution. As previously discussed, chi-square is

### Bivariate Statistics Session 2: Measuring Associations Chi-Square Test

Bivariate Statistics Session 2: Measuring Associations Chi-Square Test Features Of The Chi-Square Statistic The chi-square test is non-parametric. That is, it makes no assumptions about the distribution

### Chapter 19 The Chi-Square Test

Tutorial for the integration of the software R with introductory statistics Copyright c Grethe Hystad Chapter 19 The Chi-Square Test In this chapter, we will discuss the following topics: We will plot

### Chi Square (χ 2 ) Statistical Instructions EXP 3082L Jay Gould s Elaboration on Christensen and Evans (1980)

Chi Square (χ 2 ) Statistical Instructions EXP 3082L Jay Gould s Elaboration on Christensen and Evans (1980) For the Driver Behavior Study, the Chi Square Analysis II is the appropriate analysis below.

### Simple 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

### Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics

Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics For 2015 Examinations Aim The aim of the Probability and Mathematical Statistics subject is to provide a grounding in

### Topic 8. Chi Square Tests

BE540W Chi Square Tests Page 1 of 5 Topic 8 Chi Square Tests Topics 1. Introduction to Contingency Tables. Introduction to the Contingency Table Hypothesis Test of No Association.. 3. The Chi Square Test

### Chi 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

### Crosstabulation & Chi Square

Crosstabulation & Chi Square Robert S Michael Chi-square as an Index of Association After examining the distribution of each of the variables, the researcher s next task is to look for relationships among

### Fairfield 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

### Normality 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

### Topic 19: Goodness of Fit

Topic 19: November 24, 2009 A goodness of fit test examine the case of a sequence if independent experiments each of which can have 1 of k possible outcomes. In terms of hypothesis testing, let π = (π

### Business 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, McGraw-Hill/Irwin, 2008, ISBN: 978-0-07-331988-9. Required Computing

### Chi Square Distribution

17. Chi Square A. Chi Square Distribution B. One-Way Tables C. Contingency Tables D. Exercises Chi Square is a distribution that has proven to be particularly useful in statistics. The first section describes

### WHICH TYPE OF GRAPH SHOULD YOU CHOOSE?

PRESENTING GRAPHS WHICH TYPE OF GRAPH SHOULD YOU CHOOSE? CHOOSING THE RIGHT TYPE OF GRAPH You will usually choose one of four very common graph types: Line graph Bar graph Pie chart Histograms LINE GRAPHS

### Adverse Impact Ratio for Females (0/ 1) = 0 (5/ 17) = 0.2941 Adverse impact as defined by the 4/5ths rule was not found in the above data.

1 of 9 12/8/2014 12:57 PM (an On-Line Internet based application) Instructions: Please fill out the information into the form below. Once you have entered your data below, you may select the types of analysis

### Class 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

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

### 2 Tests for Goodness of Fit:

Tests for Goodness of Fit: General Notion: We often wish to know whether a particular distribution fits a general definition Example: To use t tests, we must suppose that the population is normally distributed

### Contingency Tables and the Chi Square Statistic. Interpreting Computer Printouts and Constructing Tables

Contingency Tables and the Chi Square Statistic Interpreting Computer Printouts and Constructing Tables Contingency Tables/Chi Square Statistics What are they? A contingency table is a table that shows

### SAS Software to Fit the Generalized Linear Model

SAS Software to Fit the Generalized Linear Model Gordon Johnston, SAS Institute Inc., Cary, NC Abstract In recent years, the class of generalized linear models has gained popularity as a statistical modeling

### INTERPRETING THE ONE-WAY ANALYSIS OF VARIANCE (ANOVA)

INTERPRETING THE ONE-WAY ANALYSIS OF VARIANCE (ANOVA) As with other parametric statistics, we begin the one-way ANOVA with a test of the underlying assumptions. Our first assumption is the assumption of

### Examination 110 Probability and Statistics Examination

Examination 0 Probability and Statistics Examination Sample Examination Questions The Probability and Statistics Examination consists of 5 multiple-choice test questions. The test is a three-hour examination

### The Big 50 Revision Guidelines for S1

The Big 50 Revision Guidelines for S1 If you can understand all of these you ll do very well 1. Know what is meant by a statistical model and the Modelling cycle of continuous refinement 2. Understand

### A Primer on Mathematical Statistics and Univariate Distributions; The Normal Distribution; The GLM with the Normal Distribution

A Primer on Mathematical Statistics and Univariate Distributions; The Normal Distribution; The GLM with the Normal Distribution PSYC 943 (930): Fundamentals of Multivariate Modeling Lecture 4: September

### How 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

### Section 12 Part 2. Chi-square test

Section 12 Part 2 Chi-square test McNemar s Test Section 12 Part 2 Overview Section 12, Part 1 covered two inference methods for categorical data from 2 groups Confidence Intervals for the difference of

### Statistics I for QBIC. Contents and Objectives. Chapters 1 7. Revised: August 2013

Statistics I for QBIC Text Book: Biostatistics, 10 th edition, by Daniel & Cross Contents and Objectives Chapters 1 7 Revised: August 2013 Chapter 1: Nature of Statistics (sections 1.1-1.6) Objectives

### Chi-square test Fisher s Exact test

Lesson 1 Chi-square 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

### Math 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

### 2. Simple Linear Regression

Research methods - II 3 2. Simple Linear Regression Simple linear regression is a technique in parametric statistics that is commonly used for analyzing mean response of a variable Y which changes according

### Once saved, if the file was zipped you will need to unzip it.

1 Commands in SPSS 1.1 Dowloading data from the web The data I post on my webpage will be either in a zipped directory containing a few files or just in one file containing data. Please learn how to unzip

### Chi-square (χ 2 ) Tests

Math 442 - Mathematical Statistics II May 5, 2008 Common Uses of the χ 2 test. 1. Testing Goodness-of-fit. Chi-square (χ 2 ) Tests 2. Testing Equality of Several Proportions. 3. Homogeneity Test. 4. Testing

### Odds ratio, Odds ratio test for independence, chi-squared statistic.

Odds ratio, Odds ratio test for independence, chi-squared 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

### Statistical Impact of Slip Simulator Training at Los Alamos National Laboratory

LA-UR-12-24572 Approved for public release; distribution is unlimited Statistical Impact of Slip Simulator Training at Los Alamos National Laboratory Alicia Garcia-Lopez Steven R. Booth September 2012

### Testing on proportions

Testing on proportions Textbook Section 5.4 April 7, 2011 Example 1. X 1,, X n Bernolli(p). Wish to test H 0 : p p 0 H 1 : p > p 0 (1) Consider a related problem The likelihood ratio test is where c is

### Unit 29 Chi-Square Goodness-of-Fit Test

Unit 29 Chi-Square Goodness-of-Fit Test Objectives: To perform the chi-square hypothesis test concerning proportions corresponding to more than two categories of a qualitative variable To perform the Bonferroni

### Course 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, McGraw-Hill/Irwin, 2010, ISBN: 9780077384470 [This

### CONTINGENCY TABLES ARE NOT ALL THE SAME David C. Howell University of Vermont

CONTINGENCY TABLES ARE NOT ALL THE SAME David C. Howell University of Vermont To most people studying statistics a contingency table is a contingency table. We tend to forget, if we ever knew, that contingency

### 11. Chi Square. Go to Data/Weight Cases and select Freq as the weights. Select Analyze/Nonparametric Tests/Chi Square.

11. Chi Square Objectives Calculate goodness of fit Chi Square Calculate Chi Square for contingency tables Calculate effect size Save data entry time by weighting cases A Chi Square is used to analyze

### Testing Research and Statistical Hypotheses

Testing Research and Statistical Hypotheses Introduction In the last lab we analyzed metric artifact attributes such as thickness or width/thickness ratio. Those were continuous variables, which as you

### Chapter 13. Chi-Square. Crosstabs and Nonparametric Tests. Specifically, we demonstrate procedures for running two separate

1 Chapter 13 Chi-Square This section covers the steps for running and interpreting chi-square analyses using the SPSS Crosstabs and Nonparametric Tests. Specifically, we demonstrate procedures for running

### STT315 Chapter 4 Random Variables & Probability Distributions KM. Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables

Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables Discrete vs. continuous random variables Examples of continuous distributions o Uniform o Exponential o Normal Recall: A random

### Statistical Models in R

Statistical Models in R Some Examples Steven Buechler Department of Mathematics 276B Hurley Hall; 1-6233 Fall, 2007 Outline Statistical Models Structure of models in R Model Assessment (Part IA) Anova

### Using SPSS to perform Chi-Square tests:

Using SPSS to perform Chi-Square tests: Graham Hole, January 2006: page 1: Using SPSS to perform Chi-Square tests: This handout explains how to perform the two types of Chi-Square test that were discussed

### Continuous Random Variables. and Probability Distributions. Continuous Random Variables and Probability Distributions ( ) ( ) Chapter 4 4.

UCLA STAT 11 A Applied Probability & Statistics for Engineers Instructor: Ivo Dinov, Asst. Prof. In Statistics and Neurology Teaching Assistant: Neda Farzinnia, UCLA Statistics University of California,

### Elementary 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

### SCHOOL OF HEALTH AND HUMAN SCIENCES DON T FORGET TO RECODE YOUR MISSING VALUES

SCHOOL OF HEALTH AND HUMAN SCIENCES Using SPSS Topics addressed today: 1. Differences between groups 2. Graphing Use the s4data.sav file for the first part of this session. DON T FORGET TO RECODE YOUR

### DATA INTERPRETATION AND STATISTICS

PholC60 September 001 DATA INTERPRETATION AND STATISTICS Books A easy and systematic introductory text is Essentials of Medical Statistics by Betty Kirkwood, published by Blackwell at about 14. DESCRIPTIVE

### Density 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

### Introduction to Analysis of Variance (ANOVA) Limitations of the t-test

Introduction to Analysis of Variance (ANOVA) The Structural Model, The Summary Table, and the One- Way ANOVA Limitations of the t-test Although the t-test is commonly used, it has limitations Can only

### Probability Distributions

CHAPTER 5 Probability Distributions CHAPTER OUTLINE 5.1 Probability Distribution of a Discrete Random Variable 5.2 Mean and Standard Deviation of a Probability Distribution 5.3 The Binomial Distribution

### HISTOGRAMS, CUMULATIVE FREQUENCY AND BOX PLOTS

Mathematics Revision Guides Histograms, Cumulative Frequency and Box Plots Page 1 of 25 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Higher Tier HISTOGRAMS, CUMULATIVE FREQUENCY AND BOX PLOTS

### IBM SPSS Statistics for Beginners for Windows

ISS, NEWCASTLE UNIVERSITY IBM SPSS Statistics for Beginners for Windows A Training Manual for Beginners Dr. S. T. Kometa A Training Manual for Beginners Contents 1 Aims and Objectives... 3 1.1 Learning

### Is it statistically significant? The chi-square test

UAS Conference Series 2013/14 Is it statistically significant? The chi-square test Dr Gosia Turner Student Data Management and Analysis 14 September 2010 Page 1 Why chi-square? Tests whether two categorical

### THE BINOMIAL DISTRIBUTION & PROBABILITY

REVISION SHEET STATISTICS 1 (MEI) THE BINOMIAL DISTRIBUTION & PROBABILITY The main ideas in this chapter are Probabilities based on selecting or arranging objects Probabilities based on the binomial distribution

### Exploratory Data Analysis

Exploratory Data Analysis Johannes Schauer johannes.schauer@tugraz.at Institute of Statistics Graz University of Technology Steyrergasse 17/IV, 8010 Graz www.statistics.tugraz.at February 12, 2008 Introduction

### Randomized Block Analysis of Variance

Chapter 565 Randomized Block Analysis of Variance Introduction This module analyzes a randomized block analysis of variance with up to two treatment factors and their interaction. It provides tables of

### Chapter 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

### Poisson Models for Count Data

Chapter 4 Poisson Models for Count Data In this chapter we study log-linear models for count data under the assumption of a Poisson error structure. These models have many applications, not only to the

### Tests of Hypotheses Using Statistics

Tests of Hypotheses Using Statistics Adam Massey and Steven J. Miller Mathematics Department Brown University Providence, RI 0292 Abstract We present the various methods of hypothesis testing that one

### 12.5: CHI-SQUARE GOODNESS OF FIT TESTS

125: Chi-Square Goodness of Fit Tests CD12-1 125: CHI-SQUARE 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

### MULTIPLE REGRESSION EXAMPLE

MULTIPLE REGRESSION EXAMPLE For a sample of n = 166 college students, the following variables were measured: Y = height X 1 = mother s height ( momheight ) X 2 = father s height ( dadheight ) X 3 = 1 if

### SPSS: Descriptive and Inferential Statistics. For Windows

For Windows August 2012 Table of Contents Section 1: Summarizing Data...3 1.1 Descriptive Statistics...3 Section 2: Inferential Statistics... 10 2.1 Chi-Square Test... 10 2.2 T tests... 11 2.3 Correlation...

### CHAPTER 5 COMPARISON OF DIFFERENT TYPE OF ONLINE ADVERTSIEMENTS. Table: 8 Perceived Usefulness of Different Advertisement Types

CHAPTER 5 COMPARISON OF DIFFERENT TYPE OF ONLINE ADVERTSIEMENTS 5.1 Descriptive Analysis- Part 3 of Questionnaire Table 8 shows the descriptive statistics of Perceived Usefulness of Banner Ads. The results

### MBA 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

### Hypothesis Testing & Data Analysis. Statistics. Descriptive Statistics. What is the difference between descriptive and inferential statistics?

2 Hypothesis Testing & Data Analysis 5 What is the difference between descriptive and inferential statistics? Statistics 8 Tools to help us understand our data. Makes a complicated mess simple to understand.