# Elementary Statistics

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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

2 lementary Statistics Chap10 Dr. Ghamsary Page Chi-Square Test Generally speaking, the chi-square test is a statistical test used to examine differences with categorical variables. The chi-square test is used in two similar but distinct circumstances: 1. for estimating how closely an observed distribution matches an expected distribution - we'll refer to this as the goodness-of-fit test. for estimating whether two random variables are independent (Contingency Tables) Goodness of Fit Test One of the more interesting goodness-of-fit applications of the chi-square test is to examine issues of fairness and cheating in games of chance, such as coins, cards, dice, and roulette. Since such games usually involve wagering, there is significant incentive for people to try to rig the games and allegations of missing cards, "loaded" dice, and "sticky" roulette wheels are all too common. So how can the goodness-of-fit test be used to examine cheating in gambling? It is easier to describe the process through an example. Take the example of dice. Most dice used in wagering have six sides, with each side having a value of one, two, three, four, five, or six. If the die being used is fair, then the chance of any particular number coming up is the same: 1 in 6. However, if the die is loaded, then certain numbers will have a greater likelihood of appearing, while others will have a lower likelihood So we would like to test and see if a given data set will match the hypothesized distribution. The following is the test statistics used for this purpose. where, χ = ( O ) O: is the observed data : the expected value.

3 lementary Statistics Chap10 Dr. Ghamsary Page 3 Clearly if the data matches the claimed distribution, this chi-square value will be small and we cannot reject the null hypothesis. Otherwise this value, χ, will be large and we must reject the H 0. xample 1: The simplest example is to flip a coin 100 times and record the outcomes. Suppose we observed 40 heads. Test the claim that the coin is fair, which means the outcomes are equally likely. Use 5% level of significance. Solution: Let us write the outcome in the following table. The expected number of heads is Step1: R S T 0.50(100)=50 H :The Coin isfair 0 H :The Coin isnot fair 1 Step: Calculate the test statistics as follows: Step3: Decision: So we reject H 0. Head Tail Observed xpected df=-1=1 α = 005. ( O ) ( 50) ( 50) by using Table III CV= χ = = + = Conclusion: This means the coin is biased..

4 lementary Statistics Chap10 Dr. Ghamsary Page 4 xample : The next simplest example is to roll a die 10 times and record the outcomes. Suppose we have observed 18 one s, 3 two s, 15 three s, four s, 17 five s, and 5 six s. Test the claim that the die is fair, which means the outcomes are equally likely again. Use 5% level of significance. Solution: Let us write the outcome in the following table. The expected number of outcomes is all equal 0, under the assumption of equality. So we have =10/6= Observed Step1: R S T xpected H :The Die isfair 0 H :The Die is not fair 1 Step: Calculate the test statistics as follows: df=6-1=5 α = 005. by using Table III CV=11.07 ( O ) χ = = ( 18 0) ( 3 0) ( 15 0) ( 0) ( 17 0) ( 5 0) + + = Step3: Decision: fail to reject H 0 Conclusion: This means the die is unbiased

5 lementary Statistics Chap10 Dr. Ghamsary Page 5 xample 3: An ice cream shop would like to know which flavor is preferred by the customers. The past record shows that 50% prefer vanilla, 0% prefer chocolate, 10% prefer vanilla fudge, 15% prefer strawberry, and 5% prefer other kinds. A random sample of 500 customers revealed the following results. Test the claim that the observed numbers and the percentage match. Flavor Vanilla Chocolate Strawberry Vanilla Fudge Others Customers Solution: Let us calculate the expected value as follows: Vanilla: 50% of 500 = 0.50*500=50 Chocolate: 0% of 500 = 0.0*500 =100 Strawberry: 15% of 500 = 0.15*500= 75 Vanilla Fudge: 10% of 500 = 0.10*500=50 Others: 5% of 500 = 0.05*500=5 Flavor Vanilla Chocolate Strawberry Vanilla Fudge Other Observed xpected Step1: H 0:The Observed and expected match H 1:TheObservedandexpected donot match df=5-1=4α = 005. CV=9.49 Step: Calculate the test statistics as follows: ( O ) χ = = ( 40 50) ( ) ( 70 75) ( ) ( 30 5 ) Step3: Decision: fail to reject H 0 Conclusion: This means the die is unbiased

6 lementary Statistics Chap10 Dr. Ghamsary Page 6 xample 4: Affirmative Action Problem A large organization in a city is accused of being racist in one or more race group. If in that city, there are 45% White, 15% Black, 0% Hispanic, 5% Asian, and the rest are others. A random sample of 50 from the whole corporation is collected with the following results. Test to see if the frequency of the observed and the percentage in the population are the same. H 0: The frequency observed matches the percentage of population Step1: H 1: The frequency observed does not match the percentage of population α = 005. df=5-1=4 CV=9.49 Race Observed xpected % White % 11.5 Black 30 15% 37.5 Hispanic 40 0% 50 Asian 0 5% 1.5 Other 60 15% 37.5 Total: 50 CV=9.49 Step: ( O ) χ = = ( ) ( ) ( 40 50) ( ) ( ) Step3: Decision: reject H 0 Conclusion: This means the frequency of observed and the % of the population do not match.

7 lementary Statistics Chap10 Dr. Ghamsary Page 7 xample5: In a study in Alameda County, California, researchers compared the demographic characteristics of members of grand juries to determine how closely these juries reflected the population of the county. If the juries were selected randomly or impartially, then the characteristics of the jurors should closely match those of the larger county; however, if attorneys were tilting the jury selection process, then the jurors' characteristics would be quite different from the county. (figures taken from UCLA Law Review, vol 0, as shown at: Age Country-Wide % # of Jurors > Questions Based on the figures shown in the table above, use the chi-square test to evaluate whether there is evidence of jury fixing in terms of the age of jurors in Alameda County. a. What is the null hypothesis? What is the alternative hypothesis? b. What figures do you need to calculate for this test? c. How many degrees of freedom are there? d. What is the value of the chi-square statistic for this table? What is the p-value of this statistic? e. From this value, what can you conclude about the age of jurors in Alameda County?

8 lementary Statistics Chap10 Dr. Ghamsary Page 8 Test of Independence The other primary use of the chi-square test is to examine whether two variables are independent or not. What does it mean to be independent, in this sense? It means that the two factors are not related. Typically in any research such as epidemiology and social science research, we're interested in finding factors that are related - education and income, occupation and prestige, age and voting behavior. In this case, the chi- square can be used to assess whether two variables are independent or not. More generally, we say that factor A is "not correlated with" or "independent of" the factor B if more of one is not associated with more of another. If two categorical variables are correlated their values tend to move together, either in the same direction or in the opposite. In practice there are many data comes in the following format. They are called two way frequency table and some other text book call it contingency tables. The test of dependency is a test to see if row factor and the column factor are related. Test Statistics: is the same as before, namely: χ ( O ) =, Where b the O is the observed cells and is the expected cells which is can be find from the following: Row Total gb Colunm Total g = Grand total Also we have degrees of freedom = (r-1)(c-1), Where, r = Number of rows, c = Number of column

9 lementary Statistics Chap10 Dr. Ghamsary Page 9 xample 6: Dr. Ghamsary and colleagues are testing to see if the habits of smoking and gender are independent. They have collected a random sample of 50 people as they appear in the following table. Test their claim by using the 0.05 level of significance. Sex Male Smoking Yes No Total Female 150 Total Solution: H 0: Sex and smoking are independent Step1: H 1: Sex and smoking are not independent df = ( 1)( 1) = 1 α = 005. CV= * * = = 56 1 = = * * = = 84 = = Smoking Total Yes No Sex Male Female Total Step: Calculate the test statistics as follows: ( O ) χ = = Step3: Decision: fail to reject H 0 ( 60 56) ( 40 44) ( 80 84) ( 70 66) Conclusion: This means the sex and smoking are independent.

10 lementary Statistics Chap10 Dr. Ghamsary Page 10 Cautionary Note It is important to keep in mind that the chi-square test only tests whether two variables are independent. It cannot address questions of which is greater or less. Using the chi-square test, we cannot evaluate directly the hypothesis that men smoke more than women; rather, the test (strictly speaking) can only test whether the two variables are independent or not. xample 7: Ghamsary and others have done some research (Kashan, Vol 1, ) on income and level of education. They are interested to know if people with more education have higher income. They collected a random sample of 00 people from a large population and they found the following results.test the claim that the education and income are independent factors. Use α = 001. Income\ducation None High School 4 year college Graduate School Less than 30K K-50K Above 50K Solution: Income\ducation N HS College Graduate Total Less than 30K K-50K Above 50K Total * 40 61* 60 61* * = = = = = = = = * 40 90* 60 90* * 50 1 = =14.40 = = = = = = * 40 99* 60 99* * = = = = = = = =

11 lementary Statistics Chap10 Dr. Ghamsary Page 11 Step1: H 0:Income and ducation are independent H 1:Income and ducation are not independent Step: Calculate the test statistics as follows: ( O ) χ = = ( ) ( ) ( ) ( 6 1.0) df = ( 3 1)( 4 1) = 6 at 0.01, CV= ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) MINITAB: Chi-Square Test xpected counts are printed below observed counts C1 C C3 C4 Total Total Chi-Sq = = DF = 6, P-Value = Step3: Decision: Reject H 0 Conclusion: This means the income and education are not independent.

12 lementary Statistics Chap10 Dr. Ghamsary Page 1 xample 8: In a recent research taken from a random sample of 500 student, show in the following table by two factors, Study on time for the tests and School areas. Is there an association between the type of school area and the student goals? School Area Study on time on the tests Rural Suburban Urban Total Always Some Times Never Total

13 lementary Statistics Chap10 Dr. Ghamsary Page 13 xample 9: A chocolate manufacturing company conducted a survey of 300 customers. The research question is: Is there a significant relationship between packaging preference (size of the bottle purchased) and economic status? There were four packaging sizes: small, medium, large, and jumbo. conomic status was: lower, middle, and upper. The following data was collected. Test the claim that the size of the packaging and economic status are independent by using 0.10 level of significance. conomic Status Size Lower Middle Upper Total Small Medium Large Jumbo Total

14 lementary Statistics Chap10 Dr. Ghamsary Page 14 xample 10: A random sample of 1500 persons is questioned regarding their political affiliation and opinion on the war in IRAQ. Test if the political affiliation and their opinion on the war in IRAQ are dependent using 5% level of significance. The observed data is given in the following table. War in IRAQ Party Affiliation Favor Indifferent Opposed Total Democrat Republican Independent Total

### Chapter 11. Chapter 11 Overview. Chapter 11 Objectives 11/24/2015. Other Chi-Square Tests

11/4/015 Chapter 11 Overview Chapter 11 Introduction 11-1 Test for Goodness of Fit 11- Tests Using Contingency Tables Other Chi-Square Tests McGraw-Hill, Bluman, 7th ed., Chapter 11 1 Bluman, Chapter 11

### Mind on Statistics. Chapter 15

Mind on Statistics Chapter 15 Section 15.1 1. A student survey was done to study the relationship between class standing (freshman, sophomore, junior, or senior) and major subject (English, Biology, French,

### 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

### 13.2 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)

### Chi-Square Tests and the F-Distribution. Goodness of Fit Multinomial Experiments. Chapter 10

Chapter 0 Chi-Square Tests and the F-Distribution 0 Goodness of Fit Multinomial xperiments A multinomial experiment is a probability experiment consisting of a fixed number of trials in which there are

### 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

### 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

### 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

### AP Statistics for Friday 4/15/16. 2) turn in work; hand back work. 3) lesson on Chi Square tests for Association with handouts

AP Statistics for Friday 4/15/16 1) go over schedule 2) turn in work; hand back work 3) lesson on Chi Square tests for Association with handouts 4) Assign #105 p 631 / 19, 23, 31 schedule... Normal Distribution

### 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

### 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

### MATH Chapter 23 April 15 and 17, 2013 page 1 of 8 CHAPTER 23: COMPARING TWO CATEGORICAL VARIABLES THE CHI-SQUARE TEST

MATH 1342. Chapter 23 April 15 and 17, 2013 page 1 of 8 CHAPTER 23: COMPARING TWO CATEGORICAL VARIABLES THE CHI-SQUARE TEST Relationships: Categorical Variables Chapter 21: compare proportions of successes

### Math 108 Exam 3 Solutions Spring 00

Math 108 Exam 3 Solutions Spring 00 1. An ecologist studying acid rain takes measurements of the ph in 12 randomly selected Adirondack lakes. The results are as follows: 3.0 6.5 5.0 4.2 5.5 4.7 3.4 6.8

### MATH 10: Elementary Statistics and Probability Chapter 11: The Chi-Square Distribution

MATH 10: Elementary Statistics and Probability Chapter 11: The Chi-Square Distribution Tony Pourmohamad Department of Mathematics De Anza College Spring 2015 Objectives By the end of this set of slides,

### Chi-Square Test for Qualitative Data

Chi-Square Test for Qualitative Data For qualitative data (measured on a nominal scale) * Observations MUST be independent - No more than one measurement per subject * Sample size must be large enough

### 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

### Chi-Square Test (χ 2 )

Chi Square Tests Chi-Square Test (χ 2 ) Nonparametric test for nominal independent variables These variables, also called "attribute variables" or "categorical variables," classify observations into a

### Chi 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

### 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

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

Chi-square test Testing for independeny The r x c contingency tables square test 1 The chi-square distribution HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided

### 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

### 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

### Lecture 42 Section 14.3. Tue, Apr 8, 2008

the Lecture 42 Section 14.3 Hampden-Sydney College Tue, Apr 8, 2008 Outline the 1 2 the 3 4 5 the The will compute χ 2 areas, but not χ 2 percentiles. (That s ok.) After performing the χ 2 test by hand,

### 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

### 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.

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: Two-Way Tables The Problem of Multiple Comparisons Expected

### CHI SQUARE DISTRIBUTION

CI SQUARE DISTRIBUTION 1 Introduction to the Chi Square Test of Independence This test is used to analyse the relationship between two sets of discrete data. Contingency tables are used to examine the

### 4) 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

### Chi-Square Tests. In This Chapter BONUS CHAPTER

BONUS CHAPTER Chi-Square 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

### Mind on Statistics. Chapter 4

Mind on Statistics Chapter 4 Sections 4.1 Questions 1 to 4: The table below shows the counts by gender and highest degree attained for 498 respondents in the General Social Survey. Highest Degree Gender

### Having a coin come up heads or tails is a variable on a nominal scale. Heads is a different category from tails.

Chi-square Goodness of Fit Test The chi-square test is designed to test differences whether one frequency is different from another frequency. The chi-square test is designed for use with data on a nominal

### 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

### Using Stata for Categorical Data Analysis

Using Stata for Categorical Data Analysis NOTE: These problems make extensive use of Nick Cox s tab_chi, which is actually a collection of routines, and Adrian Mander s ipf command. From within Stata,

### 11-2 Goodness of Fit Test

11-2 Goodness of Fit Test In This section we consider sample data consisting of observed frequency counts arranged in a single row or column (called a one-way frequency table). We will use a hypothesis

### 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

### Solutions: Problems for Chapter 3. Solutions: Problems for Chapter 3

Problem A: You are dealt five cards from a standard deck. Are you more likely to be dealt two pairs or three of a kind? experiment: choose 5 cards at random from a standard deck Ω = {5-combinations of

### 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

### Testing Hypotheses using SPSS

Is the mean hourly rate of male workers \$2.00? T-Test One-Sample Statistics Std. Error N Mean Std. Deviation Mean 2997 2.0522 6.6282.2 One-Sample Test Test Value = 2 95% Confidence Interval Mean of the

### Simulating Chi-Square Test Using Excel

Simulating Chi-Square Test Using Excel Leslie Chandrakantha John Jay College of Criminal Justice of CUNY Mathematics and Computer Science Department 524 West 59 th Street, New York, NY 10019 lchandra@jjay.cuny.edu

### CHAPTER 15 NOMINAL MEASURES OF CORRELATION: PHI, THE CONTINGENCY COEFFICIENT, AND CRAMER'S V

CHAPTER 15 NOMINAL MEASURES OF CORRELATION: PHI, THE CONTINGENCY COEFFICIENT, AND CRAMER'S V Chapters 13 and 14 introduced and explained the use of a set of statistical tools that researchers use to measure

### Understanding and Interpreting the Chi-square Statistic (x 2 ) Rose Ann DiMaria, PhD, RN WVU-School of Nursing Charleston Division

Understanding and Interpreting the Chi-square Statistic (x 2 ) Rose Ann DiMaria, PhD, RN WVU-School of Nursing Charleston Division Inferential statistics Make judgments about accuracy of given sample in

### Pearson s 2x2 Chi-Square Test of Independence -- Analysis of the Relationship between two Qualitative

Pearson s 2x2 Chi-Square Test of Independence -- Analysis of the Relationship between two Qualitative or (Binary) Variables Analysis of 2-Between-Group Data with a Qualitative (Binary) Response Variable

### 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

### 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

### 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

### CHAPTER 11: FACTS ABOUT THE CHI- SQUARE DISTRIBUTION

CHAPTER 11: FACTS ABOUT THE CHI- SQUARE DISTRIBUTION Exercise 1. If the number of degrees of freedom for a chi-square distribution is 25, what is the population mean and standard deviation? mean = 25 and

### MONT 107N Understanding Randomness Solutions For Final Examination May 11, 2010

MONT 07N Understanding Randomness Solutions For Final Examination May, 00 Short Answer (a) (0) How are the EV and SE for the sum of n draws with replacement from a box computed? Solution: The EV is n times

### 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.

### 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

### 1. Rejecting a true null hypothesis is classified as a error. 2. Failing to reject a false null hypothesis is classified as a error.

1. Rejecting a true null hypothesis is classified as a error. 2. Failing to reject a false null hypothesis is classified as a error. 8.5 Goodness of Fit Test Suppose we want to make an inference about

### Association Between Variables

Contents 11 Association Between Variables 767 11.1 Introduction............................ 767 11.1.1 Measure of Association................. 768 11.1.2 Chapter Summary.................... 769 11.2 Chi

### statistics Chi-square tests and nonparametric Summary sheet from last time: Hypothesis testing Summary sheet from last time: Confidence intervals

Summary sheet from last time: Confidence intervals Confidence intervals take on the usual form: parameter = statistic ± t crit SE(statistic) parameter SE a s e sqrt(1/n + m x 2 /ss xx ) b s e /sqrt(ss

### 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

### χ 2 = (O i E i ) 2 E i

Chapter 24 Two-Way Tables and the Chi-Square Test We look at two-way tables to determine association of paired qualitative data. We look at marginal distributions, conditional distributions and bar graphs.

### THE ASSOCIATED PRESS POLL CONDUCTED BY IPSOS PUBLIC AFFAIRS RELEASE DATE: DECEMEBER 16, 2005 PROJECT #81-5139-71 REGISTERED VOTERS/ PARTY AFFILIATION

1101 Connecticut Avenue NW, Suite 200 Washington, DC 20036 (202) 463-7300 Interview dates: December 13-15, 2005 Interviews 1,006 adults, 813 registered voters Margin of error: +3.1 for all adults, +3.5

### Lecture 13. Understanding Probability and Long-Term Expectations

Lecture 13 Understanding Probability and Long-Term Expectations Thinking Challenge What s the probability of getting a head on the toss of a single fair coin? Use a scale from 0 (no way) to 1 (sure thing).

### AMS 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

### One-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

### Two Categorical Variables: the Chi-Square Test

CHAPTER 22 Two Categorical Variables: the Chi-Square Test 22.1 22.2 22.3 22.4 Data Analysis for Two-Way Tables Inference for Two-Way Tables Goodness of Fit Selected Exercise Solutions Introduction We saw

### Chi-Square Analysis (Ch.8) Purpose. Purpose. Examples

Chi-Square Analysis (Ch.8) Chi-square test of association (contingency) x tables rxc tables Post-hoc Interpretation Running SPSS Windows CROSSTABS Chi-square test of goodness of fit Purpose Chi-square

### Chapter 26: Tests of Significance

Chapter 26: Tests of Significance Procedure: 1. State the null and alternative in words and in terms of a box model. 2. Find the test statistic: z = observed EV. SE 3. Calculate the P-value: The area under

### COMPARISONS OF CUSTOMER LOYALTY: PUBLIC & PRIVATE INSURANCE COMPANIES.

277 CHAPTER VI COMPARISONS OF CUSTOMER LOYALTY: PUBLIC & PRIVATE INSURANCE COMPANIES. This chapter contains a full discussion of customer loyalty comparisons between private and public insurance companies

### Hypothesis Testing for a Proportion

Math 122 Intro to Stats Chapter 6 Semester II, 2015-16 Inference for Categorical Data Hypothesis Testing for a Proportion In a survey, 1864 out of 2246 randomly selected adults said texting while driving

### In 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

### AP: LAB 8: THE CHI-SQUARE TEST. Probability, Random Chance, and Genetics

Ms. Foglia Date AP: LAB 8: THE CHI-SQUARE TEST Probability, Random Chance, and Genetics Why do we study random chance and probability at the beginning of a unit on genetics? Genetics is the study of inheritance,

### People like to clump things into categories. Virtually every research

05-Elliott-4987.qxd 7/18/2006 5:26 PM Page 113 5 Analysis of Categorical Data People like to clump things into categories. Virtually every research project categorizes some of its observations into neat,

### Chapter 16: law of averages

Chapter 16: law of averages Context................................................................... 2 Law of averages 3 Coin tossing experiment......................................................

### CHI-Squared Test of Independence

CHI-Squared Test of Independence Minhaz Fahim Zibran Department of Computer Science University of Calgary, Alberta, Canada. Email: mfzibran@ucalgary.ca Abstract Chi-square (X 2 ) test is a nonparametric

### STATISTICS 8, FINAL EXAM. Last six digits of Student ID#: Circle your Discussion Section: 1 2 3 4

STATISTICS 8, FINAL EXAM NAME: KEY Seat Number: Last six digits of Student ID#: Circle your Discussion Section: 1 2 3 4 Make sure you have 8 pages. You will be provided with a table as well, as a separate

### 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

### Hypothesis Tests for 1 sample Proportions

Hypothesis Tests for 1 sample Proportions 1. Hypotheses. Write the null and alternative hypotheses you would use to test each of the following situations. a) A governor is concerned about his "negatives"

### Categorical Data Analysis

Richard L. Scheaffer University of Florida The reference material and many examples for this section are based on Chapter 8, Analyzing Association Between Categorical Variables, from Statistical Methods

### Lecture 13 - χ 2 Tests

Lecture 13 - χ 2 Tests Statistics 102 Colin Rundel March 6, 2013 Weldon s dice Weldon s dice Walter Frank Raphael Weldon (1860-1906), was an English evolutionary biologist and a founder of biometry. He

### The overall size of these chance errors is measured by their RMS HALF THE NUMBER OF TOSSES NUMBER OF HEADS MINUS 0 400 800 1200 1600 NUMBER OF TOSSES

INTRODUCTION TO CHANCE VARIABILITY WHAT DOES THE LAW OF AVERAGES SAY? 4 coins were tossed 1600 times each, and the chance error number of heads half the number of tosses was plotted against the number

### Introduction to Hypothesis Testing. Copyright 2014 Pearson Education, Inc. 9-1

Introduction to Hypothesis Testing 9-1 Learning Outcomes Outcome 1. Formulate null and alternative hypotheses for applications involving a single population mean or proportion. Outcome 2. Know what Type

### Statistics for Management II-STAT 362-Final Review

Statistics for Management II-STAT 362-Final Review Multiple Choice Identify the letter of the choice that best completes the statement or answers the question. 1. The ability of an interval estimate to

### Unit 29: Inference for Two-Way Tables

Unit 29: Inference for Two-Way Tables Summary of Video In this video, we visit the Broad Institute in Cambridge, Massachusetts, where our host, Dr. Pardis Sabeti, has a small research team investigating

### Chi-Square Tests. Part I: Testing Goodness of Fit. Testing for Goodness of Fit and Independence. Example: Counts of Suicides by Month in US in 1970

1 Chi-Square Tests Testing for Goodness of Fit and Independence Part I: Testing Goodness of Fit There is a chance model There are observed frequency counts Wish to see whether the counts are consistent

### We know from STAT.1030 that the relevant test statistic for equality of proportions is:

2. Chi 2 -tests for equality of proportions Introduction: Two Samples Consider comparing the sample proportions p 1 and p 2 in independent random samples of size n 1 and n 2 out of two populations which

### Medicare Advantage National Senior Survey 600 Senior Registered Voters in the Medicare Advantage Program February 24-28, 2015

Medicare Advantage National Senior Survey 600 Senior Registered Voters in the Medicare Advantage Program February 24-28, 2015 1. In what year were you born? 1. Before 1950 (CONTINUE TO QUESTION 2) 100

### What is a χ 2 (Chi-square) test used for? Statistical test used to compare observed data with expected data according to a hypothesis.

What is a χ 2 (Chi-square) test used for? Statistical test used to compare observed data with expected data according to a hypothesis. What does that mean? Let s look at the next slide to find out Ex.

### Unit 12 Logistic Regression Supplementary Chapter 14 in IPS On CD (Chap 16, 5th ed.)

Unit 12 Logistic Regression Supplementary Chapter 14 in IPS On CD (Chap 16, 5th ed.) Logistic regression generalizes methods for 2-way tables Adds capability studying several predictors, but Limited to

### : P(winter) = P(spring) = P(summer)= P(fall) H A. : Homicides are independent of seasons. : Homicides are evenly distributed across seasons

Chi-Square worked examples: 1. Criminologists have long debated whether there is a relationship between weather conditions and the incidence of violent crime. The article Is There a Season for Homocide?

### 1. 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 1-propZint 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

### - - Each Split Sample = ± 5.6 percentage points

- - Interview Dates: February 11 to 21, 2012 Sample Frame: Registered Voters Sample Size: TENNESSEE = 606 Split Sample Sizes: Split A = 303; Split B = 303 Margin of Error: TENNESSEE = ± 4.0 percentage

### CHAPTER 11 CHI-SQUARE AND F DISTRIBUTIONS

CHAPTER 11 CHI-SQUARE AND F DISTRIBUTIONS CHI-SQUARE TESTS OF INDEPENDENCE (SECTION 11.1 OF UNDERSTANDABLE STATISTICS) In chi-square tests of independence we use the hypotheses. H0: The variables are independent

### Elementary Statistics

Elementary Statistics Chapter 1 Dr. Ghamsary Page 1 Elementary Statistics M. Ghamsary, Ph.D. Chap 01 1 Elementary Statistics Chapter 1 Dr. Ghamsary Page 2 Statistics: Statistics is the science of collecting,

### AP Statistics 7!3! 6!

Lesson 6-4 Introduction to Binomial Distributions Factorials 3!= Definition: n! = n( n 1)( n 2)...(3)(2)(1), n 0 Note: 0! = 1 (by definition) Ex. #1 Evaluate: a) 5! b) 3!(4!) c) 7!3! 6! d) 22! 21! 20!

### CHAPTER 12. Chi-Square Tests and Nonparametric Tests LEARNING OBJECTIVES. USING STATISTICS @ T.C. Resort Properties

CHAPTER 1 Chi-Square Tests and Nonparametric Tests USING STATISTICS @ T.C. Resort Properties 1.1 CHI-SQUARE TEST FOR THE DIFFERENCE BETWEEN TWO PROPORTIONS (INDEPENDENT SAMPLES) 1. CHI-SQUARE TEST FOR

### Standard 12: The student will explain and evaluate the financial impact and consequences of gambling.

STUDENT MODULE 12.1 GAMBLING PAGE 1 Standard 12: The student will explain and evaluate the financial impact and consequences of gambling. Risky Business Simone, Paula, and Randy meet in the library every

### 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

### Chapter 20: chance error in sampling

Chapter 20: chance error in sampling Context 2 Overview................................................................ 3 Population and parameter..................................................... 4

### Test Positive True Positive False Positive. Test Negative False Negative True Negative. Figure 5-1: 2 x 2 Contingency Table

ANALYSIS OF DISCRT VARIABLS / 5 CHAPTR FIV ANALYSIS OF DISCRT VARIABLS Discrete variables are those which can only assume certain fixed values. xamples include outcome variables with results such as live

### N Mean Std. Deviation Std. Error of Mean

DEPARTMENT OF POLITICAL SCIENCE AND INTERNATIONAL RELATIONS Posc/Uapp 815 SAMPLE QUESTIONS FOR THE FINAL EXAMINATION I. READING: A. Read Agresti and Finalay, Chapters 6, 7, and 8 carefully. 1. Ignore the

### 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

### LAB : THE CHI-SQUARE TEST. Probability, Random Chance, and Genetics

Period Date LAB : THE CHI-SQUARE TEST Probability, Random Chance, and Genetics Why do we study random chance and probability at the beginning of a unit on genetics? Genetics is the study of inheritance,

### SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Ch. 10 Chi SquareTests and the F-Distribution 10.1 Goodness of Fit 1 Find Expected Frequencies Provide an appropriate response. 1) The frequency distribution shows the ages for a sample of 100 employees.