Comparing Multiple Proportions, Test of Independence and Goodness of Fit


 Richard Simpson
 1 years ago
 Views:
Transcription
1 Comparing Multiple Proportions, Test of Independence and Goodness of Fit
2 Content Testing the Equality of Population Proportions for Three or More Populations Test of Independence Goodness of Fit Test 2
3 Comparing Multiple Proportions, Test of Independence and Goodness of Fit In this chapter we introduce three additional hypothesistesting procedures. The test statistic and the distribution used are based on the chisquare ( 2 ) distribution. In all cases, the data are categorical. 3
4 Testing the Equality of Population Proportions for Three or More Populations Using the notation p 1 = population proportion for population 1 p 2 = population proportion for population 2 and p k = population proportion for population k The hypotheses for the equality of population proportions for k 3 populations are as follows: H 0 : p 1 = p 2 =... = p k H a : Not all population proportions are equal 4
5 Testing the Equality of Population Proportions for Three or More Populations If H 0 cannot be rejected, we cannot detect a difference among the k population proportions. If H 0 can be rejected, we can conclude that not all k population proportions are equal. Further analyses can be done to conclude which population proportions are significantly different from others. 5
6 Testing the Equality of Population Proportions for Three or More Populations Example: Finger Lakes Homes Finger Lakes Homes manufactures three models of prefabricated homes, a twostory colonial, a log cabin, and an Aframe. To help in productline planning, management would like to compare the customer satisfaction with the three home styles. p 1 = proportion likely to repurchase a Colonial for the population of Colonial owners p 2 = proportion likely to repurchase a Log Cabin for the population of Log Cabin owners p 3 = proportion likely to repurchase an AFrame for the population of AFrame owners 6
7 Testing the Equality of Population Proportions for Three or More Populations We begin by taking a sample of owners from each of the three populations. Each sample contains categorical data indicating whether the respondents are likely or not likely to repurchase the home. 7
8 Testing the Equality of Population Proportions for Three or More Populations Observed Frequencies (sample results) Home Owner Colonial Log AFrame Total Likely to Yes Repurchase No Total
9 Testing the Equality of Population Proportions for Three or More Populations Next, we determine the expected frequencies under the assumption H 0 is correct. Expected Frequencies Under the Assumption H 0 is True (Row i Total)(Column j Total) eij Total Sample Size If a significant difference exists between the observed and expected frequencies, H 0 can be rejected. 9
10 Testing the Equality of Population Proportions for Three or More Populations Expected Frequencies (computed) Home Owner Colonial Log AFrame Total Likely to Yes Repurchase No Total
11 Testing the Equality of Population Proportions for Three or More Populations Next, compute the value of the chisquare test statistic. ( f ) 2 ij eij e i j ij where: f ij = observed frequency for the cell in row i and column j e ij = expected frequency for the cell in row i and column j under the assumption H 0 is true 2 Note: The test statistic has a chisquare distribution with k 1 degrees of freedom, provided the expected frequency is 5 or more for each cell. 11
12 Testing the Equality of Population Proportions for Three or More Populations Computation of the ChiSquare Test Statistic. Obs. Exp. Sqd. Sqd. Diff. / Likely to Home Freq. Freq. Diff. Diff. Exp. Freq. Repurch. Owner f ij e ij (f ij  e ij ) (f ij  e ij ) 2 (f ij  e ij ) 2 /e ij Yes Colonial Yes Log Cab Yes AFrame No Colonial No Log Cab No AFrame Total =
13 Testing the Equality of Population Proportions for Three or More Populations Rejection Rule pvalue approach: Reject H 0 if pvalue < a Critical value approach: Reject H 0 if 2 2 where is the significance level and there are k  1 degrees of freedom 13
14 Testing the Equality of Population Proportions for Three or More Populations Rejection Rule (using a =.05) Reject H 0 if pvalue.05 or With =.05 and k  1 = 31 = 2 degrees of freedom Do Not Reject H 0 Reject H
15 Testing the Equality of Population Proportions for Three or More Populations Conclusion Using the pvalue Approach Area in Upper Tail Value (df = 2) Because 2 = is between and 7.378, the area in the upper tail of the distribution is between.01 and.025. The pvalue. We can reject the null hypothesis. Actual pvalue is
16 Testing the Equality of Population Proportions for Three or More Populations We have concluded that the population proportions for the three populations of home owners are not equal. To identify where the differences between population proportions exist, we will rely on a multiple comparisons procedure. 16
17 Multiple Comparisons Procedure We begin by computing the three sample proportions. Colonial: p Log Cabin: p AFrame: p We will use a multiple comparison procedure known as the Marascuillo procedure. 17
18 Multiple Comparisons Procedure Marascuillo Procedure We compute the absolute value of the pairwise difference between sample proportions. Colonial and Log Cabin: Colonial and AFrame: Log Cabin and AFrame: p 1 p p 1 p p 2 p
19 Multiple Comparisons Procedure Critical Values for the Marascuillo Pairwise Comparison For each pairwise comparison compute a critical value as follows: CV ij 2 p (1 ) i(1 p) pj p i j ; k 1 n n i j For =.05 and k = 3: 2 =
20 Multiple Comparisons Procedure Pairwise Comparison Tests Pairwise Comparison Significant if p p CV ij p p > CV ij i j i j Colonial vs. Log Cabin Not Significant Colonial vs. AFrame Not Significant Log Cabin vs. AFrame Significant 20
21 Test of Independence 1. Set up the null and alternative hypotheses. H 0 : The column variable is independent of the row variable H a : The column variable is not independent of the row variable 2. Select a random sample and record the observed frequency, f ij, for each cell of the contingency table. 3. Compute the expected frequency, e ij, for each cell. (Row i Total)(Column j Total) eij Sample Size 21
22 Test of Independence 4. Compute the test statistic. ( f ) 2 ij eij e i j ij 5. Determine the rejection rule. 2 2 Reject H 0 if p value or. where is the significance level and, with n rows and m columns, there are (n 1)(m  1) degrees of freedom. 2 22
23 Test of Independence Example: Finger Lakes Homes (B) Each home sold by Finger Lakes Homes can be classified according to price and to style. Finger Lakes manager would like to determine if the price of the home and the style of the home are independent variables. 23
24 Test of Independence Example: Finger Lakes Homes (B) The number of homes sold for each model and price for the past two years is shown below. For convenience, the price of the home is listed as either $99,000 or less or more than $99,000. Price Colonial Log SplitLevel AFrame $99, > $99,
25 Test of Independence 1. Hypotheses H 0 : Price of the home is independent of the style of the home that is purchased H a : Price of the home is not independent of the style of the home that is purchased 2. Test of Independence 2 Test 3. = Reject H 0 if 25
26 Test of Independence With =.05 and (21)(41) = 3 d.f., Reject H 0 if pvalue.05 or Calculate Test Statistic Expected Frequencies Price Colonial Log SplitLevel AFrame Total < $99K > $99K Total
27 Test of Independence Test Statistic 2 ( ) (6 11) (3 6.75)
28 Test of Independence Conclusion Using the pvalue Approach Area in Upper Tail Value (df = 3) Because 2 = is between and 9.348, the area in the upper tail of the distribution is between.05 and.025. The pvalue. We can reject the null hypothesis. Actual pvalue is
29 Test of Independence Conclusion Using the Critical Value Approach 2 = Conclusion We reject, at the.05 level of significance, the assumption that the price of the home is independent of the style of home that is purchased. 29
30 Goodness of Fit Test: Multinomial Probability Distribution 1. State the null and alternative hypotheses. H 0 : The population follows a multinomial distribution with specified probabilities for each of the k categories H a : The population does not follow a multinomial distribution with specified probabilities for each of the k categories 30
31 Goodness of Fit Test: Multinomial Probability Distribution 2. Select a random sample and record the observed frequency, f i, for each of the k categories. 3. Assuming H 0 is true, compute the expected frequency, e i, in each category by multiplying the category probability by the sample size. 31
32 Goodness of Fit Test: Multinomial Probability Distribution 4. Compute the value of the test statistic. 2 k ( fi ei) e i 1 where: f i = observed frequency for category i e i = expected frequency for category i k = number of categories Note: The test statistic has a chisquare distribution with k 1 df provided that the expected frequencies are 5 or more for all categories. i 2 32
33 Goodness of Fit Test: Multinomial Probability Distribution 5. Rejection rule: pvalue approach: Reject H 0 if pvalue < Critical value approach: Reject H 0 if 2 2 where is the significance level and there are k 1 degrees of freedom 33
34 Multinomial Distribution Goodness of Fit Test Example: Finger Lakes Homes (A) Finger Lakes Homes manufactures four models of prefabricated homes, a twostory colonial, a log cabin, a splitlevel, and an Aframe. To help in production planning, management would like to determine if previous customer purchases indicate that there is a preference in the style selected. 34
35 Multinomial Distribution Goodness of Fit Test Example: Finger Lakes Homes (A) The number of homes sold of each model for 100 sales over the past two years is shown below. Split A Model Colonial Log Level Frame # Sold
36 Multinomial Distribution Goodness of Fit Test 1. Hypotheses H 0 : p C = p L = p S = p A =.25 H a : The population proportions are not p where: C =.25, p L =.25, p S =.25, and p A =.25 p C = population proportion that purchase a colonial p L = population proportion that purchase a log cabin p S = population proportion that purchase a splitlevel p A = population proportion that purchase an Aframe 2. Goodness of Fit Test 2 Test 3. = Reject H 0 if
37 Multinomial Distribution Goodness of Fit Test Rejection Rule Reject H0 if pvalue.05 or With =.05 and k  1 = 41 = 3 degrees of freedom Do Not Reject H 0 Reject H
38 Multinomial Distribution Goodness of Fit Test 5. Calculate Test Statistic Expected Frequencies Test Statistic e 1 =.25(100) = 25 e 2 =.25(100) = 25 e 3 =.25(100) = 25 e 4 =.25(100) = 25 2 (30 25) (20 25) (35 25) (15 25) =
39 Multinomial Distribution Goodness of Fit Test Conclusion Using the pvalue Approach Area in Upper Tail Value (df = 3) Because 2 = 10 is between and , the area in the upper tail of the distribution is between.025 and.01. The pvalue. We can reject the null hypothesis. 39
40 Multinomial Distribution Goodness of Fit Test Conclusion Using the Critical Value Approach 2 = 10 > Conclusion We reject, at the.05 level of significance, the assumption that there is no home style preference. 40
41 Goodness of Fit Test: Normal Distribution 1. State the null and alternative hypotheses. H 0 : The population has a normal distribution H a : The population does not have a normal distribution 2. Select a random sample and a. Compute the mean and standard deviation. b. Define intervals of values so that the expected frequency is at least 5 for each interval. c. For each interval, record the observed frequencies 3. Compute the expected frequency, e i, for each interval. (Multiply the sample size by the probability of a normal random variable being in the interval.) 41
42 Goodness of Fit Test: Normal Distribution 4. Compute the value of the test statistic. 2 k i 1 ( f e ) i e i i Reject H0 if (where is the significance level and there are k 2 1 degrees of freedom). 42
43 Goodness of Fit Test: Normal Distribution Example: IQ Computers IQ Computers (one better than HP?) manufactures and sells a general purpose microcomputer. As part of a study to evaluate sales personnel, management wants to determine, at a.05 significance level, if the annual sales volume (number of units sold by a salesperson) follows a normal probability distribution. 43
44 Goodness of Fit Test: Normal Distribution Example: IQ Computers A simple random sample of 30 of the salespeople was taken and their numbers of units sold are listed below (mean = 71, standard deviation = 18.54) 44
45 Goodness of Fit Test: Normal Distribution 1. Hypotheses H 0 : The population of number of units sold has a normal distribution with mean 71 and standard deviation H a : The population of number of units sold does not have a normal distribution with mean 71 and standard deviation Goodness of Fit Test 2 Test 3. = Reject H 0 if
46 Goodness of Fit Test: Normal Distribution 5. Interval Definition To satisfy the requirement of an expected frequency of at least 5 in each interval we will divide the normal distribution into 30/5 = 6 equal probability intervals. 46
47 Goodness of Fit Test: Normal Distribution Interval Definition Areas = 1.00/6 = (18.54) = = (18.54) 47
48 Goodness of Fit Test: Normal Distribution Observed and Expected Frequencies i f i e i f i  e i Less than to to to to More than Total
49 Goodness of Fit Test: Normal Distribution Rejection Rule With =.05 and k  p  1 = = 3 d.f. (where k = number of categories and p = number 2 of population parameters estimated), Test Statistic Reject H 0 if pvalue.05 or (1) ( 2) (1) (0) ( 1) (1)
50 Goodness of Fit Test: Normal Distribution Conclusion Using the pvalue Approach Area in Upper Tail Value (df = 3) Because 2 = is between.584 and in the ChiSquare Distribution Table, the area in the upper tail of the distribution is between.90 and.10. The pvalue >. We cannot reject the null hypothesis. There is little evidence to support rejecting the assumption the population is normally distributed with = 71 and = Actual pvalue is
51 Goodness of Fit Test: Normal Distribution Conclusion Using the Critical Value Approach 2 = 1.6 < Conclusion We cannot reject, at the.05 level of significance, the assumption that the population is normally distributed with = 71 and =
52 Goodness of Fit Test: Poisson Distribution 1. Set up the null and alternative hypotheses. 2. Select a random sample and a. Record the observed frequency, f i, for each of the k values of the Poisson random variable. b. Compute the mean number of occurrences,. 3. Compute the expected frequency of occurrences, e i, for each value of the Poisson random variable. 52
53 Goodness of Fit Test: Poisson Distribution 4. Compute the value of the test statistic. 5. Reject H 0 if 2 k i f e 2 i e i i (where is the significance level and there are k 1 1 degrees of freedom). 53
54 Example: Troy Parking Garage Poisson Distribution Goodness of Fit Test In studying the need for an additional entrance to a city parking garage, a consultant has recommended an approach that is applicable only in situations where the number of cars entering during a specified time period follows a Poisson distribution. 54
55 Example: Troy Parking Garage Poisson Distribution Goodness of Fit Test A random sample of 100 oneminute time intervals resulted in the customer arrivals listed below. A statistical test must be conducted to see if the assumption of a Poisson distribution is reasonable. # Arrivals Frequency
56 Example: Troy Parking Garage Poisson Distribution Goodness of Fit Test 1. Hypotheses H 0 : Number of cars entering the garage during a oneminute interval is Poisson distributed. H a : Number of cars entering the garage during a oneminute interval is not Poisson distributed 2. Goodness of Fit Test 2 Test 3. = Reject H 0 if
57 Example: Troy Parking Garage Poisson Distribution Goodness of Fit Test 5. Calculation Estimate of Poisson Probability Function otal Arrivals = 0(0) + 1(1) + 2(4) (1) = 600 Total Time Periods = 100 Estimate of = 600/100 = 6 Hence, f( x) 6 x 6 e x! 57
58 Example: Troy Parking Garage Poisson Distribution Goodness of Fit Test Expected Frequencies x f (x ) nf (x ) x f (x ) nf (x ) Total
59 Example: Troy Parking Garage Poisson Distribution Goodness of Fit Test Observed and Expected Frequencies i f i e i f i  e i 0 or 1 or or more
60 Example: Troy Parking Garage Poisson Distribution Goodness of Fit Test Test Statistic ( 1.20) (1.08) (2.01) Rejection Rule With =.05 and k  p  1 = = 7 d.f. (where k = number of categories and p = number of population parameters estimated), Reject H 0 if 2 > Conclusion We cannot reject H 0. There s no reason to doubt the assumption of a Poisson distribution. 60
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 informationIs it statistically significant? The chisquare test
UAS Conference Series 2013/14 Is it statistically significant? The chisquare test Dr Gosia Turner Student Data Management and Analysis 14 September 2010 Page 1 Why chisquare? Tests whether two categorical
More informationChiSquare Test. Contingency Tables. Contingency Tables. ChiSquare Test for Independence. ChiSquare Tests for GoodnessofFit
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
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 informationMATH 10: Elementary Statistics and Probability Chapter 11: The ChiSquare Distribution
MATH 10: Elementary Statistics and Probability Chapter 11: The ChiSquare Distribution Tony Pourmohamad Department of Mathematics De Anza College Spring 2015 Objectives By the end of this set of slides,
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 information112 Goodness of Fit Test
112 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 oneway frequency table). We will use a hypothesis
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 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 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 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 informationIntroduction to Hypothesis Testing. Copyright 2014 Pearson Education, Inc. 91
Introduction to Hypothesis Testing 91 Learning Outcomes Outcome 1. Formulate null and alternative hypotheses for applications involving a single population mean or proportion. Outcome 2. Know what Type
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 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 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 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 informationChisquare test Testing for independeny The r x c contingency tables square test
Chisquare test Testing for independeny The r x c contingency tables square test 1 The chisquare distribution HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computeraided
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 informationSection 12 Part 2. Chisquare test
Section 12 Part 2 Chisquare 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
More informationOneWay Analysis of Variance (ANOVA) Example Problem
OneWay Analysis of Variance (ANOVA) Example Problem Introduction Analysis of Variance (ANOVA) is a hypothesistesting technique used to test the equality of two or more population (or treatment) means
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 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 informationMath 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
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 informationCATEGORICAL DATA ChiSquare Tests for Univariate Data
CATEGORICAL DATA ChiSquare Tests For Univariate Data 1 CATEGORICAL DATA ChiSquare Tests for Univariate Data Recall that a categorical variable is one in which the possible values are categories or groupings.
More informationHypothesis Testing. Bluman Chapter 8
CHAPTER 8 Learning Objectives C H A P T E R E I G H T Hypothesis Testing 1 Outline 81 Steps in Traditional Method 82 z Test for a Mean 83 t Test for a Mean 84 z Test for a Proportion 85 2 Test for
More informationChapter 8 Introduction to Hypothesis Testing
Chapter 8 Student Lecture Notes 81 Chapter 8 Introduction to Hypothesis Testing Fall 26 Fundamentals of Business Statistics 1 Chapter Goals After completing this chapter, you should be able to: Formulate
More informationHypothesis Testing for a Proportion
Math 122 Intro to Stats Chapter 6 Semester II, 201516 Inference for Categorical Data Hypothesis Testing for a Proportion In a survey, 1864 out of 2246 randomly selected adults said texting while driving
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 informationRecommend 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
More informationHypothesis Testing  One Mean
Hypothesis Testing  One Mean A hypothesis is simply a statement that something is true. Typically, there are two hypotheses in a hypothesis test: the null, and the alternative. Null Hypothesis The hypothesis
More informationChi 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.
More informationChiSquare Test for Qualitative Data
ChiSquare 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
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 information2 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
More informationCHAPTER 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
More informationLecture 7: Binomial Test, Chisquare
Lecture 7: Binomial Test, Chisquare Test, and ANOVA May, 01 GENOME 560, Spring 01 Goals ANOVA Binomial test Chi square test Fisher s exact test Su In Lee, CSE & GS suinlee@uw.edu 1 Whirlwind Tour of One/Two
More informationChapter 2 Probability Topics SPSS T tests
Chapter 2 Probability Topics SPSS T tests Data file used: gss.sav In the lecture about chapter 2, only the OneSample T test has been explained. In this handout, we also give the SPSS methods to perform
More informationSimulating ChiSquare Test Using Excel
Simulating ChiSquare 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
More informationStat 104: Quantitative Methods for Economists Class 26: ChiSquare Tests
Stat 104: Quantitative Methods for Economists Class 26: ChiSquare Tests 1 Two Techniques The first is a goodnessoffit test applied to data produced by a multinomial experiment, a generalization of a
More informationUsing 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,
More informationRegression Analysis: A Complete Example
Regression Analysis: A Complete Example This section works out an example that includes all the topics we have discussed so far in this chapter. A complete example of regression analysis. PhotoDisc, Inc./Getty
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 informationTest Positive True Positive False Positive. Test Negative False Negative True Negative. Figure 51: 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
More informationMATH 10: Elementary Statistics and Probability Chapter 9: Hypothesis Testing with One Sample
MATH 10: Elementary Statistics and Probability Chapter 9: Hypothesis Testing with One Sample Tony Pourmohamad Department of Mathematics De Anza College Spring 2015 Objectives By the end of this set of
More informationPart 3. Comparing Groups. Chapter 7 Comparing Paired Groups 189. Chapter 8 Comparing Two Independent Groups 217
Part 3 Comparing Groups Chapter 7 Comparing Paired Groups 189 Chapter 8 Comparing Two Independent Groups 217 Chapter 9 Comparing More Than Two Groups 257 188 Elementary Statistics Using SAS Chapter 7 Comparing
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 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 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 informationThe Basics of a Hypothesis Test
Overview The Basics of a Test Dr Tom Ilvento Department of Food and Resource Economics Alternative way to make inferences from a sample to the Population is via a Test A hypothesis test is based upon A
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 informationDEPARTMENT OF POLITICAL SCIENCE AND INTERNATIONAL RELATIONS. Posc/Uapp 816 CONTINGENCY TABLES
DEPARTMENT OF POLITICAL SCIENCE AND INTERNATIONAL RELATIONS Posc/Uapp 816 CONTINGENCY TABLES I. AGENDA: A. Crossclassifications 1. Twobytwo and R by C tables 2. Statistical independence 3. The interpretation
More informationWording of Final Conclusion. Slide 1
Wording of Final Conclusion Slide 1 8.3: Assumptions for Testing Slide 2 Claims About Population Means 1) The sample is a simple random sample. 2) The value of the population standard deviation σ is known
More informationChapter 11. Chapter 11 Overview. Chapter 11 Objectives 11/24/2015. Other ChiSquare Tests
11/4/015 Chapter 11 Overview Chapter 11 Introduction 111 Test for Goodness of Fit 11 Tests Using Contingency Tables Other ChiSquare Tests McGrawHill, Bluman, 7th ed., Chapter 11 1 Bluman, Chapter 11
More informationPearson s 2x2 ChiSquare Test of Independence  Analysis of the Relationship between two Qualitative
Pearson s 2x2 ChiSquare Test of Independence  Analysis of the Relationship between two Qualitative or (Binary) Variables Analysis of 2BetweenGroup Data with a Qualitative (Binary) Response Variable
More informationThe calculations lead to the following values: d 2 = 46, n = 8, s d 2 = 4, s d = 2, SEof d = s d n s d n
EXAMPLE 1: Paired ttest and tinterval DBP Readings by Two Devices The diastolic blood pressures (DBP) of 8 patients were determined using two techniques: the standard method used by medical personnel
More informationχ 2 = (O i E i ) 2 E i
Chapter 24 TwoWay Tables and the ChiSquare Test We look at twoway tables to determine association of paired qualitative data. We look at marginal distributions, conditional distributions and bar graphs.
More informationLAB 4 ASSIGNMENT CONFIDENCE INTERVALS AND HYPOTHESIS TESTING. Using Data to Make Decisions
LAB 4 ASSIGNMENT CONFIDENCE INTERVALS AND HYPOTHESIS TESTING This lab assignment will give you the opportunity to explore the concept of a confidence interval and hypothesis testing in the context of a
More informationData Science and Statistics in Research: unlocking the power of your data
Data Science and Statistics in Research: unlocking the power of your data Session 2.4: Hypothesis testing II 1/ 25 OUTLINE Independent ttest Paired ttest Chisquared test 2/ 25 Independent ttest 3/
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 informationHypothesis testing: Examples. AMS7, Spring 2012
Hypothesis testing: Examples AMS7, Spring 2012 Example 1: Testing a Claim about a Proportion Sect. 7.3, # 2: Survey of Drinking: In a Gallup survey, 1087 randomly selected adults were asked whether they
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 informationANOVA  Analysis of Variance
ANOVA  Analysis of Variance ANOVA  Analysis of Variance Extends independentsamples t test Compares the means of groups of independent observations Don t be fooled by the name. ANOVA does not compare
More informationTopic 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
More informationCase l: Watchdog, Inc: Keeping hospitals honest
Case l: Watchdog, Inc: Keeping hospitals honest Insurance companies hire independent firms like Watchdog, Inc to review their hospital bills. They pay Watchdog a percentage of the overcharges it finds.
More informationStatistical Impact of Slip Simulator Training at Los Alamos National Laboratory
LAUR1224572 Approved for public release; distribution is unlimited Statistical Impact of Slip Simulator Training at Los Alamos National Laboratory Alicia GarciaLopez Steven R. Booth September 2012
More informationMATH Chapter 23 April 15 and 17, 2013 page 1 of 8 CHAPTER 23: COMPARING TWO CATEGORICAL VARIABLES THE CHISQUARE TEST
MATH 1342. Chapter 23 April 15 and 17, 2013 page 1 of 8 CHAPTER 23: COMPARING TWO CATEGORICAL VARIABLES THE CHISQUARE TEST Relationships: Categorical Variables Chapter 21: compare proportions of successes
More information6. Statistical Inference: Significance Tests
6. Statistical Inference: Significance Tests Goal: Use statistical methods to check hypotheses such as Women's participation rates in elections in France is higher than in Germany. (an effect) Ethnic divisions
More informationSydney Roberts Predicting Age Group Swimmers 50 Freestyle Time 1. 1. Introduction p. 2. 2. Statistical Methods Used p. 5. 3. 10 and under Males p.
Sydney Roberts Predicting Age Group Swimmers 50 Freestyle Time 1 Table of Contents 1. Introduction p. 2 2. Statistical Methods Used p. 5 3. 10 and under Males p. 8 4. 11 and up Males p. 10 5. 10 and under
More informationThe alternative hypothesis,, is the statement that the parameter value somehow differs from that claimed by the null hypothesis. : 0.5 :>0.5 :<0.
Section 8.28.5 Null and Alternative Hypotheses... The null hypothesis,, is a statement that the value of a population parameter is equal to some claimed value. :=0.5 The alternative hypothesis,, is the
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 information: P(winter) = P(spring) = P(summer)= P(fall) H A. : Homicides are independent of seasons. : Homicides are evenly distributed across seasons
ChiSquare 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?
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 11 Chi square Tests
11.1 Introduction 259 Chapter 11 Chi square Tests 11.1 Introduction In this chapter we will consider the use of chi square tests (χ 2 tests) to determine whether hypothesized models are consistent with
More informationBasic Statistics Self Assessment Test
Basic Statistics Self Assessment Test Professor Douglas H. Jones PAGE 1 A sodadispensing machine fills 12ounce cans of soda using a normal distribution with a mean of 12.1 ounces and a standard deviation
More informationAn Introduction to Statistics Course (ECOE 1302) Spring Semester 2011 Chapter 10 TWOSAMPLE TESTS
The Islamic University of Gaza Faculty of Commerce Department of Economics and Political Sciences An Introduction to Statistics Course (ECOE 130) Spring Semester 011 Chapter 10 TWOSAMPLE TESTS Practice
More informationChiSquare Tests and the FDistribution. Goodness of Fit Multinomial Experiments. Chapter 10
Chapter 0 ChiSquare Tests and the FDistribution 0 Goodness of Fit Multinomial xperiments A multinomial experiment is a probability experiment consisting of a fixed number of trials in which there are
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 informationUsing CrunchIt (http://bcs.whfreeman.com/crunchit/bps4e) or StatCrunch (www.calvin.edu/go/statcrunch)
Using CrunchIt (http://bcs.whfreeman.com/crunchit/bps4e) or StatCrunch (www.calvin.edu/go/statcrunch) 1. In general, this package is far easier to use than many statistical packages. Every so often, however,
More informationChapter 8 Hypothesis Testing Chapter 8 Hypothesis Testing 81 Overview 82 Basics of Hypothesis Testing
Chapter 8 Hypothesis Testing 1 Chapter 8 Hypothesis Testing 81 Overview 82 Basics of Hypothesis Testing 83 Testing a Claim About a Proportion 85 Testing a Claim About a Mean: s Not Known 86 Testing
More informationCHAPTER 12. ChiSquare Tests and Nonparametric Tests LEARNING OBJECTIVES. USING STATISTICS @ T.C. Resort Properties
CHAPTER 1 ChiSquare Tests and Nonparametric Tests USING STATISTICS @ T.C. Resort Properties 1.1 CHISQUARE TEST FOR THE DIFFERENCE BETWEEN TWO PROPORTIONS (INDEPENDENT SAMPLES) 1. CHISQUARE TEST FOR
More informationChapter 9: Hypothesis Testing GBS221, Class April 15, 2013 Notes Compiled by Nicolas C. Rouse, Instructor, Phoenix College
Chapter Objectives 1. Learn how to formulate and test hypotheses about a population mean and a population proportion. 2. Be able to use an Excel worksheet to conduct hypothesis tests about population means
More informationBusiness Statistics. Lecture 8: More Hypothesis Testing
Business Statistics Lecture 8: More Hypothesis Testing 1 Goals for this Lecture Review of ttests Additional hypothesis tests Twosample tests Paired tests 2 The Basic Idea of Hypothesis Testing Start
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 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 informationHaving a coin come up heads or tails is a variable on a nominal scale. Heads is a different category from tails.
Chisquare Goodness of Fit Test The chisquare test is designed to test differences whether one frequency is different from another frequency. The chisquare test is designed for use with data on a nominal
More informationNull Hypothesis H 0. The null hypothesis (denoted by H 0
Hypothesis test In statistics, a hypothesis is a claim or statement about a property of a population. A hypothesis test (or test of significance) is a standard procedure for testing a claim about a property
More informationstatistics Chisquare 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
More informationAugust 2012 EXAMINATIONS Solution Part I
August 01 EXAMINATIONS Solution Part I (1) In a random sample of 600 eligible voters, the probability that less than 38% will be in favour of this policy is closest to (B) () In a large random sample,
More informationChisquare (χ 2 ) Tests
Math 442  Mathematical Statistics II May 5, 2008 Common Uses of the χ 2 test. 1. Testing Goodnessoffit. Chisquare (χ 2 ) Tests 2. Testing Equality of Several Proportions. 3. Homogeneity Test. 4. Testing
More informationIntroduction to Hypothesis Testing. Hypothesis Testing. Step 1: State the Hypotheses
Introduction to Hypothesis Testing 1 Hypothesis Testing A hypothesis test is a statistical procedure that uses sample data to evaluate a hypothesis about a population Hypothesis is stated in terms of the
More informationGoodness of Fit. Proportional Model. Probability Models & Frequency Data
Probability Models & Frequency Data Goodness of Fit Proportional Model Chisquare Statistic Example R Distribution Assumptions Example R 1 Goodness of Fit Goodness of fit tests are used to compare any
More informationUnit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression
Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression Objectives: To perform a hypothesis test concerning the slope of a least squares line To recognize that testing for a
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 informationChapter 8. Hypothesis Testing
Chapter 8 Hypothesis Testing Hypothesis In statistics, a hypothesis is a claim or statement about a property of a population. A hypothesis test (or test of significance) is a standard procedure for testing
More informationChapter 19 The ChiSquare Test
Tutorial for the integration of the software R with introductory statistics Copyright c Grethe Hystad Chapter 19 The ChiSquare Test In this chapter, we will discuss the following topics: We will plot
More informationWe 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
More informationStatistics 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 1.11.6) Objectives
More information