Chi-Square Test (χ 2 ) χ – greek symbol “chi”. Chi-Square Test (χ 2 ) When is the Chi-Square Test used? The chi-square test is used to determine whether.

Slides:



Advertisements
Similar presentations
The mystery of the CHI SQUARE Is it CHEE square Or CHAI Square?!
Advertisements

CHI-SQUARE(X2) DISTRIBUTION
Chi Square Test X2.
Chi square.  Non-parametric test that’s useful when your sample violates the assumptions about normality required by other tests ◦ All other tests we’ve.
Basic Statistics The Chi Square Test of Independence.
Statistical Inference for Frequency Data Chapter 16.
Chapter 10 Chi-Square Tests and the F- Distribution 1 Larson/Farber 4th ed.
Please turn in your signed syllabus. We will be going to get textbooks shortly after class starts. Homework: Reading Guide – Chapter 2: The Chemical Context.
Introduction to Chi-Square Procedures March 11, 2010.
Ch 15 - Chi-square Nonparametric Methods: Chi-Square Applications
Chapter 26: Comparing Counts. To analyze categorical data, we construct two-way tables and examine the counts of percents of the explanatory and response.
Chi-Square Test.
Diversity and Distribution of Species
The Kruskal-Wallis Test The Kruskal-Wallis test is a nonparametric test that can be used to determine whether three or more independent samples were.
Goodness of Fit Test for Proportions of Multinomial Population Chi-square distribution Hypotheses test/Goodness of fit test.
1 1 Slide IS 310 – Business Statistics IS 310 Business Statistics CSU Long Beach.
The Chi-square Statistic. Goodness of fit 0 This test is used to decide whether there is any difference between the observed (experimental) value and.
Hypothesis Testing:.
GOODNESS OF FIT TEST & CONTINGENCY TABLE
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 1 Understandable Statistics S eventh Edition By Brase and Brase Prepared by: Lynn Smith.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 10.7.
Section 10.1 Goodness of Fit. Section 10.1 Objectives Use the chi-square distribution to test whether a frequency distribution fits a claimed distribution.
1 1 Slide IS 310 – Business Statistics IS 310 Business Statistics CSU Long Beach.
1 1 Slide © 2005 Thomson/South-Western Chapter 12 Tests of Goodness of Fit and Independence n Goodness of Fit Test: A Multinomial Population Goodness of.
Two Variable Statistics
Quantitative Methods Partly based on materials by Sherry O’Sullivan Part 3 Chi - Squared Statistic.
Chapter Chi-Square Tests and the F-Distribution 1 of © 2012 Pearson Education, Inc. All rights reserved.
Chi-Square X 2. Parking lot exercise Graph the distribution of car values for each parking lot Fill in the frequency and percentage tables.
1 In this case, each element of a population is assigned to one and only one of several classes or categories. Chapter 11 – Test of Independence - Hypothesis.
Chi-Square Procedures Chi-Square Test for Goodness of Fit, Independence of Variables, and Homogeneity of Proportions.
Chapter 10 Chi-Square Tests and the F- Distribution 1 Larson/Farber 4th ed.
Chi-Square Test.
Chapter 10 Chi-Square Tests and the F-Distribution
GOODNESS OF FIT Larson/Farber 4th ed 1 Section 10.1.
Learning Objectives Copyright © 2002 South-Western/Thomson Learning Statistical Testing of Differences CHAPTER fifteen.
Chi-Square Test James A. Pershing, Ph.D. Indiana University.
Non-parametric tests (chi-square test) Dr. Omar Al Jadaan Assistant Professor – Computer Science & Mathematics.
Section 10.2 Independence. Section 10.2 Objectives Use a chi-square distribution to test whether two variables are independent Use a contingency table.
© Copyright McGraw-Hill CHAPTER 11 Other Chi-Square Tests.
Chapter Outline Goodness of Fit test Test of Independence.
The table shows a random sample of 100 hikers and the area of hiking preferred. Are hiking area preference and gender independent? Hiking Preference Area.
Copyright © Cengage Learning. All rights reserved. Chi-Square and F Distributions 10.
11.2 Tests Using Contingency Tables When data can be tabulated in table form in terms of frequencies, several types of hypotheses can be tested by using.
Chi-Square X 2. Review: the “null” hypothesis Inferential statistics are used to test hypotheses Whenever we use inferential statistics the “null hypothesis”
Chi-Square X 2. Review: the “null” hypothesis Inferential statistics are used to test hypotheses Whenever we use inferential statistics the “null hypothesis”
Science Practice 2: The student can use mathematics appropriately. Science Practice 5: The student can perform data analysis and evaluation of evidence.
Chapter 14 – 1 Chi-Square Chi-Square as a Statistical Test Statistical Independence Hypothesis Testing with Chi-Square The Assumptions Stating the Research.
1 1 Slide © 2008 Thomson South-Western. All Rights Reserved Chapter 12 Tests of Goodness of Fit and Independence n Goodness of Fit Test: A Multinomial.
+ Section 11.1 Chi-Square Goodness-of-Fit Tests. + Introduction In the previous chapter, we discussed inference procedures for comparing the proportion.
11.1 Chi-Square Tests for Goodness of Fit Objectives SWBAT: STATE appropriate hypotheses and COMPUTE expected counts for a chi- square test for goodness.
Section 10.2 Objectives Use a contingency table to find expected frequencies Use a chi-square distribution to test whether two variables are independent.
Statistics for Psychology CHAPTER SIXTH EDITION Statistics for Psychology, Sixth Edition Arthur Aron | Elliot J. Coups | Elaine N. Aron Copyright © 2013.
CHI SQUARE DISTRIBUTION. The Chi-Square (  2 ) Distribution The chi-square distribution is the probability distribution of the sum of several independent,
Chi Square Analysis. What is the chi-square statistic? The chi-square (chi, the Greek letter pronounced "kye”) statistic is a nonparametric statistical.
Section 10.1 Goodness of Fit © 2012 Pearson Education, Inc. All rights reserved. 1 of 91.
Chapter 10 Chi-Square Tests and the F-Distribution
Chi-Square Test.
Other Chi-Square Tests
Test of independence: Contingency Table
Hypothesis Testing Review
Chapter 10 Chi-Square Tests and the F-Distribution
Section 10-1 – Goodness of Fit
Elementary Statistics: Picturing The World
Chi-Square Test.
Chi-Square Test.
Contingency Tables: Independence and Homogeneity
Chi-Square Test.
Assistant prof. Dr. Mayasah A. Sadiq FICMS-FM
Chapter Outline Goodness of Fit test Test of Independence.
Presentation transcript:

Chi-Square Test (χ 2 ) χ – greek symbol “chi”

Chi-Square Test (χ 2 ) When is the Chi-Square Test used? The chi-square test is used to determine whether there is a significant difference between the expected frequencies and the observed frequencies in one or more categories. Also, the chi-square test is used to test for independence of two or more different categories. If there is a significant difference, it basically implies that χ 2 > σ, where σ is the stated significance level with usual values of 1%, 5% or 10%. Take note that the significance level (σ) is always given in a problem.

Chi-Square Test (χ 2 ) Chi-Square Test Requirements 1. Quantitative data. 2. One or more categories. 3. Independent observations. 4. Adequate sample size (at least 10). 5. Simple random sample. 6. Data in frequency form. 7. All observations must be used.

Chi-Square Test (χ 2 ) How to find the value of χ 2 ? Consider this problem: Carl, the manager of a car dealership, did not want to stock cars that were bought less frequently because of their unpopular color. The five colors that he ordered were red, yellow, green, blue, and white. According to Carl, the expected frequencies or number of customers choosing each color should follow the percentages of last year. She felt 20% would choose yellow, 30% would choose red, 10% would choose green, 10% would choose blue, and 30% would choose white. She now took a random sample of 150 customers and asked them their color preferences. Is there a significant difference between the observed and expected frequencies? σ = 5%

Chi-Square Test (χ 2 ) Color Preferences for 150 Customers: Category ColorObserved Frequency Yellow35 Red50 Green30 Blue10 White25

Chi-Square Test (χ 2 ) We are testing if Carl’s expected frequencies “fit” with the observed frequencies. That is why a chi square test is sometimes called the goodness of fit or how good some expected frequency fits into observed data.

Chi-Square Test (χ 2 ) We must first state our hypotheses, (H o and H a ) Null hypothesis -There is no significant difference between the expected and observed frequencies. Alternative hypothesis -There is a significant difference between the expected and observed frequencies. In other words, if the probability of getting the observed frequency is within our area of rejection (bounded by our chi critical value), we are going to reject our null hypothesis. Otherwise, we are going to approve.

Chi-Square Test (χ 2 ) Formula for Calculating χ 2 where O is the observed frequency; E is the expected frequency; We already know the observed frequencies which were listed in the previous slide. We need to find out the expected frequencies.

Chi-Square Test (χ 2 ) How to get Expected Frequency (E) We get the total number of customers and multiply it to its corresponding percentage. For Yellow: 150 x 0.2 = 30 For Red: 150 x 0.3 = 45 … and so on. Values are tabulated in the next slide.

Chi-Square Test (χ 2 ) Category ColorObserved FrequencyExpected Frequency Yellow3530 Red5045 Green3015 Blue1015 White2545 Total:150

Chi-Square Test (χ 2 ) Getting χ 2 To get χ 2, we take the summation of the squares of the differences between observed frequencies and expected frequencies all over each corresponding expected frequency. Therefore, for our first data set, = 5  5^2 = 25  25 / 30 (E) = 5/6 Second data set, 50-45=5  5^2 = 25  25 / 45 (E) = 5/9 Following with the remaining three sets, we add all of those values. That is our chi squared statistic.

Chi-Square Test (χ 2 ) Category Color Observed Frequency Expected Frequency O-E(O-E) 2 (O-E) 2 /E Yellow Red Green Blue White X^2 = 26.95

Chi-Square Test (χ 2 ) Calculating our Chi Critical Value (χ c 2 ) To get χ c 2, we get our Chi table and locate our critical value with degrees of freedom (Df) and significance level ( σ). Df = 5 – 1 = 4 σ = 0.05

Chi-Square Test (χ 2 ) Df = 4 σ = 0.05 χ c 2 = 9.49 χ 2 = 26.95

Chi-Square Test (χ 2 ) Conclusion: χ 2 is a lot bigger from our Chi critical value χ c 2. In the Chi distribution graph, the area bounded by our Chi critical value (area of rejection) definitely overlaps with the area bounded by our Chi statistic. We are therefore inclined to reject our null hypothesis at 5% significance level and Carl’s distribution is incorrect. (does not fit)

Chi-Square Test (χ 2 ) TEST FOR INDEPENDENCE Problem: In a certain town, there are about one million eligible voters. A simple random sample of eligible voters was chosen to study the relationship between sex and participation in the last election. The contingency table is shown in the 2 nd slide after this slide :P. We want to find out if gender and voting are independent. σ = 0.05

Chi-Square Test (χ 2 ) Null and Alternate Hypotheses H o = Sex is independent from voting. H a = Sex and voting are dependent.

Chi-Square Test (χ 2 ) OBSERVED FREQUENCIES MenWomenTotal Voted Didn’t vote Total Contingency Table

Chi-Square Test (χ 2 ) Formula for Expected Frequency In order to get the expected frequency, it is defined by the formula: Expected frequency = RowTotal x ColumnTotal / GRAND TOTAL

Chi-Square Test (χ 2 ) EXPECTED FREQUENCIES MenWomenTotal Voted Didn’t vote Total

Chi-Square Test (χ 2 ) Summary of Frequencies ObservedExpected((O-E)^2)/E Men Voted Men didn’t vote Women Voted Women didn’t vote χ 2 = 6.53

Chi-Square Test (χ 2 ) Computing D f (degrees of freedom) Degrees of freedom for chi square test of independence is equal to: (rows-1)*(columns-1) = D f (2-1) = 1 * (2-1) = 1 * 1 = D f D f = 1 We then get our chi critical value by getting Chi table and locating significance level 0.05 and D f of 1.

Chi-Square Test (χ 2 ) χ c 2 = 3.84 χ 2 = 6.53

Chi-Square Test (χ 2 ) Conclusion: Since our χ c 2 > χ 2, we are going to reject that our null hypothesis is true, and approve of the fact that sex and voting are dependent in the town.

Chi-Square Test (χ 2 ) Summary of Formulas: Goodness of Fit: Df = number of categories – 1 Test for Independence (Contingency Table) Df = (rows-1)(columns-1)