TOPICSLIDEWhen is the chi-square test used?2Formula for the one-way and two-way chi-square test3The one-way chi-square test4Example: One-way chi-square test – Grade Distribution5• Tutorial: Performing a one-way chi-square test in Excel 2007The two-way chi-square test13• Tutorial: Performing a two-way chi-square test in Excel 2007THE CHI-SQUARE TESTChapter 10➊ When data represent frequency or percentage in each category (i.e., Nominal scale data)➋ Subjects in each category are independent from each other➌ There are at least 5 observations (i.e., scores) per categoryTHE CHI-SQUARE TESTChapter 10Chapter 10EEO22Means to sum all for all categories/cellsRefers to the observed frequencyRefers to the expected frequencyRefers to the expected frequencySymbol for the chi-square testNOTE: This formula is used for both one-way and two-way chi-square testsTHE CHI-SQUARE TEST➊ When there is only one independent variable• With two or more levels (or categories)➋ When the data are nominal scale➌ The null hypothesis is rejected when the obtained chi-square value is equal to or greater than the critical chi-square value The degrees of freedom for the one-way chi-square test is the number of categories minus one or•df= r – 1, where r is the number of categoriesTHE ONE-WAY CHI-SQUARE TESTChapter 10THE ONE-WAY CHI-SQUARE TESTChapter 10An instructor makes out his final grades for 200 students in his introductory statistics class. He is curious to see if his grade distribution resembles the “normal curve” and notes from the college catalog that in a normal distribution of grades 45% of them would be C’s, 24% of them would be B’s, 24% D’s, 3.5% of them would be A’s, and 3.5% F’s. The instructor compared the frequency of grades given in his class to the normal curve. The frequency of each grade is given on the following slide.THE ONE-WAY CHI-SQUARE TESTChapter 10➊ Null Hypothesis:• The grade distribution used by the instructor will be the same as the distribution described in the college catalog. Any observed differences between the frequencies of grades given by the instructor versus what is recommended in the college catalog is assumed to be solely due to random error.THE ONE-WAY CHI-SQUARE TESTChapter 10Observed Frequency Expected % Expected Frequency15 3.5% 753 24.0% 4887 45.0% 9033 24.0% 4812 3.5% 7Total: 200 200➊ The ‘Observed Frequency’ column represents the frequency of grades given by the instructor➋ The ‘Expected %’ column represents the percentage of grades to be given for each category➌ The ‘Expected Frequency’ column is the expected frequency of grades suggested by the college catalogTHE ONE-WAY CHI-SQUARE TESTChapter 10Observed Frequency Expected % Expected Frequency15 3.5% 753 24.0% 4887 45.0% 9033 24.0% 4812 3.5% 7Total: 200 200➊ The observed values and expected values must be in the same unit of measurement - either percentage or frequency➋ The expected percentage will be converted to frequency by multiplying each expected percentage (expressed as a proportion) by the total number of observationsTHE ONE-WAY CHI-SQUARE TESTChapter 10Observed Frequency Expected % Expected Frequency15 3.5% 753 24.0% 4887 45.0% 9033 24.0% 4812 3.5% 7Total: 200 200➊ For example, the expected percentage of ‘A’s to be given is 3.5%• The expected frequency was obtained by multiplying .035 by 200• This gave an expected frequency of 7THE ONE-WAY CHI-SQUARE TESTChapter 10Observed Frequency Expected % Expected Frequency15 3.5% 753 24.0% 4887 45.0% 9033 24.0% 4812 3.5% 7Total: 200 200➊ Once both the observed and expected frequencies are in the same unit of measurement, it is important to check that both columns have the same totals• Notice that both have the same total of 200• This is a check to ensure that the expected frequency calculations were done correctlyTHE ONE-WAY CHI-SQUARE TESTChapter 10Observed Frequency Expected Frequency15 753 4887 9033 4812 7Total: 200 200771522obt4848532909087248483327712202.182obtdf = 5 – 1 = 4488.9205.Chi-square Obtained Critical Chi-square from tableTHE ONE-WAY CHI-SQUARE TESTChapter 10➊ Statistical Conclusion:• “Since X 2(4) = 18.02, p < .05; Reject the null hypothesis”➋ If this problem is done in Excel, we can obtain the exact p –value and re-write the statistical conclusion like this:• “Since X 2(4) = 18.02, p = .001; Reject the null hypothesis”➌ Interpretation:“It appears that the instructor’s grade distribution is significantly different from the distribution suggested by the college’s catalog (p= .05). The instructor gave more than twice as many A’s than is suggested by the college catalog. This observed difference in frequencies is not solely due to random error, but suggests that the instructor assigns grades in a slightly different distribution.”➊ When there are two independent variables• Each IV has two or more levels (or categories)➋ When the data are nominal scale➌ The null hypothesis is rejected when the obtained chi-square value is equal to or greater than the critical chi-square value The degrees of freedom for the two-way chi-square test is:•df= (r – 1)(c – 1)• where ris the number of rows for IV #1 and•cis the number of columns for IV #2THE TWO-WAY CHI-SQUARE TESTChapter 10➊ When there are two independent variables, the chi-square test determines if group membership on the first IV is contingenton group membership on the second IV• Gender (males vs females) and Type of vehicle owned (Trucks vs cars)• Political Party Affiliation (republican vs democrat) and Position on death penalty (For or against)➋ If the chi-square test is significant (p = .05) for the Gender and Type of Vehicle owned example, an interpretation would be that type of vehicle owned is contingent on gender (i.e., males own more trucks than females)THE TWO-WAY CHI-SQUARE TESTChapter 10➌ If the chi-square test is significant (p = .05) for the Political Party Affiliation and Position on Death Penalty example, an interpretation would be that being for or against the death penalty is contingent on political party affiliation (i.e., Republicans are more likely for the death penalty) When the chi-square test is non-significant, it suggests that membership on the first IV is not contingent on membership on the second IVTHE TWO-WAY CHI-SQUARE TESTChapter 10THE CHI-SQUARE TESTChapter 10That’s it for chapter