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Chapter 13: Chi-Square Test 1

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Motivating Example Research Question: Among all adults in the U.S. who were in a car accident, is there a relationship between cell phone use and injury severity? Sample: 200 randomly-selected U.S. adults who were in car accidents Results: See Table 1 1 2 1 This example is entirely fictitious

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Table 1: Bivariate Table InjuriesUsedCell?Total SustainedNoYes None82 (82%)66 (66%)148 Minor12 (12%)18 (18%)30 Severe6 (6%)16 (16%)22 Total100 200 3 Relationship: There is a relationship in the sample; cell phone users are less likely than non-users to sustain no injuries (66% vs. 82%)

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Table 2: No Association Table InjuriesUsedCell?Total SustainedNoYes None74 (74%) 148 (74%) Minor15 (15%) 30 (15%) Severe11 (11%) 22 (11%) Total100 200 4 No Relationship: Cell phone users are just as likely as non-users to sustain no injuries (74%)

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Relationship in Sample Vs. Population Sample: We found a relationship in the sample of 200 accident victims Population: We want to know whether there is a relationship in the population ALL adults in the U.S. who were in car accidents We can use hypothesis testing procedures The chi-square test is used to test hypotheses involving bivariate tables 5

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Chi-Square ( χ 2 ) Test Procedure State the null and research hypotheses Compute a χ 2 statistic Determine the degrees of freedom Find the p-value for the χ 2 statistic Decide whether there is evidence to reject the null hypothesis Interpret the results 6

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χ 2 Test Assumptions Assumption 1: The sample is selected at random from a population Assumption 2: The variables are nominal or ordinal Note: In this class, you wont have to determine whether the assumptions have been met 7

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χ 2 Test: Hyptheses Null Hypothesis (H 0 ): The two variables are not related in the population Research Hypothesis (H 1 ): The two variables are related in the population Alpha ( α ): This will be given to you in every problem (when its not given, assume α = 0.05) 8

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χ 2 Test: Hyptheses Cell Phone – Injuries Example Null Hypothesis (H 0 ): Cell phone use and injury severity are not related among all adults in the U.S. who were in a car accident Research Hypothesis (H 1 ): Cell phone use and injury severity are related among all adults in the U.S. who were in a car accident Alpha ( α ): Use α = 0.05) 9

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χ 2 Test: Calculating the χ 2 Statistic Formula: Two Components Observed Frequencies (f o ) Expected Frequencies (f e ) 10

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Calculating the χ 2 Statistic: f o and f e Observed Frequencies (f o ) Definition: The actual frequencies in the sample Example: In the cell phone – injuries example, these are given in Table 1 Expected Frequencies (f e ) Definition: The frequencies we would expect assuming the two variables were independent In other words, assuming the null hypothesis was true Example: In the cell phone – injuries example, these are given in Table 2 11

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Calculating the χ 2 Statistic: Logic Behind the Formula We are comparing the observed and expected frequencies We are comparing the results in our sample with what we would expect if the two variables were independent (i.e., assuming H 0 is true) We are doing this because we are testing the null hypothesis (H 0 ), which assumes that the two variables are independent in the population 12

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Calculating the χ 2 Statistic: Size of Difference Small Difference If the differences between the observed and expected frequencies are small, the χ 2 statistic will be small As a result, we will likely fail to reject H 0 Large Difference If the differences between the observed and expected frequencies are large, the χ 2 statistic will be large As a result, we will likely reject H 0 What is Small or Large? We will use Appendix D to decide what is small or large 13

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Calculating the χ 2 Statistic: Computing f e Procedure: For each cell, multiply the corresponding column marginal and row marginal, then divide by the sample size: Huh?!?!? Lets do this for the cell phone – injuries example (next several slides) 14

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Calculating the χ 2 Statistic: Computing f e 15 InjuriesUsedCell?Total SustainedNoYes None148 Minor30 Severe22 Total100 200 Begin with a table containing only the row and column totals

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Calculating the χ 2 Statistic: Computing f e 16 InjuriesUsedCell?Total SustainedNoYes None148 Minor30 Severe22 Total100 200 For each cell, multiply the corresponding row and column total, then divide by the total sample size (200 here)

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Calculating the χ 2 Statistic: Computing f e 17 InjuriesUsedCell?Total SustainedNoYes None148 Minor30 Severe22 Total100 200

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Calculating the χ 2 Statistic: Computing f e 18 InjuriesUsedCell?Total SustainedNoYes None148 Minor30 Severe22 Total100 200 For each cell, multiply the corresponding row and column total, then divide by the total sample size (200 here)

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Calculating the χ 2 Statistic: Computing f e 19 InjuriesUsedCell?Total SustainedNoYes None148 Minor30 Severe22 Total100 200

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Calculating the χ 2 Statistic: Computing f e 20 InjuriesUsedCell?Total SustainedNoYes None148 Minor30 Severe22 Total100 200

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Calculating the χ 2 Statistic: Computing f e 21 InjuriesUsedCell?Total SustainedNoYes None148 Minor30 Severe22 Total100 200

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Calculating the χ 2 Statistic: Computing f e 22 InjuriesUsedCell?Total SustainedNoYes None74 148 Minor15 30 Severe11 22 Total100 200 This is the complete table of expected frequencies (f e )

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Calculating the χ 2 Statistic 23 InjuriesUsedCell?Total SustainedNoYes None8266148 Minor121830 Severe61622 Total100 200 InjuriesUsedCell?Total SustainedNoYes None74 148 Minor15 30 Severe11 22 Total100 200 Observed Frequencies (f o ) Expected Frequencies (f e )

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χ 2 Test: Degrees of Freedom (df) Formula: r = number of rows c = number of columns Interpretation: The number of cells in the table that need to have numbers before we can fill in the remaining cells Cell Phone – Injury Example 24

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χ 2 Test: Determining the P-Value χ 2 Distribution The p-value will be based on the χ 2 distribution The χ 2 distribution is positively skewed This means that our hypothesis tests will always be one-tailed Values of the χ 2 statistic are always positive Minimum = 0 (variables are completely independent) Maximum = The shape of the χ 2 distribution is dictated by its df See figure on next slide 25

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χ 2 Test: Determining the P-Value 26

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χ 2 Test: Determining the P-Value Steps Find df in the first column of Appendix D Read across the row until you find the χ 2 value you computed Read up to the first row to find the p-value Cell Phone – Injury Example χ 2 = 7.46, df = 2 Reading across the row where df = 2, a value of 7.46 is between 5.991 and 7.824 Reading up to the top row, the p-value is between 0.05 and 0.02 27

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χ 2 Test: Determining the P-Value Additional practice finding p-values χ 2 = 0.446, df = 2 P-value = 0.80 χ 2 = 4.09, df = 1 P-value is between 0.02 and 0.05 χ 2 = 0.01, df = 2 P-value is greater than 0.99 χ 2 = 15.00, df = 4 P-value is between 0.001 and 0.01 28

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χ 2 Test: Evidence to Reject H 0 ? Decision Rule If the p-value is less than α, we have evidence to reject H 0 in favor of H 1 If the p-value is greater than α, we do not have evidence to reject H 0 in favor of H 1 Cell Phone – Injury Example The p-value (which is between 0.02 and 0.05) is less than α = 0.05 We have evidence to reject H 0 in favor of H 1 29

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χ 2 Test: Interpretation If We Reject H 0 : We have evidence to suggest that the two variables are related in the population If We Do Not Reject H 0 : We do not have evidence to suggest that the two variables are related in the population Cell Phone – Injury Example: We have evidence that cell phone use and injury severity are related among all adults in the U.S. who were in a car accident 30

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