Chi-Square Test For nominal/qualitative data 1-variable: ‘goodness of fit test’ 2-variable: ‘test of independence Observations must be independent No more than one measurement per subject Large enough sample size Expected frequencies must be ≥ 5

∑ “Goodness of Fit” χ2 (fO-fE)2 fE Null: frequencies given by chance Calculate expected frequencies Compute χ2 Compare to Critical df = (# categories - 1) (fO-fE)2 fE ∑

1 2 3 4 5 6 18 19 21 23 22 17 B 1 2 3 4 5 6 8 9 15 16 57 D

χ 2 Ben’s 120 Rolls 120 120 =∑ = 1.4 Num Obs. Exp. O-E (O-E)2 1 18 20 -2 4 .20 2 19 -1 .05 3 21 23 9 .45 5 22 6 17 -3 χ 2 (O-E)2 E 120 120 =∑ = 1.4

χ2 Damien’s 120 Rolls 120 120 =∑ = 85 Num Obs. Exp. O-E (O-E)2 1 8 20 -12 144 7.20 2 9 -11 121 6.05 3 15 -5 25 1.25 4 5 16 -4 0.80 6 57 37 1369 68.45 χ2 (O-E)2 E 120 120 =∑ = 85

Finding Critical χ2 χ2 critical = 11.07 df = (# categories - 1) = 5 Ben: χ2 = 1.4 not significant Damien: χ2 = 85 significant!

χ2 Test for Independence H0: No association between variables χ2 Compare to Critical χ2 df = (# rows - 1) x (# columns - 1) Expected Frequency of Each Cell = Row Total x Col. Total Grand Total = (fO-fE)2 fE ∑

200 Students Vegetarian Non-Vegetarian Male 10 60 Female 50 80 70 130 60 140 200 Is there an association between gender and vegetarianism? Ho: no, they are independent. Ha: yes there is, Ho is false.

= Expected Frequency of Each Cell Row Total x Col. Total Grand Total 200 Students Vegetarian Non-Vegetarian Male 10 [ ] = 21 60 [ ]= 49 Female 50 [ ]= 39 80 [ ]= 91 70x60 70x140 70 200 200 130x60 130x140 130 200 200 60 140 200 Expected Frequency of Each Cell Row Total x Col. Total Grand Total =

χ2 ∑ = 200 200 =∑ = 12.66 Obs. Exp. O-E (O-E)2 10 21 -11 121 5.76 60 Male Veg 10 21 -11 121 5.76 Non-Veg 60 49 11 2.47 Female 50 39 3.10 80 91 1.33 χ2 (O-E)2 E ∑ = 200 200 =∑ = 12.66

Finding Critical χ2 df = (# Rows - 1) x (# Columns -1) = (2-1) x (2-1) = 1 12.66 > 3.841 = significant: gender & diet not independent

Ch. 19 hw *What is your favorite season? Here are 100 people’s answers: 23 Winter, 24 Spring, 34 Summer, 19 Fall Test the null hypothesis that the seasons are equally popular. 1. State Ho and Ha. What are the expected values? 2. Calculate χ2, find its critical value, & interpret the results. *Are lefties more artistic? Test if handedness is independent of college major: Business: 35 R, 5 L; Sciences: 50 R, 10 L; Arts/Humanities: 15 R, 5 L 1. State Ho & Ha. 2. Find the row & column totals, and calculate each expected value. 3. Calculate χ2, find its critical value, & interpret the results.