1. 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

2. ∑ “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 ∑

3. 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

4. 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

5. 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

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

7. 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 ∑

8. 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.

9. = 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 =

10. 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

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

12. 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.