1. Chi-Squared Distribution
Multinomial Distribution - Qualitative Data (Nominal) k - Classes
Class
… k
p1 p2 p3 p4 … pk Population Proportions
pj Proportion of All Observations in the jth Class
∑pj = 1
foj = Observed Number in jth Class in a Sample of Size N
fej = Expected Number in jth Class based upon pj
fej = N•pj

2. Example 1: Favorite Ice Cream (Rule of 5: fe ≥5)
Van Choc Straw N fo fe
Van Choc Straw N fo fe

3. Example 2: Business Major
BUSA ACCT MKTG Other N fo fe
H0: BUSA:ACCT:MKTG:Other=8:3:3:6
HA; Ratio Not 8:3:3:6 R:

4. Example 3: Gender of Business Majors
Male Female N H0: pM=.60 pF=.40
fo HA: pM≠.60 pF≠.40
fe R:
Binomial: Test Population Proportion

5. Goodness of Fit for a Distribution
"Fair Die" = Uniform Distribution N fo fe
H0: Die Fits Uniform Distribution
HA: Die Does Not Fit Uniform Distribution R:

6. Goodness of Fit for Exam Scores to a Normal Distribution
Score fo Z pj fe (fo - fe) χ2
17 - <26 9
26 - <32 7
32 - <38 16 38
Exam: Mean = 32, Std Dev = 6

7. Contingency Tables - Cross-Tabulation of Nominal Data (X-Tabs)
Want to Test for the Independence of 2 Categorical Factors
Factor 1
1 2 3 …. c Sum
1 f11 f12 f13 … f1c R1
2 f21 f22 f23 … f2c R2
3 f31 f32 f33 … f3c R3 Factor 2
… … … … … … …
r fr1 fr2 fr3 … frc Rr
Sum C1 C2 C3 … Cc N
Independence - P(A and B) = P(A)•P(B) p11=pR1•pC1=(R1/N)(C1/N)
fe11=N•p11=N(R1/N)(C1/N)=(R1•C1)/N

8. Example 6: Income vs Cookie Mix
Cookie <30K K >60K Total
Pre-Mix
Mix Own
Total
H0: Independence
HA: Dependence R:

9. Ex 7: Product
Use Don't Use
Non-Athletic
Athletic
H0: Indep
HA: Dep R:
Compare 2 Pop Proportions: Ath Non-Ath n x