Question

The following contingency table presents the analysis of 300 persons according to

hair colour and eye colour:

Hair Colour Eye Colour

Blue Grey Brown

Burgundy 30 10 40

Brown 40 20 40

Black 50 30 40



Test the hypothesis that there is an association between hair colour and eye colour at

5% level of significance

Answer 1

H₀ : Hair and eye colour are independent

H₁ : Hair and eye colour are associated

χ²=$\sum$^r_i=1 $\sum$ ^c_j=1=(O_ij-E_ij)²/E_ij

E_ij=R_iC_j/N, R_i - sum of i^-th row, C_j - sum of j^-th column, N - total sample size

E₁₁=(80*120)/300=32, (O₁₁-E₁₁)²=4

E₁₂=(80*60)/300=16, (O₁₂-E₁₂)²=36

E₁₃=(80*120)/300=32, (O₁₃-E₁₃)²=64

E₂₁=(100*120)/300=40, (O₂₁-E₂₁)²=0

E₂₂=(100*60)/300=20, (O₂₂-E₂₂)²=0

E₂₃=(100*120)/300=40, (O₂₃-E₂₃)²=0

E₃₁=(120*120)/300=48, (O₃₁-E₃₁)²=4

E₃₂=(120*60)/300=24, (O₃₂-E₃₂)²=36

E₃₃=(120*120)/300=48, (O₃₃-E₃₃)²=64

χ²=7.29

degrees of freedom: (r-1)(c-1)=4

the critical value of chi square with 4df at 5% level significance: 9.488

χ²<critical value, so we accept H₀ ,so hair colour is independent of eye colour