Answer to Question #118922 in Statistics and Probability for simran

Question

manufacturer wishes to determine the effectiveness of four types of machines
A, B, C and D in the production of bolts. For this, the number of defective bolts
produced by each machine on two alternative days are shown in the following table:
Day I Day II
Machine A 24 30
Machine B 41 44
Machine C 32 31
Machine D 28 38

Perform an analysis of variance method to determine whether there is a difference
between machines, at 5% level of significance.

Answer 1

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=(54*125)/268=25, (O₁₁-E₁₁)²/E₁₁=0.0416

E₁₂=(54*143)/268=29, (O₁₂-E₁₂)²/E₁₂=0.033

E₂₁=(85*125)/268=40, (O₂₁-E₂₁)²/E₂₁=0.024

E₂₂=(85*143)/268=45, (O₂₂-E₂₂)²/E₂₂=0.0227

E₃₁=(63*125)/268=29, (O₃₁-E₃₁)²/E₃₁=0.281

E₃₂=(63*143)/268=34, (O₃₂-E₃₂)²/E₃₂=0.29

E₄₁=(66*125)/268=31, (O₄₁-E₄₁)²/E₄₁=0.32

E₄₂=(66*143)/268=35, (O₄₂-E₄₂)²/E₄₂=0.2368

χ²=1.2491

degrees of freedom: (r-1)(c-1)=3

the critical value of chi square with 3df at 5% level significance: 7.82

χ²<critical value, so we accept that there is no difference between machines