Question

Exercise 7. Machine Errors and Days. A manufacturer wishes to see if there is a difference in the distribution for the number of defective bolts produced by at least one of several machines through the work week on the day shift. Six randomly selected machines are evaluated each day for a week. The findings are listed in the table. Let α = 0.05. Using the Friedman Test, can you support the researcher’s claim?

Machine	Monday	Tuesday	Wednesday	Thursday	Friday
A	6	4	5	5	4
B	10	8	7	7	9
C	7	5	6	5	9
D	8	4	6	5	5
E	5	7	4	6	8
F	7	9	12	8	8

Answer 1

Null hypothesis	H₀: All treatment effects are zero
Alternative hypothesis	H₁: Not all treatment effects are zero

Ranks T1
5 5 4 5 2 1 Sum: 22

Ranks T2
1.5 3 1.5 1 4 4 Sum: 15

Ranks T3
3.5 1.5 3 4 1 5 Sum: 18

Ranks T4
3.5 1.5 1.5 2.5 3 2.5 Sum: 14.5

Ranks T5

1.5 4 5 2.5 5 2.5 Sum: 20.5

X²_r = (12/(nk(k+1)) * (∑R²) - 3n(k+1)

X²_r = 0.067 * 1663.5 - 108

X²_r = 2.9

The X²_r statistic is 2.9 (4, N = 6).

The p-value is .5747.

The result is not significant at p < .05. We fail to reject null hypothesis and conclude that there are no differences in the distribution for the number of defective bolts produced by several machines through the work week on the day shift.