A manufacturer wishes to determine the effectiveness of four types of machines (A, B, C and D) in the production of bolts. To accumulate this, the numbers of defective bolts produced for each of two shifts in the results are shown in the following table:
| Machine | First shift | Second Shift | ||||||||
| M | T | W | Th | F | M | T | W | Th | F | |
| A | 6 | 4 | 5 | 5 | 4 | 5 | 7 | 4 | 6 | 8 |
| B | 10 | 8 | 7 | 7 | 9 | 7 | 9 | 12 | 8 | 8 |
| C | 7 | 5 | 6 | 5 | 9 | 9 | 7 | 5 | 4 | 6 |
| D | 8 | 4 | 6 | 5 | 5 | 5 | 7 | 9 | 7 | 10 |
Perform an analysis of variance to determine at 5% level of significance, whether there is a difference (a) Between the machines and (b) Between the shifts.
To perform an analysis of variance (ANOVA) for this scenario, we will conduct two separate tests:
1. Test for the difference between the machines (A, B, C, D).
2. Test for the difference between the shifts (First Shift, Second Shift).
Let's start with the ANOVA for the difference between the machines:
ANOVA for Machines:
1. **Calculate the Sum of Squares Total (SST):**
SST = Σ(x_ij - x_bar)^2
Where:
- x_ij is the number of defective bolts for each observation.
- x_bar is the overall mean of all observations.
2. **Calculate the Sum of Squares Between (SSB):**
SSB = Σ(n_i * (x_bar_i - x_bar)^2)
Where:
- n_i is the number of observations for each machine.
- x_bar_i is the mean of each machine.
- x_bar is the overall mean of all observations.
3. **Calculate the Sum of Squares Within (SSW):**
SSW = SST - SSB
4. **Calculate the Degrees of Freedom (DF):**
- DF Between (k-1), where k is the number of machines.
- DF Within (N-k), where N is the total number of observations.
5. **Calculate the Mean Squares:**
- MSB = SSB / DF Between
- MSW = SSW / DF Within
6. **Calculate the F-Statistic:**
- F = MSB / MSW
7. **Determine the Critical Value from the F-distribution table for α = 0.05 and degrees of freedom (DF Between, DF Within).**
8. **Compare F-Statistic with Critical Value:**
- If F > Critical Value, reject the null hypothesis.
Let's calculate the ANOVA table for the machines:
| Factor | Source | SS | df | Mean Square | F Value | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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| Between | SSB | k - 1 | MSB | F-Statistic | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Within | SSW | N - k | MSW | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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