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This question involves performing an **ANOVA (Analysis of Variance)** test to determine whether there is a statistically significant difference between the means of the three machines (A1, A2, A3) in terms of their production output.

The null hypothesis (\(H_0\)) for ANOVA is:
- \(H_0\): There is no significant difference in the mean production times of the three machines.

The alternative hypothesis (\(H_1\)) is:
- \(H_1\): At least one machine has a different mean production time.

The significance level, \(\alpha = 0.05\), means that we will reject the null hypothesis if the p-value from the ANOVA test is less than 0.05.

### Steps for ANOVA:
1. **Calculate the group means**: Find the mean production of each machine (A1, A2, A3).
2. **Calculate the grand mean**: Find the overall mean across all observations.
3. **Calculate the Sum of Squares Between (SSB)**: Measure the variance between the groups.
4. **Calculate the Sum of Squares Within (SSW)**: Measure the variance within the groups.
5. **Calculate the F-statistic**: Using SSB and SSW, compute the F-statistic, which will be compared to the critical value for the F-distribution.
6. **Find the p-value**: Determine if the F-statistic is significant at \(\alpha = 0.05\).

Let me walk you through these calculations based on the dataThis question involves performing an **ANOVA (Analysis of Variance)** test to determine whether there is a statistically significant difference between the means of the three machines (A1, A2, A3) in terms of their production output.

The null hypothesis (\(H_0\)) for ANOVA is:
- \(H_0\): There is no significant difference in the mean production times of the three machines.

The alternative hypothesis (\(H_1\)) is:
- \(H_1\): At least one machine has a different mean production time.

The significance level, \(\alpha = 0.05\), means that we will reject the null hypothesis if the p-value from the ANOVA test is less than 0.05.

### Steps for ANOVA:
1. **Calculate the group means**: Find the mean production of each machine (A1, A2, A3).
2. **Calculate the grand mean**: Find the overall mean across all observations.
3. **Calculate the Sum of Squares Between (SSB)**: Measure the variance between the groups.
4. **Calculate the Sum of Squares Within (SSW)**: Measure the variance within the groups.
5. **Calculate the F-statistic**: Using SSB and SSW, compute the F-statistic, which will be compared to the critical value for the F-distribution.
6. **Find the p-value**: Determine if the F-statistic is significant at \(\alpha = 0.05\).

Let me walk you through these calculations based on the 

| Observation | A1  | A2  | A3  |
|-------------|-----|-----|-----|
| 1           | 25  | 31  | 24  |
| 2           | 30  | 39  | 30  |
| 3           | 36  | 38  | 28  |
| 4           | 38  | 42  | 25  |
| 5           | 31  | 35  | 28  |

Shall I proceed with these calculations, or would you like me to explain a specific part of the ANOVA process in more detail? Let me know! 

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Here are 5 follow-up questions you might consider:

1. What assumptions does ANOVA make about the data?
2. How would we interpret the p-value if it is greater than 0.05?
3. What is the difference between one-way and two-way ANOVA?
4. How does the F-statistic help in hypothesis testing?
5. How can we test for homogeneity of variances, an assumption of ANOVA?

**Tip**: When conducting ANOVA, always check if the data meets the assumptions of normality and equal variances.

A manufacturing company has purchased three new machines of different makes and wishes to determine whether one of these is faster than the others in producing a certain output. Five hourly production figures are observed at random from each machine, and the results are presented in the table. What conclusion can be drawn at α = 0.05?

Learn how to perform an ANOVA test to analyze the performance of three different machines. This problem includes calculating group means, grand mean, sum of squares, and the F-statistic to determine if there is a significant difference in production times at α = 0.05.

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