To test whether the mean time needed to mix a batch of material is the same for machines produced by three manufacturers, the Jacobs Chemical Company obtained the following data on the time (in minutes) needed to mix the material.

\begin{tabular}{cccc}
& Manufacturer 1 & Manufacturer 2 & Manufacturer 3 \\
15 & 31 & 21 \\
21 & 29 & 20 \\
19 & 34 & 24 \\
17 & 30 & 23 \\
\end{tabular}

a. Use these data to test whether the population mean times for mixing a batch of material differ for the three manufacturers. Use [tex]$\alpha=0.05$[/tex].

Compute the values below (to 2 decimals, if necessary):

- Sum of Squares, Treatment: [tex]\square[/tex]
- Sum of Squares, Error: [tex]\square[/tex]
- Mean Squares, Treatment: [tex]\square[/tex]
- Mean Squares, Error: [tex]\square[/tex]
- Calculate the value of the test statistic (to 2 decimals): [tex]\square[/tex]
- The [tex]$p$[/tex]-value is: [tex]\square[/tex]

What is your conclusion?

- [tex]\square[/tex]

Question

03-08-2024
Mathematics

Answered

To test whether the mean time needed to mix a batch of material is the same for machines produced by three manufacturers, the Jacobs Chemical Company obtained the following data on the time (in minutes) needed to mix the material.

\begin{tabular}{cccc}
& Manufacturer 1 & Manufacturer 2 & Manufacturer 3 \\
15 & 31 & 21 \\
21 & 29 & 20 \\
19 & 34 & 24 \\
17 & 30 & 23 \\
\end{tabular}

a. Use these data to test whether the population mean times for mixing a batch of material differ for the three manufacturers. Use [tex]$\alpha=0.05$[/tex].

Compute the values below (to 2 decimals, if necessary):

- Sum of Squares, Treatment: [tex]\square[/tex]
- Sum of Squares, Error: [tex]\square[/tex]
- Mean Squares, Treatment: [tex]\square[/tex]
- Mean Squares, Error: [tex]\square[/tex]
- Calculate the value of the test statistic (to 2 decimals): [tex]\square[/tex]
- The [tex]$p$[/tex]-value is: [tex]\square[/tex]

What is your conclusion?

- [tex]\square[/tex]

Answer :

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GinnyAnswer GinnyAnswer · Answer 1 · 2024-08-03T21:15:24-04:00

To test whether the mean time needed to mix a batch of material is the same for machines produced by three manufacturers, we can use ANOVA (Analysis of Variance). Here is a step-by-step solution using the data provided:

Given data (mixing times in minutes):
```
Manufacturer 1: 15, 21, 19, 17
Manufacturer 2: 31, 29, 34, 30
Manufacturer 3: 21, 20, 24, 23
```

a. Use these data to test whether the population mean times for mixing a batch of material differ for the three manufacturers at [tex]$\alpha = 0.05$[/tex].

### 1. Calculate the Overall Mean
The overall mean is the average of all values from the three manufacturers combined.

### 2. Calculate the Treatment Means
- Mean of Manufacturer 1
- Mean of Manufacturer 2
- Mean of Manufacturer 3

### 3. Calculate the Sum of Squares Treatment (SST)
SST measures the variability due to the interaction between the groups (treatments).

### 4. Calculate the Sum of Squares Error (SSE)
SSE measures the variability within the groups.

### 5. Calculate the Mean Squares
- Mean Square Treatment (MST)
- Mean Square Error (MSE)

### 6. Calculate the F-statistic
The F-statistic is used to determine if the variability between the group means is more than would be expected by chance.

### 7. Determine the p-value
The p-value will help us decide whether to reject the null hypothesis.

### Given Results
Based on the calculations, we have:
- Sum of Squares, Treatment (SST): 354.67
- Sum of Squares, Error (SSE): 44.00
- Mean Squares, Treatment (MST): 177.33
- Mean Squares, Error (MSE): 4.89
- F-statistic: 36.27
- p-value: 0.00

### Conclusion
The p-value is very small (0.00), much smaller than the significance level [tex]$\alpha = 0.05$[/tex].

- Decision: Since the p-value is less than [tex]$\alpha$[/tex], we reject the null hypothesis.

- Conclusion: There is significant evidence to suggest that the population mean times for mixing a batch of material differ for the three manufacturers.

So, the final values are:
- Sum of Squares, Treatment: 354.67
- Sum of Squares, Error: 44.00
- Mean Squares, Treatment: 177.33
- Mean Squares, Error: 4.89
- F-statistic: 36.27
- p-value: 0.00
- Conclusion: There is a significant difference in the mean mixing times among the three manufacturers.