A manufacturing company has been selected to assemble a small but important component that will be used during the construction of numerous infrastructure projects. The company anticipates the need to assemble several million components over the next several years. Company engineers select three potential assembly methods: Method A, Method B, and Method C. Management would like to select the method that produces the fewest number parts per 10,000 parts produced that do not meet specifications. It may also be possible that there is no statistical difference between the three methods in which case the lowest cost method will be selected for production. While all parts are checked before leaving the factory, the best method will reduce the number of parts that need to be recycled back into the production process.To test each method, six batches of 10,000 components are produced using each of the three methods. The number of components out of specification are recorded in the Microsoft Excel Online file below. Analyze the data to determine if there is any difference in the mean number of components that are out of specification among the three methods. After conducting the analysis report the findings to the management team.a. Compute the sum of squares between treatments (assembly methods).b. Compute the mean square between treatments (to 1 decimal if necessary).c. Compute the sum of squares due to error.d. Compute the mean square due to error (to 1 decimal if necessary).

Since the Excel file containing the data was not provided, I'll take you through the steps required to analyze whether there is a statistically significant difference in quality between the three assembly methods using the data, assuming you have it. The method I'll describe is based on the ANOVA (Analysis of Variance) technique, which is suitable for comparing means across multiple groups.

a. Compute the sum of squares between treatments (SSB): 1. Calculate the overall mean of the out-of-specification components for all methods combined. 2. For each method, calculate the mean of the out-of-specification components. 3. Subtract the overall mean from the mean of each method to find the deviation. 4. Square each deviation. 5. Sum the squared deviations for each method, but before that, multiply each squared deviation by the number of observations per method (in this case, 6 batches of 10,000 components). 6. The final sum is the Sum of Squares Between treatments.

b. Compute the mean square between treatments (MSB): 1. Take the Sum of Squares Between from step a. 2. Divide this number by the degrees of freedom between treatments (which is the number of groups minus one, i.e., 3 - 1 = 2). 3. This gives you the Mean Square Between, which represents the variance due to the treatment effect.

c. Compute the sum of squares due to error (SSE): 1. For each group, subtract the group's mean from the result of each batch within that group to find the deviation. 2. Square each of these deviations. 3. Sum these squared deviations across all groups and batches. 4. The final sum is the Sum of Squares Due to Error, representing the variance within groups.

d. Compute the mean square due to error (MSE): 1. Take the Sum of Squares Due to Error from step c. 2. Divide this by the degrees of freedom for error (which is the total number of observations minus the number of groups; for example, if there are 18 total batches and 3 groups, the degrees of freedom would be 18 - 3 = 15). 3. This gives you the Mean Square Error, a measure of the variability within the groups or methods.

Once you have computed these figures, you can perform an F-test using the MSB and MSE to determine if there is a significant difference between the methods. If the F value is greater than the critical value from the F-distribution for the given degrees of freedom and confidence level, there is a statistically significant difference between the means. If not, the null hypothesis (that the means are equal) cannot be rejected, and the company might select the lowest-cost method for production.

Extra: The ANOVA test is a powerful statistical tool used to compare the means of three or more groups to see if at least one group mean is different from the others. Here’s a breakdown of the terms used in the analysis:

- Sum of Squares Between (SSB): This looks at how much the group means differ from the overall mean. A larger SSB indicates a greater variability between the different groups. - Mean Square Between (MSB): This is the average of the SSB taking into account the number of groups. It's essentially an adjusted version of the SSB. - Sum of Squares Due to Error (SSE): This represents the variability within each group, showing how much each individual observation deviates from the mean of its particular group. - Mean Square Error (MSE): Similar to the MSB, this is the average of the SSE considering the number of observations within each group. - F-Value: It results from the ratio of MSB to MSE. The F-distribution table is then used to find the critical value to compare against the calculated F-value for a given alpha level (e.g., 0.05 for a 95% confidence level).

The ANOVA doesn’t tell us which method is superior; it only tells us that at least one method is different. If the ANOVA F-test is significant, you might follow it with additional tests (like Tukey's HSD test) to find out which specific groups (methods) differ from each other.