Question

Problem Details:

Sharona Medical Equipment, Ltd (fictional name) manufactures dental drills. The company has been experiencing problems with a specific part on the production line. Management suspects a machining problem has resulted in the length of a particular part varying outside of target specification limits. Management believes that the machine setting (1, 2, 3) and/or the shift (Morning, Afternoon, Night) during which the part is machined may explain the length errors. Management is particularly interested in the role the shift may play, as new hires are typically scheduled for night shifts. To investigate, four parts machined with each of the 3 settings were randomly selected from each of the 3 shifts. The deviation in length from the specified size was measured in microns and the collected data can be found in the “Machine_Shift_ProdErrors” worksheet in the Excel file titled Group Case #2 Data.

Requirements:

Your consulting firm has been hired to conduct a thorough analysis of the production error data and provide management of Sharona Medical Equipment, Ltd with a detailed summary of your analysis, conclusions reached, and any recommendations you feel qualified to make.

Data:

Size Error	Machine Setting	Shift
2	1	Afternoon
1.8	1	Afternoon
2.1	1	Afternoon
2.5	1	Afternoon
2.4	2	Afternoon
4.3	2	Afternoon
3.9	2	Afternoon
5	2	Afternoon
5	3	Afternoon
3.2	3	Afternoon
3.5	3	Afternoon
2.3	3	Afternoon
1.1	1	Morning
2.1	1	Morning
1.3	1	Morning
0.6	1	Morning
3.6	2	Morning
0.9	2	Morning
2.3	2	Morning
2.3	2	Morning
3.3	3	Morning
2.6	3	Morning
3	3	Morning
3.2	3	Morning
3.8	1	Night
2.9	1	Night
3.2	1	Night
2.8	1	Night
5.5	2	Night
6.7	2	Night
5.1	2	Night
3	2	Night
5	3	Night
5.8	3	Night
5.3	3	Night
5.4	3	Night

Answer 1

Looking to the problem as a data analyst one could apply a two way ANOVA analysis here. There are two factors, “machine setting” (that has three levels 1, 2, 3) and “the shift” (that has 3 levels: Morning, Afternoon,Night). The machining problem that has resulted in the length of a particular part of product varying outside of target specification limits, may be explained by this two factors.

The factors leads to a fixed effect model , with the mathematical form :

  

Where yijk is the size error of kth observation under ith level of factor “machine setting” and jth level of factor “shift”. is the additive effect , is the fixed effect due to ith level of machine setting and is the fixed effect due to the jth level of shift. is the random error with zero expectation and fixed(say ) variance.

Now after analyzing the entire dataset using MINITAB software we obtain the following output:

The above table shows the source of variation of the undersize manufacture.,and their statistical significance.

Suppose we want to test ,
vs : at least one is unequal ,

In words, meaning we want to test that whether the mean size error of the manufactured items at 3 different levels of the factor "machine setting" are all same vs at least on level has significantly different group mean.

The p-value of the test statistic for testing this ( corresponding to the source "Machine setting " row of ANOVA table ) is 0 ,implying at 5% level of significance we can reject the null hypothesis of each level under "Machine setting" has equal group mean and accept that at least one group is different.

Similarly we can test ,whether the variation due to different levels of shift are significant.
so we test vs : at least one is unequal ,

In words, meaning we want to test that whether the mean size error of the manufactured items at 3 different levels of the factor "Shift" are all same vs at least on level has significantly different group mean.

The p-value of the test statistic for testing this ( corresponding to the source "Machine setting " row of ANOVA table ) is 0 << 0.05 ,implying at 5% level of significance we can reject the null hypothesis of each level under "Shift" has equal group mean and accept that at least one group is significantly different at 5% level as per the data.

Thus overall one may comment that, the different shifts of work as well as different machine settings play vital role in this Size error problems on the production line.