Question 15.45

15.45 A manufacturing problem.

One step in the manufacture of large engines requires that holes of very precise dimensions be drilled. The tools that do the drilling are regularly examined and are adjusted to ensure that the holes meet the required specifications. Part of the examination involves measurement of the diameter of the drilling tool. A team studying the variation in the sizes of the drilled holes selected this measurement procedure as a possible cause of variation in the drilled holes. They decided to use a designed experiment as one part of this examination. Some of the data are given in Table 15.1. The diameters in millimeters (mm) of five tools were measured by the same operator at three times (8:00 A.M., 11:00 A.M., and 3:00 P.M.). Three measurements were taken on each tool at each time. The person taking the measurements could not tell which tool was being measured, and the measurements were taken in random order.17

drill

Table 15.38: TABLE 15.1 Tool diameter data
Tool Time Diameter
1 1 25.030 25.030 25.032
1 2 25.028 25.028 25.028
1 3 25.026 25.026 25.026
2 1 25.016 25.018 25.016
2 2 25.022 25.020 25.018
2 3 25.016 25.016 25.016
3 1 25.005 25.008 25.006
3 2 25.012 25.012 25.014
3 3 25.010 25.010 25.008
4 1 25.012 25.012 25.012
4 2 25.018 25.020 25.020
4 3 25.010 25.014 25.018
5 1 24.996 24.998 24.998
5 2 25.006 25.006 25.006
5 3 25.000 25.002 24.999
  1. Make a table of means and standard deviations for each of the combinations of the two factors.
  2. Plot the means and describe how the means vary with tool and time. Note that we expect the tools to have slightly different diameters. These will be adjusted as needed. It is the process of measuring the diameters that is important.
  3. Use a two-way ANOVA to analyze these data. Report the test statistics, degrees of freedom, and -values for the significance tests.
  4. Write a short report summarizing your results.

15.45

(a)

Diameter
Level of
Tool		 Level of
Time		        N        Mean Std Dev
1 1 3 25.030667 0.0011547
1 2 3 25.028 0
1 3 3 25.026 0
2 1 3 25.016667 0.0011547
2 2 3 25.02 0.002
2 3 3 25.016 0
3 1 3 25.006333 0.0015275
3 2 3 25.012667 0.0011547
3 3 3 25.009333 0.0011547
4 1 3 25.012 0
4 2 3 25.019333 0.0011547
4 3 3 25.014 0.004
5 1 3 24.997333 0.0011547
5 2 3 25.006 0
5 3 3 25.000333 0.0015275

(b) There are differences in diameter among the five different tools as expected. There are also differences in diameter in the different shifts, although not as large as the tool differences. There also appears to be some interaction between tool and time. Particularly, tools 3, 4, and 5 are fairly consistent across time differences, with time 2 having the largest diameters and time 1 the smallest diameters. For tool 2, however, time 1 diameters get larger than time 3s, and for tool 1, time 1 diameters get the largest, bigger than both time 2 and time 3 diameters. (c) The Tool effect is extremely significant, The Time effect is also very significant, The interaction effect is significant, (d) Because all terms are significant we would interpret effects as stated in part (b).