Question 1 - Perform a simulation study comparing the short-run performance of the Standards given p-chart with a matching (known p) Binomial Q chart described in Quesenberry (1991, reproduced in appendix). You may assume a production length of 50 samples for the entire production period.
Question 2 - Consider the Pencil lead data given below (description of variables follows):
|
Subgroup
|
Ash
|
Por
|
Subgroup
|
Ash
|
Por
|
Subgroup
|
Ash
|
Por
|
Subgroup
|
Ash
|
Por
|
|
1
|
41.9
|
13.2
|
6
|
41.6
|
13.0
|
11
|
40.5
|
11.7
|
16
|
42.4
|
13.9
|
|
1
|
42.2
|
13.9
|
6
|
42.1
|
14.1
|
11
|
42.3
|
14.5
|
16
|
41.6
|
13.0
|
|
1
|
43.0
|
14.2
|
6
|
41.1
|
12.9
|
11
|
41.4
|
13.7
|
16
|
42.0
|
14.0
|
|
1
|
41.5
|
13.9
|
6
|
42.2
|
13.5
|
11
|
41.0
|
12.5
|
16
|
42.2
|
14.3
|
|
1
|
43.1
|
14.5
|
6
|
42.9
|
14.2
|
11
|
42.7
|
14.0
|
16
|
42.9
|
14.3
|
|
2
|
40.7
|
12.2
|
7
|
43.5
|
14.7
|
12
|
41.4
|
13.2
|
17
|
42.5
|
14.0
|
|
2
|
42.8
|
14.4
|
7
|
41.0
|
12.7
|
12
|
41.9
|
14.1
|
17
|
42.1
|
13.1
|
|
2
|
42.6
|
14.1
|
7
|
43.0
|
14.3
|
12
|
41.3
|
12.8
|
17
|
42.3
|
14.5
|
|
2
|
41.9
|
13.6
|
7
|
42.0
|
13.6
|
12
|
42.7
|
14.1
|
17
|
40.9
|
12.9
|
|
2
|
42.1
|
13.6
|
7
|
42.2
|
13.8
|
12
|
42.1
|
13.6
|
17
|
42.9
|
14.3
|
|
3
|
41.4
|
13.4
|
8
|
41.5
|
13.5
|
13
|
41.6
|
13.4
|
18
|
41.7
|
13.6
|
|
3
|
41.5
|
13.6
|
8
|
42.2
|
13.2
|
13
|
42.1
|
13.8
|
18
|
41.7
|
12.7
|
|
3
|
43.0
|
13.9
|
8
|
42.7
|
14.0
|
13
|
40.5
|
12.8
|
18
|
42.0
|
15.0
|
|
3
|
42.9
|
14.5
|
8
|
43.3
|
14.5
|
13
|
42.0
|
14.0
|
18
|
42.3
|
14.0
|
|
3
|
42.7
|
14.2
|
8
|
44.0
|
14.7
|
13
|
41.3
|
13.4
|
18
|
42.9
|
14.8
|
|
4
|
41.8
|
13.7
|
9
|
43.3
|
15.1
|
14
|
43.9
|
15.7
|
19
|
40.2
|
11.8
|
|
4
|
41.2
|
12.3
|
9
|
40.3
|
11.8
|
14
|
41.3
|
13.2
|
19
|
40.6
|
12.6
|
|
4
|
42.5
|
14.0
|
9
|
41.1
|
12.4
|
14
|
42.8
|
14.1
|
19
|
41.9
|
13.3
|
|
4
|
41.9
|
14.1
|
9
|
42.8
|
14.2
|
14
|
41.7
|
13.2
|
19
|
41.2
|
12.7
|
|
4
|
41.1
|
12.3
|
9
|
42.4
|
13.7
|
14
|
41.3
|
12.9
|
19
|
42.3
|
13.5
|
|
5
|
41.7
|
13.6
|
10
|
42.7
|
13.7
|
15
|
41.1
|
12.9
|
20
|
41.9
|
14.1
|
|
5
|
41.3
|
13.1
|
10
|
42.1
|
13.2
|
15
|
43.2
|
14.2
|
20
|
41.8
|
13.2
|
|
5
|
42.8
|
15.1
|
10
|
42.3
|
13.9
|
15
|
42.9
|
14.9
|
20
|
41.2
|
12.8
|
|
5
|
41.6
|
13.1
|
10
|
41.3
|
12.9
|
15
|
43.4
|
14.5
|
20
|
40.4
|
12.5
|
|
5
|
40.4
|
11.8
|
10
|
41.9
|
13.2
|
15
|
41.7
|
12.8
|
20
|
42.6
|
14.0
|
The variable Ash is a standard ash test used to monitor the composition of Pencil lead, which is a combination of clay and graphite. The higher the ash content, the more clay in the formulation.
Por is a standard porosity test. Ceramic materials become more porous during the initial phases of the firing cycle, and then less porous later. The smaller the porosity, the more complete the firing and the stronger the ceramic body (also the more brittle). Too high a porosity indicates under-firing, and too low a porosity indicates over-firing.
For the above retrospective data, construct a bivariate control chart. You need to show all steps in computing the points plotted and control limits. Interpret the chart and compare it with the corresponding univariate charts.
Question 3 - Perform a simulation study to compare the effectiveness of various statistical intervals for the standard beta distribution. Discuss your conclusions.
Attachment:- Assignment.rar