Question

Table (every row is a pencil and its corresponding data) :

Ash	Porosity	Outer Diameter	Strength	Location
42.2	12.9	0.087	1.25	1
43.8	13.7	0.09	1.2	1
42.1	15.6	0.087	0.85	1
42	13.3	0.086	1.15	1
45	12.2	0.088	1.55	1
42.5	14.3	0.085	1	1
41.9	13.1	0.085	1	1
42.4	13.8	0.086	1.05	1
41.9	14.4	0.085	1.15	1
42.1	15.5	0.086	0.9	1
42.2	12.7	0.087	1.3	1
43.4	13.4	0.089	1.2	1
41.9	12.9	0.084	1.35	1
42.4	15.5	0.085	1	1
43.3	13.8	0.085	1.1	1
42.2	13.3	0.085	1.25	1
40	14	0.088	1.2	1
42.7	16	0.09	0.95	1
41.9	12.7	0.084	1.05	1
42.5	16	0.087	0.85	1
41.1	14.3	0.086	1.25	1
41.9	15.2	0.087	0.85	1
42.5	14	0.085	0.8	1
41.4	16	0.085	0.6	1
41	14.4	0.085	0.7	1
38.9	15.8	0.086	0.6	2
42.5	14.9	0.087	0.9	2
42.2	12.5	0.083	1.05	2
41.6	15	0.087	1.1	2
42.6	15.4	0.091	1.2	2
42.7	13.4	0.083	0.7	2
44.1	15.8	0.088	0.6	2
43.2	13.3	0.083	1.25	2
43	13.6	0.088	1.3	2
42.7	13.9	0.09	1.35	2
41.6	14.1	0.09	1.3	2
42.4	15.6	0.083	0.6	2
42.9	14.2	0.086	0.9	2
43.5	15.4	0.088	0.75	2
43.3	15.2	0.086	0.75	2
41.4	15.5	0.093	1	2
42.7	14.3	0.087	0.75	2
42.8	14.1	0.085	1	2
42	13.6	0.084	1.1	2
42.2	13.4	0.092	1.3	2
42.4	14.1	0.091	1.3	2
42.8	16.1	0.087	0.6	2
42.5	14.9	0.086	0.95	2
42.3	15.5	0.091	1.15	2
43.3	15.9	0.089	0.65	2
40.4	13	0.084	1.35	3
41.7	14.2	0.083	1.2	3
42.3	12.8	0.083	1.25	3
42.9	14.3	0.09	1.15	3
40.3	14.2	0.084	0.95	3
41.3	15	0.088	0.8	3
42	12.5	0.086	1.2	3
42.2	15.6	0.084	0.6	3
42.5	13.7	0.088	1.05	3
42.8	17.8	0.088	0.4	3
40.5	14.2	0.084	0.65	3
40.1	14.3	0.085	0.95	3
42.9	17.9	0.085	0.55	3
41.7	12.9	0.092	1.45	3
41	15.5	0.092	1.2	3
42.5	13.9	0.93	1.5	3
43	14.7	0.084	0.85	3
43.4	12.6	0.086	1	3
41.1	16.3	0.088	1.1	3
42.7	13	0.085	1.2	3
42.1	13	0.084	0.9	3
43.7	13.5	0.087	1.15	3
39.5	16	0.088	0.64	3
42.1	14.1	0.087	1.2	3
42.7	13.2	0.091	1.4	3

a) Is the strength of the pencils that we manufacture consistent across all three manufacturing locations? Use descriptive statistics, a graphical measure, and hypothesis testing to support your decision. Satisfy assumptions prior to conducting any tests.

b) Determine the model that best explains pencil strength. (In other words, what variables impact pencil strength?) Use model selection techniques, evaluation of model assumptions, and hypothesis testing to support your decision.

Answer 1

a)

Descriptive Statistics: Strength
Variable	Location	Mean	StDev
Strength	1	1.062	0.2195
	2	0.966	0.2633
	3	1.0276	0.295

From the box-plot, we know that the strength of the pencils that manufacture at three locations may be consistent.

The most suitable test is One way ANOVA.

One-way ANOVA: Strength versus Location

Source	DF	SS	MS	F	P
Location	2	0.1183	0.0591	0.87	0.424
Error	72	4.9081	0.0682
Total	74	5.0263

S = 0.2611 R-Sq = 2.35% R-Sq(adj) = 0.00%

Comment: The estimated p-value is 0.424. Hence, we can conclude that manufacture consistent across all three manufacturing locations at 0.05 level of significance.

b) The model that best explains pencil strength is completely randomized design.

From the PP plot and p-value, we can conclude that the normal assumption on the residual is satisfied.