23-Ind-B1 Reliability and Maintainability · December 2017

Question 9 of 10: Photocopier Gluing Power — Three-Factor ANOVA with Replication

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

National Exams — December 2017 — 98-Ind-B1 Applied Probability & Statistics. Three-hour, closed-book exam; one of two permitted calculators (Sharp or Casio), one 8.5″×11.0″ aid sheet (both sides), statistical tables supplied. Format: three sections — Section A: do 2 of 4 (30 marks); Section B: do 2 of 3 (30 marks); Section C: do 2 of 3 (40 marks) — a 6-question, 100-mark paper as printed. All ten questions across the three sections are solved below for completeness.

Reference texts: Montgomery & Runger, Applied Statistics and Probability for Engineers (7th ed., Wiley) — discrete/continuous distributions (ch. 3–4), joint distributions (ch. 5), point/interval estimation and sample size (ch. 8), hypothesis testing incl. two-sample tests (ch. 9–10), simple linear regression (ch. 11), design and analysis of single-factor experiments (ch. 13). Montgomery, Peck & Vining, Introduction to Linear Regression Analysis (6th ed., Wiley) — multiple regression by matrices (ch. 2–3). Montgomery, Design and Analysis of Experiments (9th ed., Wiley) — multi-factor and $2^k$ factorial designs (ch. 5–6).

Question 9 (Section C.2): Photocopier Gluing Power — Three-Factor ANOVA with Replication (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Three fixed factors, three levels each — Temperature (Low/Medium/High), Surface state (Soft/Medium/Hard), Press (20/40/60 lb/in²) — 27 cells, 4 replicates/cell ($N=108$). Partial computer ANOVA output:

Source	DF	SS	MS	F	P
Temp	X	0.166169	0.083084	SSS.S	0.000
Surface	X	0.078252	0.039126	6.70	0.002
Press	X	0.019469	0.009734	1.67	0.195
Temp×Surface	Y	0.128454	0.032113	5.50	0.001
Temp×Press	Y	0.062804	0.015701	2.69	0.037
Surface×Press	Y	0.126437	0.031609	5.41	0.001
Temp×Surf×Press	Z	0.141224	0.017653	3.02	0.005
Error	81	QQQQQQQQ	R.RRRRRR
Total	AAA	1.196032

$S=0.0764348$, $R\text{-}Sq=TT.T\%$, $R\text{-}Sq(\text{adj})=47.73\%$.

Find. (d) $X,Y,Z,AAA,QQQQQQQQ,R.RRRRRR,SSS.S,TT.T$; (e) significant interactions; (f) significant main effects and where uncertain; (g) goodness of fit; (h) optimization advice.

Approach. Each factor has 3 levels, so main-effect $df=2$ and two-way interaction $df=4$; the three-way interaction $df=8$; total $df=N-1=107$; error is recovered as $SST$ minus the sum of every effect's $SS$, and every $F=MS_{effect}/MS_{Error}$. $R^2=SS_{model}/SST$ and $R^2_{adj}$ cross-checks the printed 47.73%.

(d) Filling the ANOVA table. With $a=b=c=3$ levels each: $$X=a-1=\boxed{2}\ (\text{each main effect}),\qquad Y=(a-1)(b-1)=\boxed{4}\ (\text{each 2-way}),\qquad Z=(a-1)(b-1)(c-1)=\boxed{8}$$ $$AAA=N-1=108-1=\boxed{107}$$ Summing the seven given/derived effect $SS$ (Temp$+$Surface$+$Press$+$T×S$+$T×P$+$S×P$+$T×S×P$=0.722807$): $$QQQQQQQQ=SSE=SST-0.722807=1.196032-0.722807=\boxed{0.473225}$$ $$df_{Error}=N-27=108-27=81\ (\text{27 cells, one mean each}),\qquad R.RRRRRR=MSE=0.473225/81=\boxed{0.005842}$$ $$SSS.S=F_{Temp}=\frac{MS_{Temp}}{MSE}=\frac{0.083084}{0.005842}=\boxed{14.22}$$ $$TT.T=R^2=\frac{SS_{model}}{SST}\times100=\frac{0.722807}{1.196032}\times100=\boxed{60.4\%}$$ (Cross-check: $R^2_{adj}=1-\dfrac{SSE/81}{SST/107}=1-\dfrac{0.005842}{0.011178}=47.73\%$, matching the printed value exactly.)
(e) Significant interactions ($\alpha=0.05$, compare each $P$ to 0.05). All three two-way interactions are significant (Temp×Surface $p=0.001$; Temp×Press $p=0.037$; Surface×Press $p=0.001$), and the three-way Temp×Surface×Press interaction is also significant ($p=0.005$). $$\boxed{\text{Every interaction term in the model is statistically significant}}$$
(f) Significant main effects. Temp ($p=0.000$) and Surface ($p=0.002$) are both significant. Press, taken alone, is not significant ($p=0.195$) — but Press appears in three separate significant interactions (with Temp, with Surface, and in the three-way term), so its effect on gluing power is real but entirely conditional on the levels of the other two factors: $$\boxed{\text{Temp and Surface are significant main effects; Press's main effect is not significant in isolation, and its true impact is uncertain without specifying Temp and Surface (masked by significant interactions)}}$$
(g) Goodness of fit. $R^2=60.4\%$ means the full 27-cell model (all main effects and interactions) explains about 60% of the variability in gluing power; $R^2_{adj}=47.73\%$ is noticeably lower, reflecting the large number of estimated parameters (27 cell means from only 108 observations) relative to the variance explained. $$\boxed{\text{A moderate, not excellent, fit} - \text{roughly half the variability in gluing power remains unexplained by these three factors}}$$
(h) Optimization advice (larger gluing power is better). The 27 cell means range from a low of 0.373 (High temperature, Hard surface, 40 lb/in² press) up to a high of 0.773 (Low temperature, Medium surface, 40 lb/in² press) — well above the grand mean of 0.567. Because the significant interactions mean factor settings cannot be optimized independently, the recommendation must specify a full combination rather than "the best level of each factor separately": $$\boxed{\text{Operate at Low temperature, Medium surface hardness, and a moderate (40\ lb/in}^2\text{) press setting}}$$ Since Temp and Surface both drive the response directly AND interact strongly with Press, any change to one factor's setting should be re-validated against the others rather than adjusted in isolation.

Summary
Quantity	Value
$X,Y,Z,AAA$	2, 4, 8, 107
$SSE,MSE$	0.473225, 0.005842
$F_{Temp}$	14.22
$R^2,\ R^2_{adj}$	60.4%, 47.73%
Significant interactions	All four (T×S, T×P, S×P, T×S×P)
Best combination	Low Temp, Medium Surface, 40 lb/in² Press (mean 0.773)

Topic: Three-factor fixed-effects ANOVA with replication — degrees of freedom, sum-of-squares decomposition, and interaction interpretation.

Key relations: $df_{main}=levels-1$; $df_{2way}=\prod(levels_i-1)$; $df_{Error}=N-(\text{number of cells})$; $SSE=SST-\sum SS_{effects}$; $R^2_{adj}=1-\dfrac{SSE/df_E}{SST/df_T}$.

Why this works: With every interaction term significant, the three factors cannot be reasoned about independently — the "best" level of Press genuinely depends on which Temp/Surface combination is in use, which is exactly why the main-effect table alone (Press non-significant) would be a misleading guide to optimization on its own.

Common pitfall: Concluding "Press doesn't matter" from its non-significant main effect $p=0.195$ — a factor can have no significant AVERAGE effect across all other factor levels while still mattering a great deal at specific combinations, which is precisely what the three significant Press interactions show here.

Source: Montgomery, Design and Analysis of Experiments (9th ed.), ch. 5–6 (multi-factor factorial designs with replication, interaction interpretation).

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