23-Ind-B1 Reliability and Maintainability · December 2016
Question 8 of 11: Frozen-Pie Fill Mass — Two-Way ANOVA with Interaction
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Exams — December 2016 — 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 4 (40 marks) — a 6-question, 100-mark paper as printed. All eleven questions across the three sections are solved below for completeness.
Reference texts: Montgomery & Runger, Applied Statistics and Probability for Engineers (7th ed., Wiley) — joint distributions and covariance (ch. 5), point/interval estimation (ch. 8), hypothesis testing incl. two-sample and goodness-of-fit tests (ch. 9–10), simple linear regression (ch. 11), design and analysis of single-factor and factorial experiments (ch. 13–14). Montgomery, Peck & Vining, Introduction to Linear Regression Analysis (6th ed., Wiley) — multiple regression by matrices, confidence/prediction intervals (ch. 2–3). Montgomery, Design and Analysis of Experiments (9th ed., Wiley) — two-way factorial ANOVA and $2^k$ designs (ch. 5, 6–7).
Question 8 (Section C.1): Frozen-Pie Fill Mass — Two-Way ANOVA with Interaction (20 marks)
Given. 3 pie types × 3 speeds, $n=4$ replicate masses (g) per cell ($N=36$):
Speed \ Type
Apple
Strawberry
Blueberry
Fast
395, 390, 401, 400
392, 392, 394, 401
402, 405, 399, 399
Normal
396, 399, 400, 402
390, 392, 395, 420
404, 403, 400, 399
Slow
410, 408, 408, 407
404, 406, 401, 400
415, 412, 413, 415
Find. (a) is pie type and/or agitator speed significant at $\alpha=0.05$; (b) is there a significant type×speed interaction, and what does it mean operationally.
Approach. A balanced two-factor fixed-effects ANOVA with replication: partition the given $SST$ into pie-type, speed, interaction, and error, then $F$-test each against $MSE$.
Confirm the given summary values. $N=36$, grand sum $=14469$, grand mean $=401.9167$ (all match). $$SST=(N-1)\times\text{variance}=35\times54.82143=\boxed{1918.75}$$
Factor sums of squares. Type means: Apple $=401.333$, Strawberry $=398.917$, Blueberry $=405.5$. Speed means: Fast $=397.5$, Normal $=400.0$, Slow $=408.25$. With $a=b=3$ levels and $n=4$ reps/cell: $$SS_{Type}=b\,n\sum(\bar x_{i\cdot}-\bar x)^2=\boxed{266.17}\qquad SS_{Speed}=a\,n\sum(\bar x_{\cdot j}-\bar x)^2=\boxed{759.50}$$ Interaction (from the 9 cell means) $SS_{Inter}=n\sum(\bar x_{ij}-\bar x_{i\cdot}-\bar x_{\cdot j}+\bar x)^2=\boxed{79.83}$, and these three plus the given $SSE=813.25$ reproduce $SST$: $266.17+759.50+79.83+813.25=1918.75$ ✓.
ANOVA table and $F$-tests. $df$: Type $=2$, Speed $=2$, Interaction $=4$, Error $=27$. $MSE=813.25/27=30.120$.
Source
$SS$
$df$
$MS$
$F$
$F_{crit}$
Pie type
266.17
2
133.08
4.418
3.354
Speed
759.50
2
379.75
12.608
3.354
Interaction
79.83
4
19.958
0.663
2.728
Error
813.25
27
30.120
—
—
Both main effects exceed their critical value: $$\boxed{\text{Pie type: significant }(F=4.418\gt3.354);\quad\text{Speed: significant }(F=12.608\gt3.354)}$$
(b) Interaction and operating-procedure interpretation. $F_{Inter}=0.663\lt F_{crit}=2.728$, so $$\boxed{\text{no significant type}\times\text{speed interaction}}$$ — the effect of agitator speed on fill mass is essentially the same regardless of which pie type is running. Operationally this is good news: the firm does not need a separate speed setting tuned per pie type. Looking at the speed means against the 400 g target, Normal speed ($\bar x=400.0$g) sits right on target, Fast slightly under-fills ($397.5$g) and Slow noticeably over-fills ($408.25$g, a give-away of raw material) — a single Normal-speed setting is the recommended standard across all three pie types.
Summary
Source
$F$
Verdict
Pie type
4.418
Significant
Agitator speed
12.608
Significant
Type × speed
0.663
Not significant
Recommended setting
Normal speed (400.0g mean, on target), all pie types
