23-Ind-B1 Reliability and Maintainability · December 2018
Question 7 of 9: Edison Motors — Bartlett's Test, One-Way ANOVA, Tukey's Test
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Exams — December 2018 — 17-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 (20 marks); Section B: do 2 of 3 (40 marks); Section C: do 1 of 2 (20 marks) — a 9-question, printed-total-80-mark paper as actually structured (the front page's own summary line states “Exam: 5 Questions. Total marks: 100”, but 20+40+20=80 by the section instructions and per-question mark values printed beside each question — the front page's 100 is internally inconsistent with its own section table; the 80-mark reading is used throughout, consistent with the section-by-section split printed on pages 2, 4 and 6). All nine questions across the three sections are solved below for completeness.
Reference texts: Montgomery & Runger, Applied Statistics and Probability for Engineers (7th ed., Wiley) — normal distribution and sums of normals (ch. 4–5), point/interval estimation (ch. 8), two-sample hypothesis testing (ch. 9–10), simple linear regression (ch. 11), single-factor ANOVA and multiple comparisons (ch. 13). 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) — Bartlett's test, $2^k$ factorial designs (ch. 3, 6).
Given. Labour hours per vehicle, 5 vehicles sampled per plant.
Plant A
118
118
135
126
130
Plant B
158
155
152
167
157
Plant C
227
215
215
221
217
Find. (a) Bartlett's test for equal variances, $\alpha=0.05$; (b) one-way ANOVA — does plant affect build time? (c) Tukey's HSD to identify which pairs differ.
Approach. Bartlett's test checks the homogeneity-of-variance assumption that one-way ANOVA relies on; then run the ANOVA $F$-test, and if significant, Tukey's HSD to localize which plant pairs differ.
(a) Bartlett's test, $\alpha=0.05$. Pooled variance $s_p^2=\dfrac{\sum(n_i-1)s_i^2}{N-k}=\dfrac{4(55.8+31.7+26.0)}{12}=37.83$.
$$q=(N-k)\ln s_p^2-\sum(n_i-1)\ln s_i^2=0.654,\qquad c=1+\frac{\sum\frac{1}{n_i-1}-\frac{1}{N-k}}{3(k-1)}=1.111$$
$$\chi^2=\frac{q}{c}=\boxed{0.588}$$
Critical value $\chi^2_{0.05,2}=5.991$ ($p=0.745$). Since $0.588<5.991$, fail to reject equal variances — the ANOVA homogeneity-of-variance assumption is supported.
(b) One-way ANOVA, $\alpha=0.05$. Direct calculation from the raw data gives $SST=23047.6$ (matching the value the question offers as a shortcut — a good cross-check that the data table itself is trustworthy). Treatment (between-plant) sum of squares:
$$SS_{treat}=\sum n_i(\bar x_i-\bar x_{grand})^2=22593.6,\qquad SSE=SST-SS_{treat}=\boxed{454.0}$$
(close to, but more precise than, the question's offered approximation of 460). With $df_{treat}=2$, $df_E=12$:
$$MS_{treat}=\frac{22593.6}{2}=11296.8,\qquad MSE=\frac{454.0}{12}=37.83,\qquad F=\frac{MS_{treat}}{MSE}=\boxed{298.6}$$
Critical value $F_{0.05,2,12}=3.885$. Since $F=298.6\gg3.885$, reject $H_0$: plant has a highly significant effect on build time (using the question's own approximate $SSE=460$ instead gives $MSE=38.3$, $F\approx294.7$ — the same conclusion either way).
(c) Tukey's HSD, $\alpha=0.05$.
$$q_{0.05}(k=3,df=12)=3.773,\qquad HSD=q\sqrt{\frac{MSE}{n}}=3.773\sqrt{\frac{37.83}{5}}=\boxed{10.38\text{ h}}$$
Pairwise mean differences: $|\bar x_A-\bar x_B|=32.4$, $|\bar x_A-\bar x_C|=93.6$, $|\bar x_B-\bar x_C|=61.2$ — all exceed $HSD=10.38$, so all three plants differ significantly from one another (Plant A fastest, Plant C slowest).
