23-Ind-B1 Reliability and Maintainability · December 2018

Question 6 of 9: Comparing Two Marathon Routes — Variance and Mean Tests

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).

Question 6 (Section B.2): Comparing Two Marathon Routes — Variance and Mean Tests (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. Finishing times (min), 10 runners per year/route-order.

Halifax-first	203	209	225	239	255	260	261	262	263	279
Dartmouth-first	207	222	229	240	247	247	250	252	260	265

Find. (a) Are the two population variances equal ($\alpha=0.01$)? (b) is the mean difference 0 ($\alpha=0.1$)? (c) assumptions; (d) $p$-value and its interpretation; (e) sample-bias sources and fixes.

Approach. Run an $F$-test on the variance ratio first (it determines whether the pooled or Welch $t$-test is appropriate), then the corresponding two-sample $t$-test on the means.

Sample statistics. $$\bar x_H=245.6,\ s_H=25.48\ (n_H=10)\qquad \bar x_D=241.9,\ s_D=17.84\ (n_D=10)$$
(a) $F$-test for equal variances, $\alpha=0.01$ (two-sided). $$F=\frac{s_H^2}{s_D^2}=\frac{649.16}{318.32}=\boxed{2.039}$$ Two-sided critical values at $\alpha=0.01$, $df=(9,9)$: $F_{0.005,9,9}=6.541$ (upper), $F_{0.995,9,9}=0.153$ (lower). Since $0.153<2.039<6.541$ ($p=0.303$), fail to reject — the two population variances can be treated as equal.
(b) Pooled two-sample $t$-test, $H_0:\mu_H-\mu_D=0$, $\alpha=0.1$. $$s_p^2=\frac{9(649.16)+9(318.32)}{18}=483.74,\qquad SE=\sqrt{s_p^2\left(\tfrac1{10}+\tfrac1{10}\right)}=9.837$$ $$t=\frac{245.6-241.9}{9.837}=\boxed{0.376},\qquad df=18$$ Critical value (two-sided, $\alpha=0.1$): $t_{0.05,18}=1.734$. Since $|t|=0.376\ll1.734$, fail to reject $H_0$: no statistically significant difference in mean finishing time between route orders.
(c) Assumptions (essay). Both samples are independent random draws of finishers (the Halifax-first and Dartmouth-first years are treated as two independent populations, not paired by runner); finishing times in each population are approximately normally distributed, as stated; the two population variances are equal (justified by part a); and the 10 sampled finishers each year are representative of that year's full finisher field, not a biased subset (e.g. only fast finishers).
(d) $p$-value and interpretation. $p=2P(T_{18}>0.376)=\boxed{0.711}$. Interpretation: if the true mean finishing time really were identical regardless of which loop is run first, there would be about a 71% chance of observing a sample difference at least as large as the $3.7$-minute difference actually seen, purely from random sampling variation. That is very weak evidence against $H_0$ — entirely consistent with the “fail to reject” conclusion in (b).
(e) Bias sources and mitigation (essay). (1) Weather is explicitly noted as variable year to year and is completely confounded with route order under the strict yearly-alternation scheme — any weather effect (e.g. a hot, calm year vs. a windy, cool one) shows up indistinguishably as a “route order” effect. Mitigation: pool data across many more years so weather variation averages out on both sides, or record and statistically control for a weather covariate (temperature, wind). (2) Runner population differs year to year (different entrants, different fitness levels, no guarantee the same people ran both years), so the comparison mixes a route-order effect with a population effect. Mitigation: track a panel of the same runners across both years (a matched-pairs design), which removes between-runner variability entirely and greatly increases test power at the same sample size. (3) The sampling procedure for “10 finishers” is not described — if it excludes DNFs, injuries, or is otherwise a convenience sample rather than a documented random draw from the full finisher list, the sample may not represent the true finishing-time distribution. Mitigation: define and document an explicit random-sampling protocol from the complete, verified finisher list each year.

Final Results
Quantity	Value
$F$ (variance-equality, $\alpha=0.01$)	2.039 (crit. $[0.153,6.541]$) — equal
Pooled $t$ (mean diff.$=0$, $\alpha=0.1$)	0.376 (crit. $\pm1.734$) — fail to reject
$p$-value	0.711

Topic: Two-sample inference — $F$-test for variances followed by a pooled $t$-test for means; confounding and sampling bias.

Key relations: $F=s_1^2/s_2^2\sim F_{n_1-1,n_2-1}$; pooled $t$ as in Q3; $p$-value $=P(|T_{df}|\ge|t_{obs}|)$ under $H_0$.

Why this works: Testing the variance ratio first is the standard gatekeeper step that determines which mean-comparison procedure (pooled vs. Welch) is theoretically justified; here it clears the way for the simpler pooled test.

Common pitfall: Confusing a large $p$-value with proof that the effect is zero — a $p=0.711$ result says the data are consistent with no difference, not that no difference exists; a confounded design (route order tied to calendar year) can also mask a real effect that a better design would detect.

Source: Montgomery & Runger, Applied Statistics and Probability for Engineers, 7th ed., §9.3–9.5 (two-sample $t$ and $F$ tests); §8.2 (sampling and bias considerations).

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