23-Ind-B1 Reliability and Maintainability · December 2014

Question 7 of 9: Two-Sample $F$- and $t$-Tests — Marathon Shoe Comparison

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

Notes on this paper

National Exams — December 2014 — 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: four sections — Section A: do 2 of 3 questions (20 marks); Section B: do 1 of 2 (25 marks); Section C: do 1 of 2 (25 marks); Section D: do 1 of 2 (30 marks) — a 5-question, 100-mark paper as printed. All nine questions across the four sections are solved below for completeness. Page 1's own NOTES list is mis-numbered (two items both labelled "4."), transcribed as printed.

Reference texts: Montgomery & Runger, Applied Statistics and Probability for Engineers (7th ed., Wiley) — point/interval estimation (ch. 8–9), hypothesis testing (ch. 9–10), simple/multiple linear regression (ch. 11–12), single- and two-factor ANOVA (ch. 13). Montgomery, Peck & Vining, Introduction to Linear Regression Analysis (6th ed., Wiley) — multiple regression by matrices, confidence/prediction intervals (ch. 2–3).

Question 7 (Section C.2): Two-Sample $F$- and $t$-Tests — Marathon Shoe Comparison (25 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. Two independent samples of finishing times (minutes), $n_1=n_2=10$:

Vancouver (regular)	164.20	212.69	157.75	237.43	223.75	222.39	162.54	166.89	159.79	197.26
Victoria (smart shoe)	166.15	195.49	141.97	238.87	225.77	218.82	163.31	164.96	153.46	178.69

Find. (a) equality of variances at $\alpha=0.05$; (b) is the smart shoe faster, at $\alpha=0.05$; (c) 95% CI for $\mu_1-\mu_2$; (d) relate (b) and (c); (e) critique the test design.

Approach. Test $H_0:\sigma_1^2=\sigma_2^2$ with an $F$-ratio to decide between a pooled- or unequal-variance $t$-test; run a one-tailed two-sample $t$-test for "smart shoe faster"; build the two-sided CI with the same pooled variance; then interpret consistency between the interval and the earlier test.

Part (a) — equality of variances. $s_{\text{Van}}=31.46$, $s_{\text{Vic}}=33.22$ (min), so $$F_0=\frac{s_{\text{Van}}^2}{s_{\text{Vic}}^2}=\frac{31.46^2}{33.22^2}=\boxed{0.897}.$$ Against a two-sided $F_{0.025,9,9}=4.03$ and $F_{0.975,9,9}=0.248$, $F_0=0.897$ falls inside $(0.248,\ 4.03)$: fail to reject $H_0$ — the two variances are not significantly different at $\alpha=0.05$, so a pooled-variance $t$-test is appropriate in part (b).
Part (b) — is the smart shoe faster? $\bar x_{\text{Van}}=190.47$, $\bar x_{\text{Vic}}=184.75$. Assuming (per part (a)) equal population variances, independent random samples, and approximately normal finishing times, the pooled variance is $$s_p^2=\frac{9(31.46)^2+9(33.22)^2}{18}=1046.7,\qquad se=\sqrt{s_p^2\left(\frac{1}{10}+\frac{1}{10}\right)}=14.47,$$ $$t_0=\frac{\bar x_{\text{Van}}-\bar x_{\text{Vic}}}{se}=\frac{190.47-184.75}{14.47}=\boxed{0.395}.$$ Testing $H_1:\mu_{\text{Van}}>\mu_{\text{Vic}}$ (smart shoe faster $\Rightarrow$ lower Victoria time) against $t_{0.05,18}=1.734$: since $0.395<1.734$, fail to reject $H_0$ — there is not enough evidence at $\alpha=0.05$ that the smart shoe produces a faster time.
Part (c) — 95% CI for the difference. With $t_{0.025,18}=2.101$: $$(\bar x_{\text{Van}}-\bar x_{\text{Vic}}) \pm t_{0.025,18}\,se = 5.72 \pm 2.101(14.47) = \boxed{(-24.7,\ 36.1)\ \text{min}}.$$
Part (d) — relating (b) and (c). The 95% CI for $\mu_{\text{Van}}-\mu_{\text{Vic}}$ comfortably contains 0, which is exactly consistent with failing to reject $H_0:\mu_{\text{Van}}-\mu_{\text{Vic}}=0$ in part (b): whenever a two-sided $(1-\alpha)$ CI for a difference contains zero, the corresponding two-sided test at level $\alpha$ cannot reject $H_0$ of no difference (and a one-sided test at $\alpha/2$-equivalent stringency reaches the same non-rejection here, since the interval is not even close to excluding 0 on the smart-shoe-faster side).
Part (e) — is this a good test design? No — there is real potential for bias. The two groups ran in different cities (Vancouver vs. Victoria), so course elevation profile, road surface, weather on race day, and spectator/crowd conditions are all confounded with shoe type; a faster or slower course alone could explain the (statistically insignificant) mean difference observed. There is also no blinding (runners knew which shoe they wore, which can affect pacing psychologically) and no control for runner ability beyond "varying running abilities," which is not the same as a balanced, randomized assignment. A better design would use a within-runner, randomized, crossover protocol — the same 20 runners each race once in regular shoes and once in smart shoes (order randomized, ideally on the same course under similar conditions, e.g. two laps of one course or two comparable races), analyzed as paired differences, which removes both the course confound and most of the between-runner ability variance that is currently inflating the standard error and masking any real shoe effect.

Part	Result
(a) $F_0$ vs. $F_{0.025,9,9}$	0.897, within (0.248, 4.03) — equal variances
(b) $t_0$ vs. $t_{0.05,18}$	0.395 < 1.734 — not significantly faster
(c) 95% CI, $\mu_{\text{Van}}-\mu_{\text{Vic}}$	(−24.7, 36.1) min
(e) Design flaw	city/course confounded with shoe — use a paired crossover design

Concept & Theory

Topic. Two-sample inference — $F$-test for equal variances, pooled two-sample $t$-test, and the CI–test duality.

Key relations. $F_0=s_1^2/s_2^2\sim F_{n_1-1,n_2-1}$; pooled $t_0=\dfrac{\bar x_1-\bar x_2}{\sqrt{s_p^2(1/n_1+1/n_2)}}\sim t_{n_1+n_2-2}$; a $(1-\alpha)$ CI for $\mu_1-\mu_2$ excludes 0 iff the corresponding $\alpha$-level two-sided test rejects $H_0:\mu_1=\mu_2$.

Why this works. The $F$-test result gates which $t$-test formula is valid (pooled vs. Welch/unequal-variance); running the $t$-test and the CI from the same pooled variance guarantees the duality in part (d) holds exactly, which is a useful self-check on the arithmetic.

Common pitfall. Treating this as a paired design (subtracting row-by-row) when the two samples are genuinely independent (different individuals, different cities) — pairing unrelated observations by table row is a common but invalid shortcut; and confusing "failed to reject" with "proved the shoes are the same," when the correct reading is "this test lacked the power/design to detect a difference, if one exists."

Source. Montgomery & Runger, Applied Statistics and Probability for Engineers, ch. 9–10 (two-sample tests for means and variances, experimental design considerations).

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