23-Ind-B1 Reliability and Maintainability · December 2019

Question 6 of 9: Sign Test and Independent-Month Binomial Probabilities

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

Notes on this paper

National Exams — December 2019 — 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, 80-mark paper as actually structured by the section table and each question's own printed marks (the front page's own summary line states “Exam: 5 Questions. Total marks: 100”, which is internally inconsistent with 20+40+20=80 from its own section breakdown and every question's printed mark value). 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) — probability density functions and moments (ch. 4), point/interval estimation (ch. 8), two-sample hypothesis testing and the sign test (ch. 9–10, 16), 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, Tukey's HSD, $2^k$ factorial designs (ch. 3, 6).

Question 6 (Section B.2): Sign Test and Independent-Month Binomial Probabilities (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. Hypothesized median $=18$ months; 12 monthly observed wait times: 15,19,16,17,18,12,21,17,14,19,17,16. Historical $P(\text{wait}>20\text{ months})=0.02$, months independent.

Find. (a) Sign test for a shift in the median; (b) $P(\text{exactly 2 months exceed the threshold})$; (c) $P(\text{at least 2 months exceed 20 months})$.

Approach. The sign test only needs the sign of each observation relative to the hypothesized median (nonparametric, robust to outliers, exactly matching the note in the question stem). Parts (b)/(c) are independent Bernoulli trials across 12 months, so the count of “exceeding” months is Binomial$(12,p)$.

(a) Sign test, $H_0:\text{median}=18$, $\alpha=0.05$. Differences from 18: $-3,+1,-2,-1,\mathbf{0},-6,+3,-1,-4,+1,-1,-2$. Month 5 ties the median exactly and is dropped, leaving $n=11$ usable signs: $3$ positive, $8$ negative. $$S=\min(n_+,n_-)=\min(3,8)=3$$ Exact two-sided binomial $p$-value ($n=11$, $p=0.5$): $$p=2\,P(X\le3)=\boxed{0.227}$$ (Normal approximation with continuity correction agrees: $z=-1.21$, $p=0.228$.) Since $p=0.227\gt0.05$, fail to reject $H_0$ — no significant evidence the median wait time has changed over the past 12 months.
(b) $P(\text{exactly 2 months exceed the threshold})$. Let $X=$ number of months (out of 12) exceeding the threshold, $X\sim\text{Binomial}(12,\,p=0.02)$, using the historical per-month rate given in the stem. $$P(X=2)=\binom{12}{2}(0.02)^2(0.98)^{10}=\boxed{0.0216}$$
(c) $P(\text{at least 2 months exceed 20 months})$. Same $X\sim\text{Binomial}(12,0.02)$: $$P(X\ge2)=1-P(X=0)-P(X=1)=1-0.7847-0.1922=\boxed{0.0231}$$

Final Results
Quantity	Value
(a) Sign test $p$-value	0.227 — fail to reject (no evidence of a median shift)
(b) $P(X=2)$	0.0216
(c) $P(X\ge2)$	0.0231

Check Part (b) as printed reads “exceeds 21 months,” but the stem's own given probability ($P(\text{wait}\gt20\text{ months})=2\%$) and part (c)'s own wording both consistently use the 20-month threshold, with no separate probability ever supplied for a 21-month threshold; the “21” is printed exactly this way in the paper. Read as a typo for “20 months,” matching the only probability actually given and part (c)'s parallel wording, and answered on that consistent basis.

Topic: Nonparametric sign test for a hypothesized median; binomial model for independent per-period rare-event counts.

Key relations: Sign test: $S=\min(n_+,n_-)\sim$ compare to $\text{Binomial}(n,0.5)$ tail; rare-event count: $X\sim\text{Binomial}(n,p)$, $P(X=k)=\binom nk p^k(1-p)^{n-k}$.

Why this works: The sign test discards magnitude information (exactly the robustness-to-outliers the question stem itself motivates for using medians), trading some power for a distribution-free guarantee; the binomial model in (b)/(c) is valid specifically because month-to-month independence is stated, letting each month be treated as its own independent Bernoulli trial.

Common pitfall: Dropping tied observations from $n$ but forgetting to also update $n$ when computing the exact binomial tail probability, or confusing “exactly 2” ($P(X=2)$) with “at least 2” ($P(X\ge2)$) — the two differ by every term $P(X=3),P(X=4),\ldots$

Source: Montgomery & Runger, Applied Statistics and Probability for Engineers, 7th ed., §16.4 (the sign test), §3.5 (binomial distribution).

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