16-Civ-B10 Traffic Engineering · December 2013

Question 5 of 7: Statistical Significance of a Change in Mean Spot Speed

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

Notes on this paper

Paper format. National Examinations, December 2013 — 98-Civ-B10 Traffic Engineering. Three-hour, open-book examination; any non-communicating calculator is permitted. Seven questions are printed and five complete solutions are required, all questions being of equal value. The printed grading scheme is Q1 (a) 6, (b) 6, (c) 8; Q2 (a) 10, (b) 10; Q3 (a) 10, (b) 10; Q4 20; Q5 20; Q6 (a)–(d) 5 each; Q7 (a)–(d) 5 each. The paper also states that if doubt exists as to the interpretation of a question the candidate should submit a clear statement of any assumptions made, and that any data required but not given can be assumed. All seven questions are worked below.

Reference texts.

Garber, N. J. and Hoel, L. A., Traffic and Highway Engineering, 5th ed. — Ch. 4 (traffic engineering studies: spot-speed statistics, volume and travel-time studies, the moving-vehicle method), Ch. 6 (fundamental principles of traffic flow: time-mean and space-mean speed, speed–density models, Poisson arrivals, deterministic queueing), Ch. 8 (intersection control and signalisation: progression and time–space diagrams), Ch. 9 (capacity and level of service for highway segments), Ch. 10 (capacity and level of service at signalised intersections). This is the principal reference for the subject.
Transportation Research Board, Highway Capacity Manual (HCM) — basic freeway segments and signalised-intersection saturation flow.
Transportation Association of Canada (TAC), Geometric Design Guide for Canadian Roads — Canadian lane-width, shoulder and clearance practice.
TAC, Manual of Uniform Traffic Control Devices for Canada (MUTCDC) — Canadian signal warrants, timing and coordination practice.
Institute of Transportation Engineers, Traffic Engineering Handbook — signal-system progression and alternate-system design.

Check: assumed reference-table values. Questions 2(a) and 2(b) are capacity problems whose input list — lane width, lateral obstruction, per-cent heavy vehicles, a specific grade, design speed and a target level of service — is exactly the argument list of the classical Highway Capacity Manual equations, but the paper does not reproduce the lookup tables. Consistent with the paper's own instruction that any data required but not given may be assumed, every table value used is stated explicitly at the point of use, drawn from one coherent edition family (HCM 1985/1994, ideal capacity 2,000 pc/h/ln for the freeway segment). Substituting another edition's tables rescales the final flow rate but changes neither the method nor the arithmetic chain; the sensitivity is discussed in the Question 2(a) concept note.

Question 5: Statistical Significance of a Change in Mean Spot Speed (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. Two independent spot-speed samples taken on the same approach, before and after warning signs were installed.

Sample	Size n	Mean speed (km/h)	Standard deviation (km/h)
Before signs	200	55	10
After signs	100	50	8
Observed decrease	—	5	—
Level of significance	α = 0.10	—	—

Find. Whether the observed 5 km/h reduction in mean spot speed is statistically significant at the 0.10 level, i.e. whether it is larger than sampling variability alone would plausibly produce.

Approach. Test the null hypothesis of no change against a one-sided alternative of a decrease. Both samples are large, so the sampling distribution of the difference of means is normal and the sample standard deviations may be used in place of the unknown population values; compare the resulting z statistic with the one-tailed critical value.

State the hypotheses. Let $\mu_{1}$ and $\mu_{2}$ be the true mean speeds before and after the signs. The question asks specifically about a decrease, so the test is one-tailed: $$H_{0}:\ \mu_{1}-\mu_{2}=0\qquad\text{versus}\qquad H_{1}:\ \mu_{1}-\mu_{2}>0$$ with $\alpha=0.10$ in the single upper tail. Choosing one tail rather than two is a decision that must be made before looking at the data, and is justified here because the warning signs could only be expected to reduce speeds.
Compute the standard error of the difference of means. Because the two samples are independent, their variances add: $$SE=\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}} =\sqrt{\frac{10^{2}}{200}+\frac{8^{2}}{100}} =\sqrt{0.500+0.640}=\sqrt{1.140}$$ $$SE=1.068\ \text{km/h}$$ The smaller "after" sample contributes the larger share of the variance (0.640 against 0.500) despite having the smaller standard deviation, because sample size enters the denominator.
Form the test statistic. With samples this large the normal approximation applies and the sample standard deviations stand in for the population values: $$z=\frac{\bar{x}_{1}-\bar{x}_{2}}{SE}=\frac{55-50}{1.068}$$ $$\boxed{z=4.68}$$ The observed difference is more than four and a half standard errors from zero.
Compare with the critical value and decide. For a one-tailed test at $\alpha=0.10$ the critical value is $z_{0.10}=1.282$. Since $$z=4.68\ >\ z_{crit}=1.282$$ the null hypothesis is rejected. The corresponding p-value is $P(Z>4.68)\approx 1.4\times 10^{-6}$, so a difference this large would arise by chance about once in seven hundred thousand studies if the signs had no effect. The conclusion is not marginal: the same decision would follow at $\alpha=0.05$, $\alpha=0.01$ and even $\alpha=0.001$ (for which $z_{crit}=3.09$).
State the engineering conclusion, and its limits. $$\boxed{\text{The decrease in mean speed is statistically significant at } \alpha=0.10}$$ A one-sided 90 % confidence statement puts the true reduction at least $5-1.282(1.068)=3.63$ km/h. Two cautions belong with this result. Statistical significance is not the same as practical significance: a 5 km/h reduction on a 55 km/h approach is a 9 % change, whose effect on collision frequency must be argued separately from crash data or a speed–severity relationship. And a before-and-after comparison attributes the whole change to the signs, whereas regression to the mean, seasonal or weather differences, and other concurrent changes at an "accident prone" location could contribute; a control site measured over the same period is the standard remedy.

Quantity	Value
Hypotheses	H₀: μ₁ = μ₂ versus H₁: μ₁ > μ₂ (one-tailed)
Standard error of the difference	1.068 km/h
Test statistic z	4.68
Critical value z at α = 0.10 (one tail)	1.282
p-value	≈ 1.4 × 10⁻⁶
Decision	Reject H₀ — the decrease IS statistically significant at α = 0.10
One-sided 90 % lower bound on the reduction	3.63 km/h

Topic. Hypothesis testing for a difference between two large-sample mean spot speeds in a before-and-after study.

Key relations. $SE=\sqrt{s_{1}^{2}/n_{1}+s_{2}^{2}/n_{2}}$; $z=(\bar x_{1}-\bar x_{2})/SE$; reject $H_{0}$ if $z>z_{\alpha}$ (one-tailed) or $|z|>z_{\alpha/2}$ (two-tailed), with $z_{0.10}=1.282$, $z_{0.05}=1.645$, $z_{0.025}=1.960$; and the spot-speed sample-size formula $n=(z\,\sigma/d)^{2}$ used when designing such a study.

Why this works. The logic is to ask how surprising the observed difference would be if the signs had changed nothing. Under that null hypothesis the difference of sample means is a random variable centred on zero whose spread is the standard error, and the central limit theorem makes it normal for samples of 100 and 200 regardless of the shape of the underlying speed distribution. Variances add rather than standard deviations because the samples are independent, which is the single most important algebraic point. Using $z$ rather than Student's $t$ is justified by the sample sizes: with more than about 30 observations the $t$ distribution is indistinguishable from the normal for practical purposes, and the sample standard deviations are reliable estimates of the population values. The one-tailed choice doubles the sensitivity to a decrease at the cost of no power at all to detect an increase, and must be justified by the engineering question rather than by the observed direction of the data.

Common pitfall. Three errors recur. Adding standard deviations instead of variances ($SE\ne s_{1}/\sqrt{n_{1}}+s_{2}/\sqrt{n_{2}}$) inflates the standard error and can flip the decision on marginal data. Using the two-tailed critical value 1.645 for $\alpha=0.10$ when the question asks about a decrease — harmless here because $z=4.68$ clears both, but wrong in principle and decisive on closer calls. And forgetting that failing to reject is not proof of no effect: with these sample sizes a genuine reduction of 1 km/h would probably have gone undetected, so a non-significant result would warrant a power calculation rather than a conclusion that the signs did nothing.

Source. Garber & Hoel, Traffic and Highway Engineering, 5th ed., Ch. 4 (Traffic Engineering Studies) — spot-speed studies, sample size, and statistical comparison of before-and-after speed data; Ch. 5 for the highway-safety framing of before-and-after evaluation.

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