14.1 Applied Statistical Distributions & Sample Size Determination

Professional Traffic Operations Engineer PTOE

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1 Introduction & PTOE Exam Overview

1.1 The PTOE Credential: Certification Overview & Industry Value 1.2 Core Industry Reference Standards in Traffic Operations 1.3 Study Guide Architecture, Blueprint Alignment & Exam-Day Strategy

2 Domain 1: Traffic Flow Theory & Fundamental Concepts

2.1 Macroscopic Traffic Stream Models & Fundamental Diagram (q = k × u) 2.2 Microscopic Traffic Characteristics & Car-Following Models 2.3 Hydrodynamic Shockwave Analysis & Traffic Queuing Models

3 Domain 1: Highway & Interrupted Flow Capacity Analysis (HCM)

3.1 Basic Freeway Segments & Multilane Highways (HCM 7th Edition) 3.2 Freeway Weaving, Merging, and Diverging Segments 3.3 Signalized Intersection Capacity & Delay Modeling (HCM)

4 Domain 1: Managed Lanes, Transit Priority, TSMO & Travel Demand Management

4.1 Unsignalized Intersections (TWSC, AWSC) & Modern Roundabouts Capacity 4.2 Managed Lanes, HOV/HOT Operations, & Ramp Metering 4.3 Transit Priority & Multimodal Signal Operations (TSP) 4.4 Transportation Systems Management and Operations (TSMO) & Regional ITS 4.5 Travel Demand Management (TDM) & Trip Reduction Strategies

5 Domain 2: Geometric Design Controls, Criteria, & Sight Distance

5.1 Functional Classification, Design Controls, & Context-Sensitive Design 5.2 Sight Distance Principles: Stopping, Decision, & Passing (AASHTO) 5.3 Horizontal Alignment, Curve Radius, & Superelevation Design

6 Domain 2: Roadway Cross-Sections & Access Management

6.1 Vertical Alignment: Tangent Grades, Crest & Sag Vertical Curves 6.2 Roadway Cross-Section Elements & AASHTO Clear Zone Concepts 6.3 Access Management Principles & Driveway Spacing Criteria

7 Domain 2: Intersection & Interchange Geometric Design

7.1 Intersection Channelization, Turn Lane Design, & Storage Length 7.2 Interchange Configurations & Operational Comparisons (Diamond, Cloverleaf, SPUI) 7.3 Diverging Diamond Interchanges (DDI / DCMI) Operational Principles 7.4 Modern Roundabout Geometric Design Elements & Fastest Paths 7.5 Multimodal Geometric Design: Pedestrian Facilities, Bikeways, & ADA

8 Domain 3: Traffic Safety Concepts, Human Factors, & Safe System

8.1 The Safe System Approach & Vision Zero Principles 8.2 Human Factors, Driver Expectancy, & Perception-Reaction Time (PRT) 8.3 Crash Collision Typologies & Statistical Rate Formulations (MEV, MVMT)

9 Domain 3: Highway Safety Manual (HSM) Predictive Methods & EB

9.1 Highway Safety Manual (HSM) Predictive Methodology & Part C Models 9.2 Safety Performance Functions (SPFs) & Crash Modification Factors (CMFs) 9.3 Empirical Bayes (EB) Method & Regression-to-the-Mean (RTM) Correction 9.4 Economic Appraisal of Safety Projects (Benefit-Cost Analysis, NPV)

10 Domain 3: Safety Countermeasures & Road Safety Audits (RSA)

10.1 Roadway Departure Countermeasures (Rumble Strips, Safety Edge, Barriers) 10.2 Intersection Safety Countermeasures (Sight Triangles, Left-Turn Calming, Roundabouts) 10.3 Vulnerable Road User (VRU) Safety Countermeasures (PHB/HAWK, RRFB, Curb Extensions) 10.4 Road Safety Audits (RSAs) & Systemic Safety Analysis

11 Domain 4: Traffic Signal Warrants & Isolated Timing (MUTCD/ITE)

11.1 MUTCD Traffic Signal Warrants (Warrants 1 through 9) 11.2 Signal Phasing Principles & NEMA Dual-Ring Operations 11.3 Clearance Intervals (ITE Kinematic Equations) & Pedestrian Timing Parameters

12 Domain 4: Signal Coordination, Networks, & ATSPMs

12.1 Actuated Signal Controller Parameters (Passage Time, Gap Reduction, Max Recall) 12.2 Traffic Signal Coordination: Bandwidth, Offsets, & Progression Models 12.3 Traffic Responsive & Adaptive Traffic Signal Control Systems (ASCS) 12.4 Automated Traffic Signal Performance Measures (ATSPMs) & High-Resolution Data

13 Domain 4: Signing, Markings, Work Zones, Grade Crossings & School Zones (MUTCD)

13.1 MUTCD Regulatory, Warning, & Guide Signs Principles and Placement 13.2 Sign Retroreflectivity Standards & Nighttime Visual Maintenance 13.3 Longitudinal Pavement Markings, Delineators, & Object Markers 13.4 Temporary Traffic Control (TTC) & Work Zone Safety (MUTCD Part 6) 13.5 Highway-Rail Grade Crossings: Warning Devices, Preemption & Storage 13.6 School Zone Traffic Control & Student Walk Route Planning

14 Domain 5: Traffic Studies & Applied Statistical Methods

14.1 Applied Statistical Distributions & Sample Size Determination 14.2 Spot Speed Studies, 85th Percentile Speed, & 10-mph Pace 14.3 Volume Count Methodologies & Expansion Factoring (ADT, AADT, PHF)

15 Domain 5: Traffic Impact Studies & Trip Generation (ITE)

15.1 Travel Time, Delay, & Intersection Queue Studies 15.2 Parking Supply, Duration, & Turnover Analysis 15.3 Traffic Impact Study (TIS) Scoping, Horizon Years, & Mitigation 15.4 ITE Trip Generation Handbook: Average Rates vs. Regression Equations

16 Domain 6: Environmental, Institutional, Ethics & Equity Issues

16.1 Vehicle Emissions, Air Quality Conformity, & Decarbonization 16.2 Traffic Noise Modeling (TNM) & Noise Abatement Criteria (NAC) 16.3 Long-Range Transportation Planning & the NEPA Environmental Review 16.4 Professional Engineering Ethics, Tort Liability, & Risk Management 16.5 Transportation Equity, Access & Multimodal Community Considerations

14.1 Applied Statistical Distributions & Sample Size Determination

Key Takeaways Explain This Topic AI

The Normal distribution models continuous traffic variables such as free-flow spot speeds and high-volume vehicular headways, parameterized by mean μ \mu μ and standard deviation σ \sigma σ .

The Poisson distribution models discrete, random, independent traffic events with low-to-moderate arrival rates, defined by probability mass function P ( x ) = ( λ x ∗ e − λ ) / x ! P(x) = (\lambda^x * e^{-\lambda}) / x! P ( x ) = ( λ x ∗ e − λ ) / x ! where variance equals the mean ( σ 2 = μ = λ \sigma^2 = \mu = \lambda σ 2 = μ = λ ).

The Chi-Square ( χ 2 \chi^2 χ 2 ) goodness-of-fit test evaluates whether observed field data conforms to a theoretical distribution (Poisson or Normal), with test statistic χ 2 = ∑ [ ( O i − E i ) 2 / E i ] \chi^2 = \sum [(O_i - E_i)^2 / E_i] χ 2 = ∑ [( O i ​ − E i ​ ) 2 / E i ​ ] and degrees of freedom df = k - 1 - p.

The Student's two-sample t-test determines whether the difference between before-and-after mean operating speeds is statistically significant ( α = 0.05 \alpha = 0.05 α = 0.05 ) using pooled standard deviation s_p.

The minimum required sample size for spot speed studies is determined by N = (z * s / E)^2, where z = 1.96 for a 95% confidence level, s is estimated sample standard deviation, and E is allowable error (tolerance).

Last updated: August 2026

14.1 Applied Statistical Distributions & Sample Size Determination

PTOE Exam Focus: Applied statistics is an essential foundation for Domain 5. Candidates must master the selection and application of probability distributions (Normal vs. Poisson vs. Negative Exponential), conduct Chi-Square ( χ 2 \chi^2 χ 2 ) goodness-of-fit tests for vehicular arrivals, perform Student's t t t -tests for before-and-after operational studies, and calculate required sample sizes ( N = ( z ⋅ s E ) 2 N = \left(\frac{z \cdot s}{E}\right)^2 N = ( E z ⋅ s ​ ) 2 ) to satisfy specified confidence levels and error tolerances.

1. Statistical Distributions in Traffic Engineering

Traffic stream phenomena fall into two fundamental mathematical categories: continuous variables (such as vehicle speed, travel time, and delay) and discrete events (such as vehicle arrivals, pedestrian counts, and crash occurrences). Selecting the appropriate theoretical distribution is critical for data analysis, capacity modeling, and simulation. +-----------------------------------------------------------------------------------+

• | TRAFFIC ENGINEERING PROBABILITY DISTRIBUTIONS |
• | |
• | 1. CONTINUOUS DISTRIBUTIONS: |
• | • Normal (Gaussian) Distribution: Free-flow spot speeds, travel times. |
• | • Negative Exponential Distribution: Time headways between random arrivals. |
• | • Shifted Exponential / Lognormal: Headways with minimum tracking headway. |
• | |
• | 2. DISCRETE DISTRIBUTIONS: |
• | • Poisson Distribution: Vehicle arrivals in time t at low/moderate flow. |
• | • Binomial Distribution: Bernoulli trials, turning movement splits. |
• | • Negative Binomial Distribution: Overdispersed crash data (HSM modeling). |
• +-----------------------------------------------------------------------------------+
• 2. The Normal (Gaussian) Distribution

The Normal Distribution is the symmetric, bell-shaped continuous distribution governing random variables influenced by the additive sum of numerous independent factors (by the Central Limit Theorem ). In traffic engineering, free-flow spot speeds under uninterrupted conditions almost universally follow a Normal distribution.

Mathematical Formulation

The continuous probability density function (PDF) is given by: f ( x ) = 1 σ 2 π e − ( x − μ ) 2 2 σ 2 f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x - \mu)^2}{2\sigma^2}} f ( x ) = σ 2 π ​ 1 ​ e − 2 σ 2 ( x − μ ) 2 ​

Where:

• μ \mu μ = Population mean
• σ \sigma σ = Population standard deviation
• σ 2 \sigma^2 σ 2 = Population variance
• Standard Normal Variable ( Z Z Z -Score)

Any normally distributed variable X ∼ N ( μ , σ 2 ) X \sim \mathcal{N}(\mu, \sigma^2) X ∼ N ( μ , σ 2 ) is transformed to the standard normal distribution Z ∼ N ( 0 , 1 ) Z \sim \mathcal{N}(0, 1) Z ∼ N ( 0 , 1 ) via: Z = x − μ σ Z = \frac{x - \mu}{\sigma} Z = σ x − μ ​

Key Empirical Properties:

68.27 % 68.27\% 68.27% of observations lie within μ ± 1.0 σ \mu \pm 1.0\sigma μ ± 1.0 σ (approx. ± 1 σ \pm 1\sigma ± 1 σ )

95.00 % 95.00\% 95.00% of observations lie within μ ± 1.960 σ \mu \pm 1.960\sigma μ ± 1.960 σ (approx. ± 2 σ \pm 2\sigma ± 2 σ )

99.73 % 99.73\% 99.73% of observations lie within μ ± 3.0 σ \mu \pm 3.0\sigma μ ± 3.0 σ (approx. ± 3 σ \pm 3\sigma ± 3 σ )

85th Percentile Speed ( V 85 V_{85} V 85 ​ ): Corresponds to Z = + 1.036 ≈ 1.04 Z = +1.036 \approx 1.04 Z = + 1.036 ≈ 1.04 : V 85 = μ + 1.036 ⋅ σ V_{85} = \mu + 1.036 \cdot \sigma V 85 ​ = μ + 1.036 ⋅ σ

15th Percentile Speed ( V 15 V_{15} V 15 ​ ): Corresponds to Z = − 1.036 ≈ − 1.04 Z = -1.036 \approx -1.04 Z = − 1.036 ≈ − 1.04 : V 15 = μ − 1.036 ⋅ σ V_{15} = \mu - 1.036 \cdot \sigma V 15 ​ = μ − 1.036 ⋅ σ

3. The Poisson Distribution & Vehicle Arrivals

The Poisson Distribution is a discrete probability distribution modeling the number of events (vehicle arrivals, pedestrian actuations, collisions) occurring within a fixed interval of time or space. It applies when:

Arrivals are completely random and independent of one another.

The probability of an arrival in a tiny time increment Δ t \Delta t Δ t is proportional to the average arrival rate λ \lambda λ .

The probability of two or more arrivals in Δ t \Delta t Δ t approaches zero.

Probability Mass Function (PMF) P ( x ) = λ x e − λ x ! = ( μ ) x e − μ x ! P(x) = \frac{\lambda^x e^{-\lambda}}{x!} = \frac{(\mu)^x e^{-\mu}}{x!} P ( x ) = x ! λ x e − λ ​ = x ! ( μ ) x e − μ ​

Where:

x x x = Number of observed arrivals in time interval t t t ( x = 0 , 1 , 2 , 3 , … x = 0, 1, 2, 3, \dots x = 0 , 1 , 2 , 3 , … )

• λ = q ⋅ t \lambda = q \cdot t λ = q ⋅ t = Expected average number of arrivals in time interval t t t
• q q q = Mean arrival rate (vehicles per unit time, e.g., veh/s or veh/min)
• e ≈ 2.71828 e \approx 2.71828 e ≈ 2.71828 = Euler's constant

Unique Mathematical Property:

For a pure Poisson process, the variance equals the mean : σ 2 = μ = λ \sigma^2 = \mu = \lambda σ 2 = μ = λ

If σ 2 / μ ≈ 1.0 \sigma^2 / \mu \approx 1.0 σ 2 / μ ≈ 1.0 , data is Poisson distributed (random arrivals).

If σ 2 / μ > 1.0 \sigma^2 / \mu > 1.0 σ 2 / μ > 1.0 , data is overdispersed (clustered platoons, crash data modeled via Negative Binomial).

If σ 2 / μ < 1.0 \sigma^2 / \mu < 1.0 σ 2 / μ < 1.0 , data is underdispersed (uniform, congested flow).

Duality with Negative Exponential Headways

If vehicle arrivals in time follow a Poisson distribution with mean rate q q q (veh/s), the time headways ( h h h ) between successive vehicles are continuous and follow the Negative Exponential Distribution : P ( h ≥ t ) = e − q ⋅ t P(h \ge t) = e^{-q \cdot t} P ( h ≥ t ) = e − q ⋅ t P ( h < t ) = 1 − e − q ⋅ t P(h < t) = 1 - e^{-q \cdot t} P ( h < t ) = 1 − e − q ⋅ t

Where P ( h ≥ t ) P(h \ge t) P ( h ≥ t ) represents the probability of finding a headway greater than or equal to t t t seconds (critical for gap acceptance analysis at stop signs and unsignalized crossings).

4. Chi-Square ( χ 2 \chi^2 χ 2 ) Goodness-of-Fit Testing

The Chi-Square Goodness-of-Fit Test verifies whether empirical traffic field data (such as 1-minute arrival counts or binned spot speeds) statistically conforms to a hypothesized theoretical distribution (such as Poisson or Normal). +-----------------------------------------------------------------------------------+

• | CHI-SQUARE GOODNESS-OF-FIT WORKFLOW |
• | |
• | [ Step 1: Bin Field Observations into k Frequency Classes (O_i) ] |
• | | |
• | v |
• | [ Step 2: Compute Theoretical Expected Frequencies E_i = N * P_i ] |
• | | |
• | v |
• | [ Step 3: Pool Adjacent Bins if Expected Frequency E_i < 5 ] |
• | | |
• | v |
• | [ Step 4: Calculate Test Statistic: Chi^2 = Sum [ (O_i - E_i)^2 / E_i ] ] |
• | | |
• | v |
• | [ Step 5: Determine Degrees of Freedom: df = k - 1 - p ] |
• | | |
• | v |
• | [ Step 6: Compare Chi^2_calc with Chi^2_critical (alpha = 0.05) ] |
• | • If Chi^2_calc <= Chi^2_critical: Fail to Reject H0 (Distribution fits) |
• | • If Chi^2_calc > Chi^2_critical: Reject H0 (Distribution does not fit) |
• +-----------------------------------------------------------------------------------+

Mathematical Formulation χ calc 2 = ∑ i = 1 k ( O i − E i ) 2 E i \chi^2_{\text{calc}} = \sum_{i=1}^{k} \frac{(O_i - E_i)^2}{E_i} χ calc 2 ​ = i = 1 ∑ k ​ E i ​ ( O i ​ − E i ​ ) 2 ​

Where:

O i O_i O i ​ = Observed frequency in category/bin i i i

E i = N ⋅ P i E_i = N \cdot P_i E i ​ = N ⋅ P i ​ = Expected frequency in category/bin i i i under theoretical distribution

• k k k = Final number of categories/bins (after pooling small cells where E i < 5 E_i < 5 E i ​ < 5 )
• Degrees of Freedom ( d f df df ) d f = k − 1 − p df = k - 1 - p df = k − 1 − p

Where:

• k k k = Number of bins used in the summation
• 1 1 1 = Constraint that ∑ E i = ∑ O i = N \sum E_i = \sum O_i = N ∑ E i ​ = ∑ O i ​ = N

p p p = Number of distribution parameters estimated directly from sample data (for Poisson, p = 1 p = 1 p = 1 because λ = x ˉ \lambda = \bar{x} λ = x ˉ ; for Normal, p = 2 p = 2 p = 2 because μ = x ˉ \mu = \bar{x} μ = x ˉ and σ = s \sigma = s σ = s )

5. Student's t t t -Test for Before-and-After Studies

To evaluate whether a traffic engineering countermeasure (e.g., dynamic speed feedback signs, road diets, or signal retiming) has produced a statistically significant change in operating speeds, engineers perform a Two-Sample Independent Student's t t t -Test .

Hypotheses Formulation:

Null Hypothesis ( H 0 H_0 H 0 ​ ): μ 1 = μ 2 \mu_1 = \mu_2 μ 1 ​ = μ 2 ​ (no change in mean operating speed)

Alternative Hypothesis ( H a H_a H a ​ ): μ 1 ≠ μ 2 \mu_1 \ne \mu_2 μ 1 ​  = μ 2 ​ (two-tailed) or μ 1 > μ 2 \mu_1 > \mu_2 μ 1 ​ > μ 2 ​ (one-tailed speed reduction)

Pooled Variance ( s p 2 s_p^2 s p 2 ​ ) Formulation:

Assuming equal population variances (homoscedasticity): s p 2 = ( n 1 − 1 ) s 1 2 + ( n 2 − 1 ) s 2 2 n 1 + n 2 − 2 s_p^2 = \frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2} s p 2 ​ = n 1 ​ + n 2 ​ − 2 ( n 1 ​ − 1 ) s 1 2 ​ + ( n 2 ​ − 1 ) s 2 2 ​ ​ s p = s p 2 s_p = \sqrt{s_p^2} s p ​ = s p 2 ​ ​

Test Statistic ( t calc t_{\text{calc}} t calc ​ ): t calc = x ˉ 1 − x ˉ 2 s p 1 n 1 + 1 n 2 t_{\text{calc}} = \frac{\bar{x}_1 - \bar{x}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}} t calc ​ = s p ​ n 1 ​ 1 ​ + n 2 ​ 1 ​ ​ x ˉ 1 ​ − x ˉ 2 ​ ​

With degrees of freedom: d f = n 1 + n 2 − 2 df = n_1 + n_2 - 2 df = n 1 ​ + n 2 ​ − 2 .

Decision Rule (Two-Tailed at α = 0.05 \alpha = 0.05 α = 0.05 ):

If ∣ t calc ∣ > t crit |t_{\text{calc}}| > t_{\text{crit}} ∣ t calc ​ ∣ > t crit ​ , reject H 0 H_0 H 0 ​ and conclude the speed change is statistically significant at the 95 % 95\% 95% confidence level.

If ∣ t calc ∣ ≤ t crit |t_{\text{calc}}| \le t_{\text{crit}} ∣ t calc ​ ∣ ≤ t crit ​ , fail to reject H 0 H_0 H 0 ​ (difference could be random noise).

6. Sample Size Determination for Traffic Studies

To ensure field data achieves a designated level of statistical precision without wasting resources on unnecessary data collection, sample size N N N is derived from the margin of error of the mean.

Fundamental Sample Size Equation

For a normally distributed variable with known or estimated standard deviation s s s , confidence level 1 − α 1 - \alpha 1 − α , and maximum allowable error (tolerance) E E E : E = z α / 2 ⋅ SE = z α / 2 ⋅ s N E = z_{\alpha/2} \cdot \text{SE} = z_{\alpha/2} \cdot \frac{s}{\sqrt{N}} E = z α /2 ​ ⋅ SE = z α /2 ​ ⋅ N ​ s ​

Solving for required sample size N N N : N = ( z ⋅ s E ) 2 = z 2 ⋅ s 2 E 2 N = \left( \frac{z \cdot s}{E} \right)^2 = \frac{z^2 \cdot s^2}{E^2} N = ( E z ⋅ s ​ ) 2 = E 2 z 2 ⋅ s 2 ​

Where:

N N N = Minimum required sample size (always rounded up to the nearest integer)

z z z = Critical value corresponding to the desired confidence level:

• 90 % 90\% 90% Confidence ( α = 0.10 \alpha = 0.10 α = 0.10 ): z = 1.645 z = 1.645 z = 1.645
• 95 % 95\% 95% Confidence ( α = 0.05 \alpha = 0.05 α = 0.05 ): z = 1.960 z = 1.960 z = 1.960
• 99 % 99\% 99% Confidence ( α = 0.01 \alpha = 0.01 α = 0.01 ): z = 2.576 z = 2.576 z = 2.576

s s s = Estimated standard deviation of the population (typically 4.0 to 6.0 mph 4.0\text{ to }6.0\text{ mph} 4.0 to 6.0 mph for spot speeds if unknown; default s = 5.0 mph s = 5.0\text{ mph} s = 5.0 mph )

E E E = Allowable margin of error (tolerance), e.g., ± 1.0 mph \pm 1.0\text{ mph} ± 1.0 mph or ± 0.5 mph \pm 0.5\text{ mph} ± 0.5 mph

Finite Population Correction (FPC)

If the sample size N N N represents more than 5 % 5\% 5% of the total finite population N pop N_{\text{pop}} N pop ​ ( N / N pop > 0.05 N / N_{\text{pop}} > 0.05 N / N pop ​ > 0.05 , common in small vehicle fleets or parking turnover counts), the required sample size is adjusted downward: N adj = N 1 + N − 1 N pop N_{\text{adj}} = \frac{N}{1 + \frac{N - 1}{N_{\text{pop}}}} N adj ​ = 1 + N pop ​ N − 1 ​ N ​

Statistical Testing and Distribution Reference for Traffic Engineering

Distribution / Statistical Test Data Type / Variable Governing Formula Traffic Engineering Application Critical Decision Criteria

Normal Distribution Continuous (speeds, travel time) Z = ( x − μ ) / σ Z = (x - \mu) / \sigma Z = ( x − μ ) / σ Spot speed percentiles ( V 85 = μ + 1.04 σ V_{85} = \mu + 1.04\sigma V 85 ​ = μ + 1.04 σ ), corridor travel times Symmetric; 68-95-99.7 empirical rule applies

Poisson Distribution Discrete counts (arrivals in time t) P ( x ) = ( λ x ∗ e − λ ) / x ! P(x) = (\lambda^x * e^{-\lambda}) / x! P ( x ) = ( λ x ∗ e − λ ) / x ! Uncongested vehicle arrivals, minor street gap availability, pedestrian calls Mean equals variance ( μ = σ 2 = λ \mu = \sigma^2 = \lambda μ = σ 2 = λ )

Negative Exponential Continuous time (headways h >= t) P(h >= t) = e^{-q*t} Time headways between random vehicle arrivals in gap acceptance Memoryless property; dual to Poisson arrival process

Chi-Square ( χ 2 \chi^2 χ 2 ) Goodness-of-Fit Categorical / Binned frequencies χ 2 = ∑ [ ( O i − E i ) 2 / E i ] \chi^2 = \sum [(O_i - E_i)^2 / E_i] χ 2 = ∑ [( O i ​ − E i ​ ) 2 / E i ​ ] Validating whether arrival counts follow Poisson or speeds follow Normal Reject H0 if χ calc 2 > χ crit 2 \chi^2_{\text{calc}} > \chi^2_{\text{crit}} χ calc 2 ​ > χ crit 2 ​ (df = k - 1 - p)

Two-Sample Student's t-Test Continuous means ( x ˉ 1 \bar{x}_1 x ˉ 1 ​ vs x ˉ 2 \bar{x}_2 x ˉ 2 ​ ) t = ( x ˉ 1 − x ˉ 2 ) / [ s p ∗ 1 / n 1 + 1 / n 2 ] t = (\bar{x}_1 - \bar{x}_2) / [s_p * \sqrt{1/n_1 + 1/n_2}] t = ( x ˉ 1 ​ − x ˉ 2 ​ ) / [ s p ​ ∗ 1/ n 1 ​ + 1/ n 2 ​ ​ ] Before-and-after countermeasure evaluation (speed limit change, calming) Reject H0 if |t_calc| > t_crit (df = n_1 + n_2 - 2)

Sample Size Determination (N) Sample count estimation N = (z * s / E)^2 Determining minimum vehicles required for spot speed or volume studies Always round up to next whole vehicle; apply FPC if N/N_pop > 0.05

Loading diagram... Decision Workflow for Statistical Distribution Selection & Hypothesis Testing

Required Sample Size (N) vs Allowable Error (E, mph) at 95% Confidence (z = 1.96, s = 5.0 mph)

• 7. Worked Calculation Examples
• Example 1: Spot Speed Sample Size Determination

Problem: A traffic engineer is planning a spot speed study on an urban arterial. Historical data indicates a standard deviation of s = 5.2 mph s = 5.2\text{ mph} s = 5.2 mph . The study must achieve a 95 % 95\% 95% confidence level ( z = 1.96 z = 1.96 z = 1.96 ) with an allowable error of E = ± 1.0 mph E = \pm 1.0\text{ mph} E = ± 1.0 mph . What is the minimum required sample size?

Step-by-Step Solution:

Identify Given Parameters:

• s = 5.2 mph s = 5.2\text{ mph} s = 5.2 mph
• z = 1.96 z = 1.96 z = 1.96 ( 95 % 95\% 95% confidence)
• E = 1.0 mph E = 1.0\text{ mph} E = 1.0 mph

Apply Sample Size Formulation: N = ( z ⋅ s E ) 2 = ( 1.96 × 5.2 1.0 ) 2 = ( 10.192 ) 2 = 103.88 N = \left( \frac{z \cdot s}{E} \right)^2 = \left( \frac{1.96 \times 5.2}{1.0} \right)^2 = (10.192)^2 = 103.88 N = ( E z ⋅ s ​ ) 2 = ( 1.0 1.96 × 5.2 ​ ) 2 = ( 10.192 ) 2 = 103.88

Rounding Rule: Always round up to the nearest integer: N = 104 vehicles N = 104\text{ vehicles} N = 104 vehicles .

Example 2: Student's t t t -Test for Speed Feedback Sign Evaluation

Problem: A dynamic speed feedback sign is installed in a school zone. Spot speed studies conducted before and after installation yield the following data:

Before: n 1 = 64 n_1 = 64 n 1 ​ = 64 , x ˉ 1 = 34.2 mph \bar{x}_1 = 34.2\text{ mph} x ˉ 1 ​ = 34.2 mph , s 1 = 4.8 mph s_1 = 4.8\text{ mph} s 1 ​ = 4.8 mph

After: n 2 = 64 n_2 = 64 n 2 ​ = 64 , x ˉ 2 = 31.6 mph \bar{x}_2 = 31.6\text{ mph} x ˉ 2 ​ = 31.6 mph , s 2 = 4.2 mph s_2 = 4.2\text{ mph} s 2 ​ = 4.2 mph

Test at α = 0.05 \alpha = 0.05 α = 0.05 ( t crit = 1.980 t_{\text{crit}} = 1.980 t crit ​ = 1.980 for d f = 126 df = 126 df = 126 ) whether the sign caused a statistically significant reduction in mean operating speed.

Step-by-Step Solution:

Compute Pooled Standard Deviation ( s p s_p s p ​ ): s p 2 = ( 64 − 1 ) ( 4.8 ) 2 + ( 64 − 1 ) ( 4.2 ) 2 64 + 64 − 2 = 63 ( 23.04 ) + 63 ( 17.64 ) 126 = 1451.52 + 1111.32 126 = 2562.84 126 = 20.34 s_p^2 = \frac{(64 - 1)(4.8)^2 + (64 - 1)(4.2)^2}{64 + 64 - 2} = \frac{63(23.04) + 63(17.64)}{126} = \frac{1451.52 + 1111.32}{126} = \frac{2562.84}{126} = 20.34 s p 2 ​ = 64 + 64 − 2 ( 64 − 1 ) ( 4.8 ) 2 + ( 64 − 1 ) ( 4.2 ) 2 ​ = 126 63 ( 23.04 ) + 63 ( 17.64 ) ​ = 126 1451.52 + 1111.32 ​ = 126 2562.84 ​ = 20.34 s p = 20.34 = 4.51 mph s_p = \sqrt{20.34} = 4.51\text{ mph} s p ​ = 20.34 ​ = 4.51 mph

Compute Test Statistic ( t calc t_{\text{calc}} t calc ​ ): t calc = x ˉ 1 − x ˉ 2 s p 1 n 1 + 1 n 2 = 34.2 − 31.6 4.51 1 64 + 1 64 = 2.60 4.51 0.03125 = 2.60 4.51 × 0.17678 = 2.60 0.7973 = 3.26 t_{\text{calc}} = \frac{\bar{x}_1 - \bar{x}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}} = \frac{34.2 - 31.6}{4.51 \sqrt{\frac{1}{64} + \frac{1}{64}}} = \frac{2.60}{4.51 \sqrt{0.03125}} = \frac{2.60}{4.51 \times 0.17678} = \frac{2.60}{0.7973} = 3.26 t calc ​ = s p ​ n 1 ​ 1 ​ + n 2 ​ 1 ​ ​ x ˉ 1 ​ − x ˉ 2 ​ ​ = 4.51 64 1 ​ + 64 1 ​ ​ 34.2 − 31.6 ​ = 4.51 0.03125 ​ 2.60 ​ = 4.51 × 0.17678 2.60 ​ = 0.7973 2.60 ​ = 3.26

Statistical Decision: Since ∣ t calc ∣ = 3.26 > t crit = 1.980 |t_{\text{calc}}| = 3.26 > t_{\text{crit}} = 1.980 ∣ t calc ​ ∣ = 3.26 > t crit ​ = 1.980 , reject H 0 H_0 H 0 ​ ( p < 0.002 p < 0.002 p < 0.002 ). The speed feedback sign produced a statistically significant reduction in mean operating speed.