4. One-sample and two-sample hypothesis tests for crash metrics

Crash engineering decisions rarely rest on a single number. A sled test produces a peak chest acceleration of 52 g; a barrier test yields a HIC of 640; a paired left/right comparison shows one occupant restraint system outperforming another by 8%. The engineering question is never simply "what is the value?" but "is this value meaningfully different from a requirement, a baseline, or an alternative design?" Hypothesis testing provides the formal machinery for that judgment, and t-tests are the workhorse for the small samples typical of crash test programs.

This section covers the one-sample t-test against a regulatory or design threshold, the two-sample t-test for independent groups (e.g., driver versus passenger, or design A versus design B), and the paired t-test for matched configurations such as left/right symmetric tests or repeat runs on the same sled buck. Throughout, the emphasis is on connecting statistical significance to practical significance — the effect size that actually matters for occupant protection.

Framing the Hypotheses Around Crash Metrics

Every test begins with a null hypothesis (H₀) and an alternative (H₁). For a one-sample test on a crash metric, H₀ typically states that the population mean equals a threshold: H₀: μ = μ₀, where μ₀ might be a FMVSS limit, an internal design target, or a competitor benchmark. The alternative is usually one-sided when the concern is directional — for example, H₁: μ > 45 g for chest acceleration, because exceeding the limit is the failure mode of interest. Two-sided alternatives are appropriate when deviation in either direction is a problem, such as a HIC target where both under- and over-protection warrant review.

The test statistic for a one-sample t-test is:

t = (x̄ − μ₀) / (s / √n)

where  x̄ = sample mean of the crash metric
       μ₀ = threshold or reference value
       s  = sample standard deviation
       n  = number of tests

With n − 1 degrees of freedom, the t-distribution accounts for the uncertainty introduced by estimating σ from a small sample. In crash testing, n is often 3 to 6 per configuration, so the t-distribution's heavier tails matter: a t-statistic of 2.0 with n = 4 (df = 3) is not significant at α = 0.05 two-sided (critical value ≈ 3.18), whereas the same statistic with n = 30 would be highly significant.

Worked Example: One-Sample Test Against a Chest Acceleration Limit

Suppose a sled program runs five matched tests of a driver restraint configuration, and the peak resultant chest acceleration (3 ms clip) values are 48.2, 51.0, 49.5, 52.3, and 50.1 g. The internal design target is μ₀ = 50 g, and the team wants to know whether the configuration is statistically above target. This is a one-sided test: H₀: μ ≤ 50, H₁: μ > 50.

The sample mean is x̄ = 50.22 g and the sample standard deviation is s ≈ 1.58 g. The standard error is s/√n = 1.58/√5 ≈ 0.707 g, giving t = (50.22 − 50)/0.707 ≈ 0.31 with df = 4. The one-sided critical value at α = 0.05 is 2.132, so the result is not significant. The data are consistent with a population mean at or below 50 g — but note the confidence interval: 50.22 ± 2.776 × 0.707 gives roughly [48.3, 52.2] g. The upper bound exceeds the target, so the configuration is not proven compliant; it is simply not proven non-compliant with only five tests. This distinction drives the decision to run additional tests rather than declare success.

Two-Sample Tests for Independent Configurations

When comparing two independent groups — for example, a baseline restraint (n₁ = 4) versus a modified pretensioner (n₂ = 4) — the two-sample t-test compares their means. The pooled-variance form assumes equal population variances:

t = (x̄₁ − x̄₂) / (s_p · √(1/n₁ + 1/n₂))

s_p² = [(n₁ − 1)s₁² + (n₂ − 1)s₂²] / (n₁ + n₂ − 2)
df = n₁ + n₂ − 2

When variances differ substantially — common in crash data where one configuration may be far more variable — use Welch's t-test, which does not pool variances and computes fractional degrees of freedom. A quick check: if the ratio of sample variances exceeds about 3, prefer Welch. Most statistical software defaults to Welch for good reason.

Consider baseline chest acceleration values of 52.1, 54.0, 51.3, 53.2 g (mean 52.65, s ≈ 1.19) and modified values of 49.8, 50.5, 48.9, 51.2 g (mean 50.10, s ≈ 1.00). The pooled standard deviation is about 1.10 g, the standard error of the difference is 1.10 × √(1/4 + 1/4) ≈ 0.778 g, and t = (52.65 − 50.10)/0.778 ≈ 3.28 with df = 6. The two-sided critical value at α = 0.05 is 2.447, so the difference is statistically significant. The mean reduction of 2.55 g is real within this test set — but whether it is practically significant depends on the injury risk curve, not the p-value.

Paired Comparisons: The Right Tool for Matched Tests

Crash programs frequently produce naturally paired data: left and right occupant tests on the same vehicle, repeat runs on the same sled buck, or before/after tests on the same dummy. Pairing removes between-unit variability (buck-to-buck differences, dummy-to-dummy differences) and increases power. The paired t-test operates on the differences dᵢ = x₁ᵢ − x₂ᵢ:

t = d̄ / (s_d / √n),  df = n − 1

where d̄  = mean of the paired differences
      s_d = standard deviation of the differences

Suppose four vehicles are tested left (baseline) and right (modified), yielding differences in peak chest acceleration of 2.1, 1.8, 2.6, and 2.0 g. The mean difference is 2.13 g, s_d ≈ 0.34 g, and t = 2.13/(0.34/√4) ≈ 12.5 with df = 3 — highly significant. Note how much larger this t-statistic is than the independent-samples case: pairing exploited the correlation between left and right tests. Using an unpaired test on the same data would waste that structure and could easily miss a real effect.

From Statistical to Practical Significance

A statistically significant 2.5 g reduction in chest acceleration may correspond to a meaningful reduction in AIS 2+ chest injury risk — or it may not, depending on where the operating point sits on the risk curve. Practical significance in crash engineering is defined by the injury criterion, the regulatory margin, and the cost/benefit of the design change, not by α = 0.05. Report effect sizes alongside p-values: the mean difference in engineering units, its confidence interval, and the corresponding change in injury risk or criterion margin. A confidence interval that spans both "no meaningful benefit" and "substantial benefit" tells the decision-maker more than a bare p-value ever can.

Reporting and Decision Practice

A defensible crash test report states the hypothesis direction and rationale, the sample size and summary statistics, the test chosen and why (paired versus independent, pooled versus Welch), the test statistic with degrees of freedom, the p-value, and the effect size with confidence interval. The decision recommendation should reference the injury criterion and the regulatory margin, not the p-value alone. When the confidence interval is wide — as it will be with n = 4 or 5 — the honest conclusion is often "insufficient evidence to demonstrate compliance," which is a decision to test more, not a decision to ship.

Two failure modes recur in practice. First, treating a non-significant result as proof of equivalence: failing to reject H₀ does not establish that the means are equal, only that the data are compatible with equality. Second, ignoring pairing structure and running an unpaired test on matched data, which inflates the standard error and hides real effects. Both are avoided by stating the comparison structure before looking at the data and by reporting effect sizes with intervals alongside every test.