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SCIENCE · INVESTIGATE AND EXPLAIN

Enumerate the evidence resolution of four randomized pairs

Randomization supplies a reference distribution through the assignment mechanism, not through an assumption that every observed difference is normally distributed. Under a sharp no-intervention-effect null and independent within-pair assignment, swapping the treatment/comparison labels changes each pair contrast’s sign. Four pairs produce 16 equally likely assignments.

Learning outcome: Enumerate the sixteen signs independently and compare raw and corrected packets. Keep inclusive ties and a predeclared tail; explain why repeatedly searching choices for a smaller probability undermines the stated test.

Starting recap

Preserve packet identity, independent units, analysis recipe, outcome constraints and review authority; distinguish scientific uncertainty from stakeholder value choices.

All lessons are open directly. Your saved experiments remain when you leave and return.

Try

The corrected mean is−2.5 mg/L. All 16 label assignments are shown. Two are at least as extreme in absolute magnitude, giving 0.125; one is at least as negative, giving 0.0625. No normal approximation or simulated random seed is needed.

Open the experiment

Predict

An exact randomization tail probability describes:

Experiment

Enumerate the sixteen signs independently and compare raw and corrected packets. Keep inclusive ties and a predeclared tail; explain why repeatedly searching choices for a smaller probability undermines the stated test.

Enumerate a paired randomization reference distribution

Worked teaching model · synthetic values · Four independently randomized treatment/comparison pairs; sharp no-effect null makes all 16 within-pair label swaps equally likely. Statistic is mean change contrast, with inclusive ties. Two-sided uses absolute mean; one-sided counts means≤observed. A tail must be declared before observing results; public examples are not secure tests.

Reset keeps your prediction, captures and practice history.

Original synthetic system; units differ by row
QuantityValueUnit
Observed mean contrast-2.5mg/L
Equally likely label assignments16assignments
Assignments at least as extreme2assignments
Exact conditional tail probability0.125fraction
Independent pairs4pairs
Every contrast or label assignment (mg/L)
RecordValue
Label swap 02.5
Label swap 10.5
Label swap 21.5
Label swap 3-0.5
Label swap 41.5
Label swap 5-0.5
Label swap 60.5
Label swap 7-1.5
Label swap 81.5
Label swap 9-0.5
Label swap 100.5
Label swap 11-1.5
Label swap 120.5
Label swap 13-1.5
Label swap 14-0.5
Label swap 15-2.5
Inspect the immutable source packet cenv-basin-pairs@1

Original synthetic paired-site data authored 30 October 2026. Eight sites in four independently randomized treatment/comparison pairs. Before/after are representative interval means, not technical replicates. Calibration packet stipulates a +2 mg/L after-period sensor offset only for BASIN-P3-T; no uncertainty is supplied for that correction.

Record IDBefore (mg/L)Raw after (mg/L)Known offset (mg/L)Corrected after (mg/L)
BASIN-P1-T2018018
BASIN-P1-C1820020
BASIN-P2-T2424024
BASIN-P2-C2224024
BASIN-P3-T2830228
BASIN-P3-C2628028
BASIN-P4-T3232032
BASIN-P4-C3032032
Download original and corrected packet CSV

The CSV is a fictional teaching dataset, not actual field observations. Preserve its identity and correction column in your report.

Four pairs provide a coarse randomization distribution even when every corrected contrast is negative.

The probability describes this statistic under the sharp null and assignment mechanism, not the probability that the null is true or the size of ecological benefit. Randomization inference depends on actual assignment and no interference; the packet stipulates both for the exercise.

Predict, change one control, and retain contrasting captures. Reconstruct the analysis and decision rule before opening feedback. Display precision is not measurement precision.

Four pairs provide a coarse randomization distribution even when every corrected contrast is negative.

The probability describes this statistic under the sharp null and assignment mechanism, not the probability that the null is true or the size of ecological benefit. Randomization inference depends on actual assignment and no interference; the packet stipulates both for the exercise.

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Define the decision and evidence boundary

Randomization supplies a reference distribution through the assignment mechanism, not through an assumption that every observed difference is normally distributed. Under a sharp no-intervention-effect null and independent within-pair assignment, swapping the treatment/comparison labels changes each pair contrast’s sign. Four pairs produce 16 equally likely assignments.

Reconstruct the transparent method

The statistic is the mean paired change contrast. A two-sided comparison counts assignments whose absolute mean is at least the observed absolute mean, including ties. A predeclared decrease alternative counts means at or below the observed value. Choosing the tail after inspecting the results changes the intended evidential procedure.

Interpret the comparison

For corrected contrasts−4,−2,−2,−2, only the all-negative and all-positive assignments are as extreme in absolute mean, so the two-sided probability is 2/16=0.125. A predeclared decrease tail has 1/16=0.0625. Every observed contrast can be negative while four pairs still yield coarse evidence resolution.

Predeclare limitations and revision

These probabilities are not the probability that the null is true and do not measure ecological importance. The fictional packet stipulates randomized assignment, no interference and a known site-specific correction independent of the desired result. Raw-versus-corrected comparison illustrates measurement sensitivity; uncorrected sensor bias is not a preferred estimate of the physical endpoint. Observational matching alone would not automatically justify this assignment distribution.

Worked original packet

The corrected mean is−2.5 mg/L. All 16 label assignments are shown. Two are at least as extreme in absolute magnitude, giving 0.125; one is at least as negative, giving 0.0625. No normal approximation or simulated random seed is needed.

Investigation

Enumerate the sixteen signs independently and compare raw and corrected packets. Keep inclusive ties and a predeclared tail; explain why repeatedly searching choices for a smaller probability undermines the stated test.

Changed-case transfer

For raw contrasts−4,−2,0,−2, how many assignments are at least as extreme two-sided and why?

Feedback

The zero contrast can take either sign without changing the statistic. Two all-negative-equivalent and two all-positive-equivalent assignments give 4/16=0.25. The duplicate values still correspond to distinct equally likely assignments.

Sources and evidence boundary

Original prose, dataset and numerical exercises. Primary method/mechanism source, checked 30 October 2026. The versioned packet is synthetic, not observed fieldwork. No treatment implementation, site access, pollutant handling or real management authorization is requested. Preserve first reasoning and critique in the course workspace.

Apply

An exact randomization tail probability describes:

Public practice checks a specific claim. It does not establish independent mastery.

For raw contrasts−4,−2,0,−2, how many assignments are at least as extreme two-sided and why?

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

Previous: Challenge the counterfactual behind a paired change contrastNext: Plan precision using independent units rather than repeated readings
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