Random Confidence Interval Coverage Experiment Generator

Description

Generate repeated normal samples and compare observed coverage of delegated Student t and known-sigma z mean intervals at supported confidence levels; normalize moment/interval calculations for scale and retain full-precision summaries and endpoints, not a nominal-coverage guarantee.

Synthetic statistical data separates a specified generating model from the values observed in a finite draw. A model parameter, a sample summary, and the result of a fitted analysis answer different questions; retaining them separately makes numerical and statistical checks easier to interpret.

When to use this tool

Use this generator to create reproducible confidence interval coverage experiment cases from the supplied model or bank. Inspect the returned evidence before using cases in a test or exercise.

How to use the tool and how it works

Supply experimentCount, sampleCount, mean, standardDeviation, confidenceLevel in the field shapes below. Set seed when you need an explicit replay.
Run the tool and correct any input error. For generated controls, distinguish a successful fixture containing an invalid case from failure to generate the fixture.
Read model, experiments, coverage. Use the conventions below. Keep answer-bearing fields separate when presenting an exercise.

Repeat Gaussian sampling, estimate each sample's mean and sample deviation, and delegate t intervals using that deviation and z intervals using the known model deviation. Count intervals inclusively containing the fixed model mean and return coverage fractions.^[1]

Delegated operations used for this result: mean; sample variance. Their calculation or parsing conventions apply to the corresponding result fields.^{[2, 3]}

Input field	Meaning and units	Accepted input
experimentCount	Experiment Count	Required. Safe integer; minimum 1; maximum 128.
sampleCount	Sample Count	Required. Safe integer; minimum 2; maximum 128.
mean	Mean	Required. Finite number.
standardDeviation	Standard Deviation	Required. Finite number; minimum 0.
confidenceLevel	Confidence Level	Required. Safe integer; one of 80, 85, 90, 95, 98, 99.
seed	Optional numeric seed; takes precedence over context randomness. Without either, use the fixed tool.io default seed.	Optional. Safe integer.

Output field	Interpretation	Returned shape
model	Model	Required. Object; no unknown keys.
model.mean	Mean	Required. Finite number.
model.standardDeviation	Standard Deviation	Required. Finite number; minimum 0.
model.confidenceLevel	Confidence Level	Required. Safe integer; one of 80, 85, 90, 95, 98, 99.
model.sampleCount	Sample Count	Required. Safe integer; minimum 2; maximum 128.
experiments	Experiments	Required. Array of Object; no unknown keys.; minimum items 1; maximum items 128.
experiments[].values	Values	Required. Array of Finite number.; minimum items 2; maximum items 128.
experiments[].sampleMean	Sample Mean	Required. Finite number.
experiments[].sampleStandardDeviation	Sample Standard Deviation	Required. Finite number; minimum 0.
experiments[].tInterval	T Interval	Required. Object; no unknown keys.
experiments[].tInterval.lower	Lower	Required. Finite number.
experiments[].tInterval.upper	Upper	Required. Finite number.
experiments[].tInterval.coversMean	Covers Mean	Required. Boolean.
experiments[].zInterval	Z Interval	Required. Object; no unknown keys.
experiments[].zInterval.lower	Lower	Required. Finite number.
experiments[].zInterval.upper	Upper	Required. Finite number.
experiments[].zInterval.coversMean	Covers Mean	Required. Boolean.
coverage	Coverage	Required. Object; no unknown keys.
coverage.tCount	T Count	Required. Safe integer; minimum 0; maximum 128.
coverage.zCount	Z Count	Required. Safe integer; minimum 0; maximum 128.
coverage.experimentCount	Experiment Count	Required. Safe integer; minimum 1; maximum 128.
coverage.tFraction	T Fraction	Required. Finite number; minimum 0; maximum 1.
coverage.zFraction	Z Fraction	Required. Finite number; minimum 0; maximum 1.
randomness	Replay source and normalized 32-bit seed, or null seed for an advancing context source.	Required. Object; no unknown keys.
randomness.seed	Optional safe-integer replay seed; normalization and precedence are explained below.	Required. Safe integer. or Null.
randomness.source	Source	Required. Text; one of "context", "seed".

An explicit safe-integer seed takes precedence over injected context randomness and is normalized to an unsigned 32-bit integer. With no seed and no context source, replay uses the fixed tool.io default seed. A context source must return finite values from zero inclusive to one exclusive; its advancing state must be retained separately. randomness records the source and normalized seed, or null for context. These generators are not cryptographic random sources.

json

{
  "input": {
    "experimentCount": 2,
    "sampleCount": 2,
    "mean": 0,
    "standardDeviation": 1,
    "confidenceLevel": 80,
    "seed": 42
  },
  "output": {
    "model": {
      "mean": 0,
      "standardDeviation": 1,
      "confidenceLevel": 80,
      "sampleCount": 2
    },
    "experiments": [
      {
        "values": [
          -0.956162229384149,
          -0.2730261048826104
        ],
        "sampleMean": -0.6145941671333797,
        "sampleStandardDeviation": 0.4830501861085355,
        "tInterval": {
          "lower": -1.6658325691474236,
          "upper": 0.4366442348806643,
          "coversMean": true
        },
        "zInterval": {
          "lower": -1.5207879695702027,
          "upper": 0.29159963530344346,
          "coversMean": true
        }
      },
      {
        "values": [
          -1.8416271847835968,
          -1.1408820709540242
        ],
        "sampleMean": -1.4912546278688106,
        "sampleStandardDeviation": 0.4955016218722299,
        "tInterval": {
          "lower": -2.569590478163447,
          "upper": -0.41291877757417417,
          "coversMean": false
        },
        "zInterval": {
          "lower": -2.3974484303056336,
          "upper": -0.5850608254319875,
          "coversMean": false
        }
      }
    ],
    "coverage": {
      "tCount": 1,
      "zCount": 1,
      "experimentCount": 2,
      "tFraction": 0.5,
      "zFraction": 0.5
    },
    "randomness": {
      "seed": 42,
      "source": "seed"
    }
  }
}

Limitations and assumptions

confidenceLevel is entered as a percentage; the t calculator receives a fractional conversion. Empirical coverage varies in finite repetitions and is not the probability that a fixed interval's already-fixed parameter changes. This is model-based simulation, not evidence from a real experiment.
A replay requires the same complete input and implementation. The stated sampling convention applies within its eligible space. A finite draw does not exhaust all possible outputs; repeats remain possible unless a distinct-selection rule is explicit.
The field tables state structural bounds; cross-field constraints and bounded search or output budgets also apply. Unknown keys are rejected where the object contract disallows them. Input errors are distinct from null fields and deliberate invalid controls in a successful output.

Alternative or Complementary methods

For a supplied example, inspect the returned construction or delegated check directly. To evaluate a consumer, also test cases chosen for its own boundaries; a generated finite corpus does not establish exhaustive coverage.

References

Tool behavior and example: Random Confidence Interval Coverage Experiment Generator — tool.io
Calculation or validation: statistics/mean — tool.io
Calculation or validation: statistics/sample-variance — tool.io

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