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Worked-example setupScope and assumptions
- Each year is treated as a separate simple random sample from its stated frame.
- The supplied Student-t critical value 2.131449545559323 matches 95% two-sided coverage and fifteen degrees of freedom.
- The arithmetic does not establish independence, approximate normality, frame validity, or measurement quality; Year 2's tail remains a material condition challenge.
- Period
- Separate fictional 2025 and 2026 invoice populations
- Units
- Calendar days
- Rounding
- Full precision through endpoints; display to two decimals only after reconciliation
Year 1
[ SE_1=5.852349955359813/\sqrt{16}=1.4630874888399532. ]
[ MOE_1=2.131449545559323(1.4630874888399532) =3.1184971632014493. ]
The full-precision interval is [28.75650283679855, 34.99349716320145] days; display [28.76, 34.99] only after checking the
midpoint and half-width.
Year 2
[ SE_2=13.812434011908739/\sqrt{16}=3.4531085029771846, ]
[ MOE_2=2.131449545559323(3.4531085029771846) =7.360126549437754. ]
The interval is [22.014873450562245, 36.73512654943775] days. Its extra width
comes from larger sample spread, not a different sample size or critical value.
Challenge the warrant
The Year 1 distribution has no quartile-fence flag, but sample size remains small and customer-level dependence is unknown. Year 2 has a severe retained tail and the same unknown dependence. Both targets exclude unsettled invoices, and both rely on source and frame controls outside the interval formula.
Report the outputs as conditional calculations. Do not use interval overlap to declare the population means equal, interval separation to declare them different, or either interval as a range for individual future invoices.
Verified calculation · descriptive estimation analysis
The curriculum loader recomputed this example before it entered the site build. Expand any structured input to inspect the stated facts.
- comparisons
- 1 field
Inspect data
{
"year_2_vs_year_1": {
"direction": "right-minus-left",
"left": "year_1",
"right": "year_2"
}
}- series
- 2 fields
Inspect data
{
"year_1": {
"dispersion_convention": "sample-n-minus-one",
"inference_basis": "simple-random-sample",
"mean_confidence_interval": {
"confidence_level": 0.95,
"critical_value": 2.131449545559323,
"degrees_of_freedom": 15,
"interval_scope": "two-sided-one-sample-mean",
"reference_distribution": "student-t-supplied"
},
"missing_value_policy": "reject",
"observations": [
22,
25,
26,
27,
28,
29,
30,
31,
32,
33,
34,
35,
36,
38,
40,
44
],
"outlier_policy": "retain-and-flag",
"population_role": "sample",
"quartile_method": "median-of-halves-exclusive"
},
"year_2": {
"dispersion_convention": "sample-n-minus-one",
"inference_basis": "simple-random-sample",
"mean_confidence_interval": {
"confidence_level": 0.95,
"critical_value": 2.131449545559323,
"degrees_of_freedom": 15,
"interval_scope": "two-sided-one-sample-mean",
"reference_distribution": "student-t-supplied"
},
"missing_value_policy": "reject",
"observations": [
18,
20,
21,
22,
23,
24,
25,
26,
27,
28,
29,
30,
31,
33,
35,
78
],
"outlier_policy": "retain-and-flag",
"population_role": "sample",
"quartile_method": "median-of-halves-exclusive"
}
}Recomputed result
| Measure | Value |
|---|---|
| 1 confidence interval lower | 28.7565 |
| 1 confidence interval upper | 34.9935 |
| 1 margin of error | 3.1185 |
| 1 standard error of mean | 1.4631 |
| 2 confidence interval lower | 22.0149 |
| 2 confidence interval upper | 36.7351 |
| 2 margin of error | 7.3601 |
| 2 standard error of mean | 3.4531 |