Build a confidence interval, then challenge it

A confidence interval combines a point estimate with a sampling-precision procedure. It does not gather evidence about the source on its own.

Reproduce Year 1

Juniper supplies a two-sided 95% Student-t critical value of 2.131449545559323 for 15 degrees of freedom. The critical value is a cutoff from the Student-t distribution selected by the confidence level and degrees of freedom; here (df=n-1=15). This module supplies rather than derives that cutoff. Using the full-precision Year 1 standard error:

[ MOE=2.131449545559323(1.4630874888399532) =3.1184971632014493\text{ days}. ]

The interval is:

[ 31.875\pm3.1184971632014493 =[28.75650283679855,\ 34.99349716320145]\text{ days}. ]

Reconcile it: the endpoint midpoint is 31.875 and each endpoint is exactly one full-precision margin from that mean. Round only for display.

Say what 95% modifies

Imagine repeating the same selection and interval procedure many times. Under its conditions, about 95% of the resulting intervals would cover the fixed population mean. After one interval is computed, the fixed parameter is either inside or outside. In this foundational frequentist interpretation, do not say there is a 95% probability that this fixed parameter lies in this realized interval.

Compute Year 2, then stop

Year 2 has mean 29.375 days, standard error 3.453108503 days, and margin 7.360126549 days, producing approximately [22.0149, 36.7351] days. The wider interval reflects the sample's larger spread, driven substantially by the source-confirmed but unexplained 78-day observation.

The arithmetic is correct. The inferential warrant remains unsettled. The sample is small, the tail is severe, customer clustering is unknown, the population excludes unsettled invoices, and source-system conditions require evidence. Computing the interval teaches the mechanism; it does not force a decision-grade conclusion.

Width is not accuracy

A narrower interval can result from more rows, less sample spread, or a lower confidence level. None proves an unbiased frame or correct measurement. The margin of error does not absorb duplicates, missing units, cutoff mistakes, or future process change.

Open the complete interval build, then complete the interpretation item.

Boundary checkpoint: interval overlap

Visual overlap of two one-sample intervals is not a method for a Year 2 minus Year 1 population-mean difference. That question requires the sampling distribution of the difference under an appropriate design. The module names that missing object so learners do not misuse the displayed intervals; it deliberately defers the two-sample method.