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Reuse the known-sigma example
Use mean 50.4, known sigma 2 and n = 100 independent normal observations. The standard error is 0.2. For this exercise, use a 95% two-sided normal critical value of approximately 1.959964.
Calculate the bounds
The margin is 1.959964 × 0.2 = 0.3919928. Subtracting and adding gives approximately [50.0080072, 50.7919928]. The null value 50 lies below the lower bound.
| Interval calculation | Test calculation |
|---|---|
| |50.4 − 50| > 1.959964 × 0.2 | |z| > 1.959964 |
| 0.4 > 0.3919928 | 2 > 1.959964 |
Match all settings before reconciling
The correspondence uses the same reference model, standard error, sidedness and significance level. A one-sided test and a 95% two-sided interval are not the same configuration. Mixing a normal interval with an unknown-sigma t test can also create a discrepancy.
Interpret the interval as a procedure
The confidence level describes repeated-sampling coverage of the method under its assumptions. This realized interval is a set of parameter values that satisfy its construction. It does not put a 95% posterior probability on the fixed mean without a separate probability model for that mean.