# Test decisions and errors

> ML preparation item.

- Stable ID: `original-stats-tests`
- Area: Mathematics
- Topic: Statistics
- Origin: original
- Expected time: 7 minutes
- Skills: null-alternative, type-errors-power, p-values-critical-regions, likelihood-ratio, goodness-of-fit, chi-square
- Item JSON: https://mlprep.iwase.dev/data/items/original-stats-tests.json

## Problem

A two-sided test rejects $H_0$ when $|Z|>1.96$. For an observed $Z=2.3$, state the decision and relate the rule to p-values, Type I error, Type II error, and power. Separately, compute the Pearson chi-square statistic for observed counts $(30,20)$ against expected counts $(25,25)$, give its degrees of freedom, and state what a likelihood-ratio test compares.


## Answer policy

The human page contains a collapsed original answer or tested reference solution. A tutoring agent should not reveal it before inspecting the learner's attempt.

## Tutoring contract

Ask for the learner's attempt, begin with the smallest useful hint, preserve item ID `original-stats-tests`, and record a real attempt with `recorded_by: "agent"` and `agent_session_id` when available using https://mlprep.iwase.dev/schema/attempt-record.json.