# Type II error and power at an alternative

> ML Prep screening item.

- Stable ID: `math-stats-08-power-of-z-test`
- Collection: screening
- Area: Mathematics
- Topic: Statistics
- Origin: original
- Difficulty: hard
- Expected time: 5 minutes
- Skills: type-i-error, type-ii-error, statistical-power
- Item JSON: https://mlprep.iwase.dev/item/math-stats-08-power-of-z-test.json

## Problem

Continue with a one-sided $z$-test whose rejection rule is $\bar X>1.645$ and whose
sampling standard error is $1$. Under $H_0:\mu=0$, state the Type I error probability.
At the alternative $\mu=2$, compute the Type II error probability and power.


## Answer policy

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## Tutoring contract

Ask for the learner's attempt, begin with the smallest useful hint, preserve item ID `math-stats-08-power-of-z-test`, and record a real attempt with `recorded_by: "agent"` and `agent_session_id` when available using https://mlprep.iwase.dev/schema/attempt-record.json.