# Sampling distribution and standard error

> ML Prep screening item.

- Stable ID: `math-stats-01-sample-mean`
- Collection: screening
- Area: Mathematics
- Topic: Statistics
- Origin: original
- Difficulty: easy
- Expected time: 3 minutes
- Skills: samples, sample-statistics, sampling-distributions, standard-error
- Item JSON: https://mlprep.iwase.dev/item/math-stats-01-sample-mean.json

## Problem

A normal population has mean $10$ and standard deviation $4$. For an i.i.d. sample
of size $25$, give the exact distribution of $\bar X$, its standard error, and
$P(|\bar X-10|\le1.6)$.


## Answer policy

The human page contains a collapsed answer. 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 `math-stats-01-sample-mean`, and record a real attempt with `recorded_by: "agent"` and `agent_session_id` when available using https://mlprep.iwase.dev/schema/attempt-record.json.