Type II error and power at an alternative

Tutor me on https://mlprep.iwase.dev/item/math-stats-08-power-of-z-test/. This is a screening item. If window.mlPrepAgent is available, read attempts for item math-stats-08-power-of-z-test before tutoring. Inspect my attempt, keep the item ID, and do not reveal the full answer first. After a real attempt, append its record with recorded_by agent, include agent_session_id when available, and read it back.

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.

Reveal answer or reference solution

The cutoff was chosen so that under $H_0$ , $P_0(\bar X>1.645)=0.05$ ; this is the Type I error probability. Under $\mu=2$ ,

\beta=P_2(\bar X\le1.645)=\Phi(1.645-2)=\Phi(-0.355)\approx0.3613

Therefore the power is

1-\beta=\Phi(0.355)\approx0.6387

Previous attempts

Loading attempts saved in this browser…

Review Screening Results

Type II error and power at an alternative

Problem

Previous attempts

Appears in