ML Prep

Test decisions and errors

Problem

A two-sided test rejects H0H_0 when ∣Z∣>1.96|Z|>1.96. For an observed Z=2.3Z=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)(30,20) against expected counts (25,25)(25,25), give its degrees of freedom, and state what a likelihood-ratio test compares.

Reveal answer or reference solution

Reject H0H_0; the two-sided p-value is below 0.050.05. The rule has Type I error 0.050.05 under its assumptions. Type II error is failure to reject under a particular alternative, and power is one minus that error probability.

The chi-square statistic is

(30−25)225+(20−25)225=2,\frac{(30-25)^2}{25} + \frac{(20-25)^2}{25} = 2,

with one degree of freedom. A likelihood-ratio test compares the best likelihood under the null with the best likelihood under the larger alternative model.

Local history

Loading attempts saved in this browser…

Use with your agent

Share this URL and your attempt. Ask the agent to start with a clarifying question or the smallest useful hint.

Tutor me on https://mlprep.iwase.dev/mathematics/statistics/original-stats-tests/. If window.mlPrepAgent is available, read attempts for item original-stats-tests before tutoring. Inspect my attempt, keep the item ID, and do not reveal the full answer first. After a real attempt, append its record and read it back.

Appears in