Determine Normality of Single Observation with Z-Test

Q: Determine Normality of Single Observation with Z-Test

Evaluates hypothesis testing for a single observation against a fully specified standard normal distribution. Strong answers state hypotheses, use z = x, define critical values and p-values, and note limitations.

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Question

Hypothesis Test for One Observation Against a Standard Normal

You observe a single numeric value x and want to decide whether it could plausibly have been drawn from a standard normal distribution N(0, 1). Assume the reference distribution is fully specified and not estimated from data.

Constraints & Assumptions

You have only one observation.
The null distribution is exactly N(0, 1).
Use a two-sided test unless a directional alternative is justified.
Interpret the result as evidence about this observation, not proof about the full data-generating process.

Clarifying Questions to Ask Guidance

Is the alternative two-sided or one-sided?
What significance level alpha should be used?
Was x selected after looking at many observations or tests?
Is the goal anomaly detection, model checking, or a formal hypothesis test?

What a Strong Answer Covers Guidance

Uses a z-test or equivalent tail-probability test because the null distribution is fully specified.
States H0: X follows N(0, 1) and a two-sided H1: X is unusually extreme under N(0, 1).
Uses z = x as the test statistic.
Rejects at level alpha when absolute z exceeds the standard normal critical value z_(1 - alpha/2).
Computes the two-sided p-value as 2 times the upper-tail probability beyond absolute x.
Notes limitations of making distributional conclusions from one observation.

Follow-up Questions Guidance

How would the test change for a one-sided alternative?
What if the mean and variance were estimated from data?
How would you adjust if this observation was selected as the most extreme among many?

Determine Normality of Single Observation with Z-Test

Quick Overview

Determine Normality of Single Observation with Z-Test

Hypothesis Test for One Observation Against a Standard Normal

Constraints & Assumptions

Clarifying Questions to Ask Guidance

What a Strong Answer Covers Guidance

Follow-up Questions Guidance

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