AP®StatisticsCollege BoardRevision NotesUnit 4: Inference for Quantitative Data: MeansInference for Meanst-scores versus z-scores

t-scores versus z-scores (College Board AP® Statistics): Revision Note

Syllabus Edition

First teaching 2026

First exams 2027

Written by: Naomi C

Reviewed by: Dan Finlay

Updated on 4 June 2026

t-scores versus z-scores

How do I know when to use a t-score or a z-score?

It is important to be able to recognize which critical value to use for the inference test that you are conducting
- You will need to consider the population distribution, the population parameters given, and the sample size

What happens if the population is normally distributed?

If a population is normally distributed and the population variance is known
- then you should use a z-score
If a population is normally distributed but the population variance is unknown
- then you should use a t-score when the sample size is small, $n < 30$
In practice, if the population variance is unknown but the sample size is large, $n > 30$
- t-scores can be used
- but z-scores can also be used
  - especially as the t-scores in the table run out for high values of n
- If a z-score is used, then the sample standard deviation, $s$ , is used in place of the population standard deviation, $σ$

What happens if the population is not normally distributed?

If a population is not normally distributed but $n \geq 30$
- then, due to the Central Limit theorem, you can use z-scores
- Use the population standard deviation, $σ$ , if it is given
  - or the sample standard deviation, $s$ , if $σ$ is not given

If a population is not normally distributed and $n < 30$
- then there is no statistical procedure covered in this course

Examiner Tips and Tricks

The population standard deviation, σ, is not typically known for distributions for quantitative variables. Therefore, you will need to use t-scores for questions in this unit.

Worked Example

A medical researcher is evaluating the recovery time (in days) for patients who undergo a new type of knee surgery. The researcher selects a random sample of 25 patients and records their recovery times. The sample mean recovery time is 14 days, and the sample standard deviation is 3.2 days. The true population standard deviation of recovery times for this surgery is unknown. The researcher wishes to conduct a significance test to determine if the true mean recovery time is less than 16 days.

Which of the following best describes the appropriate test procedure and provides the correct justification?

(A) A one-sample z-test, because the researcher is testing a claim about a single population parameter.

(B) A one-sample z-test, because a standard deviation of 3.2 days is provided in the scenario.

(C) A one-sample t-test, because the population standard deviation is unknown and the sample standard deviation must be used to estimate it.

(D) A one-sample t-test, because the sample size (n=25) is less than 30, which prohibits the use of a z-test.

Answer:

t-distributions are used for inferences about a population mean when the population standard deviation (σ) is unknown and the sample standard deviation (s) must be used in its place

Because the scenario explicitly states the population standard deviation is unknown (and we only have the sample standard deviation of 3.2 days), the correct procedure is a one-sample t-test

Therefore, the correct answer is C

Why the distractors are incorrect:

(A) identifies an incorrect test
- Estimating a single population parameter does not automatically dictate a z-test (means typically require t-tests)
(B) highlights a frequently tested error
- According to AP Chief Reader reports, students often incorrectly apply a z-test by mistakenly believing that any standard deviation given in the prompt is the population standard deviation
- The 3.2 days here is the sample standard deviation (s).
(D) identifies the correct test but provides a flawed justification
- A common student misconception is that the sample size (n≥30) determines whether to use z or t
- In reality, the sample size helps satisfy the normality condition for the sampling distribution, but the decision to use z or t hinges solely on whether σ is known

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