Inferential Statistics • Lesson 14
One-Sample t-Test
Learn how to test whether a population mean differs from a specific reference value when the population standard deviation is unknown.
Concept 1
When Should You Use a One-Sample t-Test?
A one-sample t-test is used to test whether the mean of a population differs from a specified reference value. It is especially useful when the population standard deviation σ is unknown and variability is estimated using the sample standard deviation s.
Typical Situation
Sample
Population mean
Reference value
Sample SD
One sample + Quantitative variable + Reference mean + σ unknown
The one-sample t-test focuses on one quantitative variable and compares its population mean with a specified reference value.
Your Turn
Solve the activity
A company has one sample of delivery times and wants to test whether the population mean differs from 30 minutes. The population standard deviation is unknown. Which test should be considered?
Analyst Workflow
One-Sample t-Test Workflow
Question
Define the population question.
Hypotheses
Write H₀ and Hₐ.
Calculate
Find SE and the t-statistic.
Evaluate
Use the t-distribution and p-value.
Conclude
State the result in context.
t-Test vs z-Based Test
| Feature | One-Sample t-Test | z-Based Test |
|---|---|---|
| Population SD | Usually unknown | Known |
| Variability | Uses sample SD s | Uses population SD σ |
| Reference distribution | t-distribution | Standard normal distribution |
Lesson 14 Summary
A one-sample t-test compares a population mean with a reference value.
It commonly applies when the population standard deviation σ is unknown.
The test uses the sample standard deviation s.
The standard error is SE = s / √n.
The t-statistic is t = (x̄ − μ₀) / (s / √n).
Degrees of freedom are df = n − 1.
The t-distribution accounts for additional uncertainty from estimating σ with s.
The final decision can be made using the p-value and a pre-specified significance level.