Hypothesis Tests

One-Sample t-Test

Tests whether the population mean equals a hypothesized value.

Statusvalidated

Version1.0.0

Minitab equivalentStat > Basic Statistics > 1-Sample t

Description

The one-sample t-test evaluates the null hypothesis that the mean of a normally distributed population equals a specified value μ₀. The test statistic t = (x̄ − μ₀)/(s/√n) follows a t-distribution with n−1 degrees of freedom. The test requires approximately normal data; for non-normal data, the Wilcoxon signed-rank test should be used as a nonparametric alternative.

Formulas

t = \frac{x ˉ - μ _{0}}{s / n}

Test statistic: standardized deviation of the sample mean from the target value

H₀

H_{0} : μ = μ_{0}

Null hypothesis: population mean equals the target value

\overset{x}{ˉ} \pm t_{1 - α /2, n - 1} \cdot \frac{s}{n}

Confidence interval for the population mean

Assumptions

Data are approximately normally distributed (or n sufficiently large, CLT)
Observations are independent
Continuous data

Limitations

For strongly skewed distributions with small n, the test can be unreliable
Sensitive to outliers

References

Student (W. S. Gosset) (1908), The probable error of a mean, Biometrika, 6(1), 1–25
Montgomery, D. C. (2019), Introduction to Statistical Quality Control, 8th Ed., Chapter 4