Formula Used
The one-sample t-statistic is computed using the sample mean ($\bar{x}$), population mean under null hypothesis ($\mu_0$), sample standard deviation ($s$), and sample size ($n$):
$$ t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} $$
Degrees of freedom are calculated as $df = n - 1$. Confidence intervals utilize the standard error $SE = s / \sqrt{n}$ along with critical t values.
How to Use This Calculator
- Choose between Raw Data input or Summary Statistics.
- Enter your target hypothesized mean ($\mu_0$).
- Select your preferred alternative hypothesis orientation (two-tailed, left-tailed, or right-tailed).
- Set your significance level ($\alpha$) such as 0.05.
- Press the Calculate T-Test button to view instant diagnostic results.
Understanding the One-Sample T-Test in Inferential Statistics
The one-sample t-test is a fundamental parametric hypothesis testing procedure used to determine whether an unknown population mean differs significantly from a specified hypothesized value. When researchers or data analysts do not know the population standard deviation and work with smaller sample sizes, the Student's t-distribution provides the exact probabilistic distribution required for accurate hypothesis evaluation.
Key Assumptions and Requirements
For validity, observations should be independent, collected randomly, and ideally follow an approximately normal distribution, particularly when sample sizes are small. Robustness against moderate departures from normality increases as sample sizes grow larger due to the central limit theorem.