Calculate t-statistics, p-values, and critical values for statistical hypothesis testing. Supports one-sample, two-sample independent (Student's and Welch's), and paired t-tests with step-by-step calculations.
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Perform t-tests to compare means and make statistical inferences. Calculate t-statistics, p-values, degrees of freedom, and critical values for one-sample, two-sample independent, and paired t-tests. Includes Welch's t-test for unequal variances with complete step-by-step calculations.
The t-test is a statistical hypothesis test used to determine if there is a significant difference between means. Developed by William Sealy Gosset under the pseudonym 'Student', it's used when sample sizes are small and population standard deviation is unknown. The test produces a t-statistic that follows the t-distribution, allowing us to calculate the probability (p-value) of observing our results if the null hypothesis is true.
One-Sample T-Test Formula
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Use a t-test when the population standard deviation is unknown and you're estimating it from sample data, which is most real-world situations. Use a z-test only when you know the true population standard deviation or have very large samples (n > 100). The t-test is more conservative and appropriate for smaller samples.
Student's t-test assumes both groups have equal variances (pooled variance). Welch's t-test doesn't assume equal variances and adjusts degrees of freedom accordingly. Welch's test is generally recommended as it performs well even when variances are equal and is more robust when they're not.
The p-value is the probability of observing results as extreme as yours if the null hypothesis is true. If p < α (typically 0.05), reject the null hypothesis. A small p-value (< 0.05) suggests the observed difference is unlikely due to chance alone, indicating statistical significance.
Degrees of freedom (df) represent the number of independent values that can vary in your analysis. For one-sample: df = n - 1. For two-sample equal variance: df = n₁ + n₂ - 2. For Welch's test, df is calculated using the Welch-Satterthwaite equation. Higher df means the t-distribution approaches the normal distribution.
Use a two-tailed test when you want to detect any difference (could be higher or lower). Use a one-tailed test when you have a specific directional hypothesis - left-tailed (μ < μ₀) to test if mean is lower, right-tailed (μ > μ₀) to test if mean is higher. One-tailed tests have more power for their specific direction.