T-Test Calculator

Understanding T-Tests

A t-test is a statistical hypothesis test used to determine whether there is a significant difference between means. It's used when the population standard deviation is unknown and the sample size is small to moderate.

Types of T-Tests

1. One-Sample T-Test

Tests whether a sample mean differs from a known population mean.

Where: x̄ = sample mean, μ₀ = population mean, s = sample standard deviation, n = sample size

2. Two-Sample (Independent) T-Test

Compares means of two independent groups assuming equal variances.

Where: sp = pooled standard deviation, df = n₁ + n₂ - 2

3. Paired T-Test

Compares means of two related groups (before/after, matched pairs).

Where: d̄ = mean of differences, sd = standard deviation of differences, df = n - 1

4. Welch's T-Test

Compares means of two independent groups WITHOUT assuming equal variances.

Uses Welch-Satterthwaite equation for degrees of freedom.

Key Components

Hypothesis Testing

• Null Hypothesis (H₀): No difference between means (e.g., μ = μ₀, μ₁ = μ₂)
• Alternative Hypothesis (H₁):
  • Two-tailed: μ ≠ μ₀ (difference exists, either direction)
  • Right-tailed: μ > μ₀ (sample mean is greater)
  • Left-tailed: μ < μ₀ (sample mean is less)

Interpreting Results

• If p-value ≤ α: Reject H₀ (statistically significant difference)
• If p-value > α: Fail to reject H₀ (no significant difference)
• Common α levels: 0.05 (95% confidence), 0.01 (99% confidence), 0.10 (90% confidence)

Assumptions

• Normality: Data should be approximately normally distributed (less critical for large samples due to Central Limit Theorem)
• Independence: Observations should be independent
• Equal Variances: Required for standard two-sample t-test (use Welch's if violated)
• Random Sampling: Data should come from random samples

Real-World Applications

• Medicine: Compare treatment effects (drug vs. placebo)
• Psychology: Test differences in behavior or cognition
• Education: Compare teaching methods or test scores
• Business: A/B testing, comparing sales strategies
• Agriculture: Compare crop yields under different conditions
• Quality Control: Test if manufacturing process meets specifications

Effect Size

Beyond statistical significance, consider Cohen's d to measure practical significance:

• Small effect: |d| ≈ 0.2
• Medium effect: |d| ≈ 0.5
• Large effect: |d| ≈ 0.8