1. What is Two-Sample Inference?

Two-sample inference involves comparing two independent samples to determine if their population means are significantly different. This calculator provides confidence intervals and hypothesis test results for the difference between two means.

2. How the Calculator Works

The calculator uses the following formulas:

Where:

• \( \bar{x}_1, \bar{x}_2 \) — Sample means
• \( s_1^2, s_2^2 \) — Sample variances
• \( n_1, n_2 \) — Sample sizes
• \( t_{\alpha/2} \) — Critical t-value

3. Interpreting the Results

Confidence Interval: If the interval doesn't contain 0, the means are significantly different at the chosen confidence level.

p-value: If p-value < 0.05, the difference is statistically significant at the 5% level.

4. Using the Calculator

Tips: Enter comma-separated values for both samples. Choose confidence level (90%, 95%, or 99%). Each sample should have at least 2 values.

5. Frequently Asked Questions (FAQ)

Q1: What assumptions does this test make?
A: The test assumes independent samples, approximately normal distributions, and equal variances (though it's robust to moderate violations).

Q2: When should I use a paired t-test instead?
A: Use paired tests when measurements are related (e.g., before/after treatment on same subjects).

Q3: What if my data isn't normally distributed?
A: For non-normal data, consider non-parametric tests like the Mann-Whitney U test.

Q4: How large should my samples be?
A: Small samples (n < 30) require normality assumption. Larger samples can rely on the Central Limit Theorem.

Q5: What does the p-value tell me?
A: The probability of observing a difference this large or larger if there was truly no difference between populations.