1. What is a Critical Value?

A critical value is a point on the test distribution that is compared to the test statistic to determine whether to reject the null hypothesis. It represents the threshold beyond which we consider results statistically significant.

2. How Does the Calculator Work?

The calculator determines critical values using statistical distributions:

Where:

• \( t \) — t-distribution critical value (requires degrees of freedom)
• \( z \) — z-distribution critical value (standard normal distribution)
• \( \alpha \) — Significance level (typically 0.05, 0.01, or 0.10)
• Degrees of Freedom — Sample size minus parameters estimated (for t-distribution)

Explanation: The calculator uses statistical functions to determine the precise critical value based on your selected distribution, alpha level, and tail type.

3. Importance of Critical Values

Details: Critical values are essential in hypothesis testing as they define the rejection region for statistical tests. They help determine whether observed results are statistically significant or occurred by chance.

4. Using the Calculator

Tips: Select the appropriate distribution (z for known population variance, t for unknown), enter the alpha level (typically 0.05), specify degrees of freedom for t-distribution, and choose one-tailed or two-tailed test based on your hypothesis.

5. Frequently Asked Questions (FAQ)

Q1: When should I use z-distribution vs t-distribution?
A: Use z-distribution when population variance is known or sample size is large (>30). Use t-distribution when population variance is unknown and sample size is small.

Q2: What's the difference between one-tailed and two-tailed tests?
A: One-tailed tests look for an effect in one direction only, while two-tailed tests consider both directions. Two-tailed tests are more conservative.

Q3: How does degrees of freedom affect the critical value?
A: As degrees of freedom increase, the t-distribution approaches the z-distribution. With fewer degrees of freedom, the t-distribution has heavier tails.

Q4: What are common alpha values used in research?
A: α = 0.05 (5% significance) is most common, followed by α = 0.01 (1% significance) for more stringent tests.

Q5: Can I use this calculator for confidence intervals?
A: Yes, critical values are used in constructing confidence intervals. For a 95% CI, use α = 0.05 with two-tailed test.