1. What is a Critical Value?

A critical value is a point on the scale of a test statistic beyond which we reject the null hypothesis. It separates the region where the null hypothesis is rejected from where it is not rejected.

2. How Does the Calculator Work?

The calculator uses the inverse distribution function:

Where:

• \( \alpha \) — Significance level (probability of Type I error)
• distribution inverse — The inverse cumulative distribution function for the chosen distribution

Explanation: The critical value is calculated based on the chosen probability distribution and the desired significance level.

3. Importance of Critical Values

Details: Critical values are essential in hypothesis testing, determining confidence intervals, and making statistical decisions. They help determine whether observed test statistics are significant.

4. Using the Calculator

Tips: Enter the significance level (typically 0.05, 0.01, or 0.10), select the appropriate distribution, and provide degrees of freedom if required.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between one-tailed and two-tailed critical values?
A: One-tailed tests use α directly, while two-tailed tests use α/2 for each tail. This calculator provides one-tailed values.

Q2: How do I choose the right distribution?
A: Use Z for normal distributions with known variance, t for small samples or unknown variance, chi-square for variance tests, and F for ANOVA.

Q3: What are typical alpha values?
A: Common values are 0.05 (5%), 0.01 (1%), and 0.10 (10%), depending on the desired confidence level.

Q4: Why do some distributions require degrees of freedom?
A: Distributions like t, chi-square, and F have shapes that depend on sample size, which is accounted for by degrees of freedom.

Q5: Can I get exact p-values from critical values?
A: Yes, critical values and p-values are related through the cumulative distribution function.