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's determined by the significance level (α) and the probability distribution of the test statistic.

2. How Does the Calculator Work?

The calculator uses the inverse distribution function:

Where:

• \( \alpha \) — Significance level (probability of Type I error)
• \( \text{dist\_inv} \) — Inverse cumulative distribution function for the chosen distribution

Explanation: The critical value marks the threshold for statistical significance in hypothesis testing.

3. Importance of Critical Values

Details: Critical values are essential for determining whether to reject the null hypothesis in statistical tests. They define the rejection region for the test statistic.

4. Using the Calculator

Tips: Enter the significance level (α) between 0 and 1, 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.

Q2: How do degrees of freedom affect critical values?
A: For t, chi-square, and F distributions, critical values change with degrees of freedom, approaching normal distribution values as df increases.

Q3: What's a typical significance level?
A: α = 0.05 is common, but 0.01 or 0.10 may be used depending on the field and consequences of errors.

Q4: Can I find critical values for non-standard distributions?
A: This calculator handles common distributions. For others, specialized software or tables may be needed.

Q5: Why are critical values important in confidence intervals?
A: They determine the margin of error in confidence intervals (e.g., ±1.96 for 95% CI with normal distribution).