1. What is the Critical Value Formula?

The critical value (Z) is the number of standard deviations from the mean that corresponds to a given confidence level in a normal distribution. It's used in hypothesis testing and confidence interval construction.

2. How Does the Calculator Work?

The calculator uses the inverse normal distribution function:

Where:

• \( Z \) — Critical value (standard deviations)
• \( \Phi^{-1} \) — Inverse standard normal cumulative distribution function
• \( \alpha \) — Significance level (Type I error probability)

Explanation: The formula calculates the Z-score that corresponds to the (1-α/2) percentile of the standard normal distribution.

3. Importance of Critical Value

Details: Critical values determine the cutoff points for statistical significance in hypothesis tests and define the margins of confidence intervals.

4. Using the Calculator

Tips: Enter the desired significance level (α) as a value between 0 and 1 (e.g., 0.05 for 95% confidence).

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between one-tailed and two-tailed critical values?
A: Two-tailed tests use α/2 (dividing significance between both tails), while one-tailed tests use the full α in one direction.

Q2: What are common critical values?
A: For α=0.05 (95% CI), Z≈1.96; for α=0.01 (99% CI), Z≈2.576.

Q3: When should I use t-scores instead of Z-scores?
A: Use t-scores when sample sizes are small (<30) and population standard deviation is unknown.

Q4: How does this relate to p-values?
A: The critical value defines the rejection region, while the p-value is the probability of observing a test statistic at least as extreme.

Q5: Can this be used for non-normal distributions?
A: Different distributions (t, F, χ²) have their own critical values based on their specific shapes and degrees of freedom.