1. What is Critical Z-Value?

The critical z-value (Z_c) is the z-score that corresponds to a specified significance level (α) in a standard normal distribution. It is used in hypothesis testing to define the rejection region for statistical tests.

2. How Does the Calculator Work?

The calculator uses the inverse normal distribution formula:

Where:

• \( Z_c \) — Critical z-value
• \( \Phi^{-1} \) — Inverse cumulative distribution function of standard normal distribution
• \( \alpha \) — Significance level (0-0.5)

Explanation: The formula calculates the z-score that leaves α/2 probability in each tail of the standard normal distribution.

3. Importance of Critical Z-Value

Details: Critical z-values are essential for constructing confidence intervals and conducting hypothesis tests in statistics. They help determine whether to reject or fail to reject the null hypothesis.

4. Using the Calculator

Tips: Enter the significance level (α) between 0.0001 and 0.5. Common values include 0.01, 0.05, and 0.10 for 99%, 95%, and 90% confidence levels respectively.

5. Frequently Asked Questions (FAQ)

Q1: What is the relationship between α and confidence level?
A: Confidence level = 1 - α. For example, α = 0.05 corresponds to a 95% confidence level.

Q2: Why divide α by 2 in the formula?
A: For two-tailed tests, the significance level is split equally between both tails of the distribution.

Q3: What are common critical z-values?
A: For α = 0.05, Z_c ≈ 1.96; for α = 0.01, Z_c ≈ 2.576; for α = 0.10, Z_c ≈ 1.645.

Q4: When should I use one-tailed vs two-tailed critical values?
A: Use one-tailed when the alternative hypothesis is directional, and two-tailed when non-directional.

Q5: Can this calculator be used for one-tailed tests?
A: This calculator provides two-tailed critical values. For one-tailed tests, use Z_c = Φ^{-1}(1 - α).