1. What is Critical Value Zc?

The critical value Zc is the z-score that corresponds to a specific confidence level in a standard normal distribution. It represents the number of standard deviations from the mean that defines the boundaries of the confidence interval.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

• \( Z_c \) — Critical value (z-score)
• \( \alpha \) — Significance level (unitless)
• \( \text{invNorm} \) — Inverse normal distribution function

Explanation: The formula calculates the z-score that corresponds to the (1 - α/2) percentile of the standard normal distribution, which is used for two-tailed tests.

3. Importance of Critical Value Calculation

Details: Critical values are essential for hypothesis testing, constructing confidence intervals, and determining statistical significance in various research and analytical applications.

4. Using the Calculator

Tips: Enter the alpha value (significance level) between 0 and 1. Common values include 0.05, 0.01, and 0.10 for 95%, 99%, and 90% confidence levels respectively.

5. Frequently Asked Questions (FAQ)

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

Q2: Why divide alpha by 2 in the formula?
A: For two-tailed tests, the alpha is split equally between both tails of the distribution, so each tail gets α/2.

Q3: What are common critical values?
A: Common values include ±1.96 for α=0.05, ±2.576 for α=0.01, and ±1.645 for α=0.10.

Q4: When should I use one-tailed vs two-tailed critical values?
A: Use one-tailed when you have a directional hypothesis, two-tailed when you're testing for any difference regardless of direction.

Q5: Can this calculator be used for non-normal distributions?
A: This calculator is specifically for the standard normal distribution. Other distributions (t, chi-square, F) require different critical value calculations.