1. What is a Critical Z Value?

The critical Z value is the point on the standard normal distribution that corresponds to a specified significance level (α). It's used in hypothesis testing to determine rejection regions.

2. How Does the Calculator Work?

The calculator uses the inverse normal distribution function:

Where:

• \( Z \) — Critical Z value (standard deviations from mean)
• \( \alpha \) — Significance level (0 to 1)
• \( \text{invNorm} \) — Inverse standard normal distribution function

Explanation: For a two-tailed test with significance level α, the critical values are at ±Z where Z is the (1-α/2) percentile of the standard normal distribution.

3. Importance of Critical Z Values

Details: Critical Z values determine the threshold for statistical significance in Z-tests. They help establish confidence intervals and rejection regions in hypothesis testing.

4. Using the Calculator

Tips: Enter the desired significance level (α) between 0 and 1 (e.g., 0.05 for 5% significance). The calculator will return the two-tailed critical Z value.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between one-tailed and two-tailed critical values?
A: For one-tailed tests, use invNorm(1-α). For two-tailed tests (most common), use invNorm(1-α/2).

Q2: What are common critical Z 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 Z values vs t values?
A: Use Z when population standard deviation is known or sample size is large (>30). Use t for small samples with unknown σ.

Q4: How is this related to p-values?
A: The critical Z defines the threshold where p-values become statistically significant.

Q5: Can I get negative critical values?
A: The calculator returns the positive value. For two-tailed tests, the critical region includes both ±Z.