1. What is Critical Z Value?

The critical z-value (Z_crit) is the point on the standard normal distribution that defines the boundary for statistical significance. 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:

• \( \Phi^{-1} \) — Inverse standard normal cumulative distribution function
• \( \alpha \) — Significance level (typically 0.05 for 95% confidence)

Explanation: For a two-tailed test with significance level α, the critical z-values are the points where the cumulative probability equals 1-α/2 and α/2.

3. Importance of Z Critical Value

Details: Critical z-values are essential for constructing confidence intervals and conducting hypothesis tests in statistics. They define the threshold for statistical significance.

4. Using the Calculator

Tips: Enter your desired significance level (α) between 0 and 1 (e.g., 0.05 for 95% confidence). 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 Φ^-1(1-α) instead of Φ^-1(1-α/2). Two-tailed tests split α between both tails.

Q2: What are common critical z-values?
A: For α=0.05 (95% CI): ±1.96, for α=0.01 (99% CI): ±2.576, for α=0.10 (90% CI): ±1.645.

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 sample size affect critical values?
A: For z-tests, critical values remain constant regardless of sample size (unlike t-tests).

Q5: Can I calculate critical values for non-normal distributions?
A: Yes, but you'll need the inverse CDF for that specific distribution rather than the normal distribution.