1. What is a Critical Z Score?

The critical Z score is the value on the standard normal distribution that corresponds to a given significance level (α). It marks the threshold beyond which we reject the null hypothesis in hypothesis testing.

2. How Does the Calculator Work?

The calculator uses the inverse normal distribution function:

Where:

• \( Z \) — Critical Z score (dimensionless)
• \( \alpha \) — Significance level (fraction, typically 0.05)
• \( \text{invNorm} \) — Inverse standard normal distribution function

Explanation: For a two-tailed test with significance level α, the critical values are the Z scores that leave α/2 in each tail of the standard normal distribution.

3. Importance of Critical Z Scores

Details: Critical Z scores are fundamental in hypothesis testing, determining confidence intervals, and setting decision boundaries in statistical analysis.

4. Using the Calculator

Tips: Enter the desired significance level (typically 0.05 for 5% significance). The calculator will return the corresponding critical Z score for a two-tailed test.

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 (two-tailed), Z≈1.96. For α=0.01, Z≈2.576.

Q3: How does sample size affect critical values?
A: For large samples (n>30), Z scores are appropriate. For small samples, t-scores should be used instead.

Q4: Can I use this for non-normal distributions?
A: Z scores assume normality. For non-normal distributions, other methods may be needed.

Q5: How precise are these calculations?
A: The calculator provides precise values based on the standard normal distribution.