1. What is the Critical Z Value?

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

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

• \( Z_c \) — Critical z-value
• \( \alpha \) — Significance level (typically 0.05, 0.01, etc.)
• \( \text{invNorm} \) — 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 Z Calculation

Details: Critical z-values are essential for determining statistical significance in hypothesis testing, constructing confidence intervals, and making decisions about rejecting or failing to reject null hypotheses.

4. Using the Calculator

Tips: Enter the significance level (α) as a decimal value between 0 and 1. Common values include 0.05, 0.01, and 0.001.

5. Frequently Asked Questions (FAQ)

Q1: What is the difference between one-tailed and two-tailed critical values?
A: One-tailed tests use invNorm(1 - α) while two-tailed tests use invNorm(1 - α/2) to account for both tails of the distribution.

Q2: What are common critical z-values?
A: For α = 0.05 (two-tailed), Z_c ≈ 1.96; for α = 0.01, Z_c ≈ 2.576; for α = 0.001, Z_c ≈ 3.291.

Q3: When should I use critical z-values?
A: Use them when working with normally distributed data, large sample sizes, or when the population standard deviation is known.

Q4: How does sample size affect critical values?
A: For large samples (n > 30), z-values are appropriate. For smaller samples, t-distribution critical values should be used instead.

Q5: Can critical z-values be negative?
A: Critical z-values are typically positive as they represent distances from the mean, but the rejection region can be on both sides of the distribution.