1. What is Z Critical Value?

The Z critical value is the number of standard deviations from the mean that corresponds to a given confidence level in a standard normal distribution. It's used in hypothesis testing and confidence interval construction.

2. How Does the Calculator Work?

The calculator uses the inverse standard normal distribution:

Where:

• \( \alpha \) — Significance level (typically 0.05 for 95% confidence)
• NORM.S.INV — 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 distribution.

3. Importance of Z Critical Value

Details: Z critical values are essential for determining rejection regions in hypothesis tests and for constructing confidence intervals in normally distributed data.

4. Using the Calculator

Tips: Enter the desired significance level (α) as a decimal between 0 and 1. Common values are 0.10, 0.05, or 0.01 corresponding to 90%, 95%, and 99% confidence levels.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between one-tailed and two-tailed critical values?
A: For one-tailed tests, use α rather than α/2 in the calculation. Two-tailed tests split α between both tails.

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

Q3: When should I use Z critical values?
A: When population standard deviation is known and sample size is large (typically n>30), or when data is normally distributed.

Q4: What if my sample size is small?
A: For small samples (n≤30) with unknown population standard deviation, use t-distribution critical values instead.

Q5: How is this related to p-values?
A: The Z critical value defines the threshold for statistical significance. If your test statistic exceeds this value, your p-value will be less than α.