1. What is a Two-Tailed Z Value?

The two-tailed Z value is the critical value from the standard normal distribution that corresponds to a given significance level (α) in a two-tailed hypothesis test. It represents the threshold beyond which we reject the null hypothesis.

2. How Does the Calculator Work?

The calculator uses the inverse standard normal distribution:

Where:

• \( \alpha \) — Significance level (typically 0.05, 0.01, etc.)
• invNorm — Inverse standard normal distribution function
• The ± indicates we consider both tails of the distribution

Explanation: For a two-tailed test with α=0.05, we look for the Z values that leave 2.5% in each tail (hence 1-α/2 = 0.975).

3. Importance of Critical Z Values

Details: Critical Z values determine the rejection region for hypothesis tests. They are essential for constructing confidence intervals and making decisions about statistical significance.

4. Using the Calculator

Tips: Enter your desired significance level (α) between 0 and 1. Common values are 0.01, 0.05, or 0.10. The calculator will return the corresponding critical Z value for a two-tailed test.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between one-tailed and two-tailed Z 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 Z values for standard α levels?
A: For α=0.05: ±1.96; for α=0.01: ±2.576; for α=0.10: ±1.645.

Q3: When should I use a two-tailed test?
A: When you're testing for any difference (greater or lesser) without directionality specified in your hypothesis.

Q4: How precise is this calculator?
A: The calculator uses a high-precision approximation of the inverse normal function, accurate to at least 4 decimal places.

Q5: Can I use this for small sample sizes?
A: For small samples (n < 30), consider using t-distribution critical values instead of Z values.