1. What is Z Critical Value?

The Z critical value (Z_c) 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 calculations.

2. How Does the Calculator Work?

The calculator uses the inverse standard normal distribution function:

Where:

• \( \text{invNorm} \) — Inverse standard normal cumulative distribution function
• \( \text{confidence} \) — Desired confidence level (as a fraction between 0 and 1)

Explanation: The formula calculates the Z-score that leaves (1-confidence)/2 in each tail of the normal distribution.

3. Importance of Z Critical Value

Details: Z critical values are essential for constructing confidence intervals and conducting hypothesis tests in statistics. They determine the margin of error and help establish statistical significance.

4. Using the Calculator

Tips: Enter the desired confidence level as a decimal (e.g., 0.95 for 95% confidence). The value must be between 0 and 1 (exclusive).

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between Z and t critical values?
A: Z values are used when population standard deviation is known or sample size is large (>30), while t values are used for small samples with unknown population standard deviation.

Q2: What are common Z critical values?
A: Common values are 1.96 for 95% confidence, 2.576 for 99% confidence, and 1.645 for 90% confidence.

Q3: Why is the formula symmetric?
A: The standard normal distribution is symmetric, so we calculate the positive critical value which applies to both tails.

Q4: How accurate is this calculator?
A: It uses a precise approximation of the inverse normal CDF accurate to about 7 decimal places.

Q5: Can I use this for one-tailed tests?
A: For one-tailed tests, use invNorm(confidence) directly instead of dividing alpha by 2.