1. What is Critical Value Z_α/2?

The critical value Z_α/2 is the z-score that corresponds to a specified significance level (α) in a two-tailed normal distribution. It marks the boundary of the rejection region in hypothesis testing.

2. How Does the Calculator Work?

The calculator uses the inverse normal distribution function:

Where:

• \( \alpha \) — Significance level (fraction between 0 and 1)
• invNorm — Inverse standard normal cumulative distribution function

Explanation: The function finds the z-score where the cumulative probability equals 1-α/2.

3. Importance of Critical Values

Details: Critical values are essential for constructing confidence intervals and conducting hypothesis tests in statistics. They determine the threshold for statistical significance.

4. Using the Calculator

Tips: Enter the desired significance level (α) as a fraction between 0 and 1 (e.g., 0.05 for 5% significance). The calculator will return the corresponding critical z-value.

5. Frequently Asked Questions (FAQ)

Q1: Why divide alpha by 2?
A: For two-tailed tests, the significance level is split between both tails of the distribution.

Q2: What are common critical values?
A: Common values are 1.96 (α=0.05), 2.576 (α=0.01), and 1.645 (α=0.10 for one-tailed).

Q3: How is this different from t-critical values?
A: Z-values are for normal distributions with known variance or large samples; t-values are for small samples with unknown variance.

Q4: When would I use this?
A: When constructing confidence intervals or conducting z-tests with normally distributed data.

Q5: Why is mean not used in the calculation?
A: Critical values are based on the standard normal distribution (mean=0, SD=1). The mean is accounted for in the test statistic calculation.