Critical Value Calculator Two Tailed

Two-tailed Critical Value Formula:

\[ Z = \pm \text{invNorm}(1 - \alpha/2) \]

Critical Z Value:

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1. What is a Two-Tailed Critical Value?

The two-tailed critical value defines the threshold for statistical significance in two-tailed hypothesis tests. It represents the number of standard deviations from the mean at which we reject the null hypothesis, with the rejection region split between both tails of the distribution.

2. How Does the Calculator Work?

The calculator uses the inverse normal distribution function:

\[ Z = \pm \text{invNorm}(1 - \alpha/2) \]

Where:

\( Z \) — Critical value (standard deviations from mean)
\( \alpha \) — Significance level (probability of Type I error)
\( \text{invNorm} \) — Inverse standard normal distribution function

Explanation: For a two-tailed test with significance level α, we find the Z-score that leaves α/2 probability in each tail of the standard normal distribution.

3. Importance of Critical Values

Details: Critical values determine whether test results are statistically significant. They are essential for hypothesis testing, confidence intervals, and determining rejection regions.

4. Using the Calculator

Tips: Enter your desired significance level (α) between 0 and 1 (e.g., 0.05 for 5% significance). The calculator returns the positive and negative Z-scores that bound the central (1-α) region of the standard normal distribution.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between one-tailed and two-tailed critical values?
A: Two-tailed tests divide α equally between both tails, while one-tailed tests put all α in one tail (for directional hypotheses).

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

Q3: When should I use a two-tailed test?
A: When you're testing for any difference (without specifying direction) between groups or from a hypothesized value.

Q4: How does sample size affect critical values?
A: Critical Z values remain the same, but with larger samples, smaller effects become statistically significant.

Q5: What if my test statistic follows a t-distribution?
A: For small samples, use t-distribution critical values which depend on degrees of freedom.