Two Sample Critical Value Calculator

Critical Value Formula:

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

Critical Z Value:

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1. What is the Two-Sample Critical Value?

The two-sample critical value (Z) is the threshold value that determines the rejection region in hypothesis testing for comparing two sample means. It's based on the standard normal distribution and the chosen significance level (alpha).

2. How Does the Calculator Work?

The calculator uses the inverse normal distribution function:

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

Where:

\( Z \) — Critical value (standard normal deviate)
\( \alpha \) — Significance level (typically 0.05)
\( \text{invNorm} \) — Inverse standard normal cumulative distribution function

Explanation: For a two-tailed test with significance level α, the critical values are the (1-α/2) and α/2 percentiles of the standard normal distribution.

3. Importance of Critical Value Calculation

Details: The critical value determines whether a test statistic is extreme enough to reject the null hypothesis in two-sample comparisons. It's essential for hypothesis testing in research and statistical analysis.

4. Using the Calculator

Tips: Enter the desired significance level (α) between 0 and 1 (typically 0.05 for 5% significance). The calculator will return the two-tailed critical Z value.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between one-tailed and two-tailed critical values?
A: For one-tailed tests, use invNorm(1-α). For two-tailed tests (most common), use invNorm(1-α/2) as shown in this calculator.

Q2: What are typical alpha values used?
A: Common values are 0.05 (5%), 0.01 (1%), and 0.10 (10%), with 0.05 being most common in many fields.

Q3: How does sample size affect critical values?
A: Critical Z values don't change with sample size, but t-distribution critical values (for small samples) do.

Q4: When should I use Z vs t critical values?
A: Use Z when population standard deviation is known or samples are large (>30). Use t for small samples with unknown population SD.

Q5: What if I need the critical value for a one-tailed test?
A: Simply use invNorm(1-α) instead of invNorm(1-α/2). For α=0.05, the one-tailed Z critical value is 1.6449.