1. What is the Critical Value TC?

The critical value (t_c) is the cutoff point on a t-distribution that determines statistical significance for a given alpha level and degrees of freedom. It's used in hypothesis testing to define the rejection region.

2. How Does the Calculator Work?

The calculator uses the inverse t-distribution function:

Where:

• \( \alpha \) — Significance level (decimal)
• \( df \) — Degrees of freedom
• \( t_{inv} \) — Inverse t-distribution function

Explanation: The function returns the t-value that corresponds to the specified cumulative probability (1-α) in the t-distribution with the given degrees of freedom.

3. Importance of Critical Value

Details: Critical values are essential for determining whether to reject the null hypothesis in t-tests. They help establish the threshold for statistical significance.

4. Using the Calculator

Tips: Enter alpha (typically 0.05, 0.01, or 0.10) and degrees of freedom (n-1 for a sample size of n). Both values must be valid (0 < α < 1, df ≥ 1).

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between one-tailed and two-tailed critical values?
A: For one-tailed tests, use α directly. For two-tailed tests, use α/2 as the input.

Q2: How do degrees of freedom affect the critical value?
A: As df increases, the t-distribution approaches the normal distribution, and critical values decrease toward the z-value.

Q3: What are typical alpha values used?
A: Common values are 0.05 (5%), 0.01 (1%), and 0.10 (10%), depending on the desired confidence level.

Q4: When should I use t-critical values vs z-scores?
A: Use t-values when population standard deviation is unknown and sample size is small (<30). Use z-scores for large samples or when population SD is known.

Q5: Can I get negative critical values?
A: Yes, critical values can be negative for left-tailed tests or the lower bound of two-tailed tests.