1. What is the T-Distribution?

The t-distribution is a probability distribution used in statistics for estimating population parameters when the sample size is small and/or when the population standard deviation is unknown. It is similar to the normal distribution but has heavier tails.

2. How Does the Calculator Work?

The calculator uses the t-value formula:

Where:

• \( \bar{X} \) — Sample mean
• \( \mu \) — Population mean
• \( s \) — Sample standard deviation
• \( n \) — Sample size

Explanation: The t-value measures how many standard errors the sample mean is from the population mean.

3. Importance of T-Value Calculation

Details: The t-value is crucial for hypothesis testing (t-tests) and constructing confidence intervals, especially with small sample sizes where the normal distribution isn't appropriate.

4. Using the Calculator

Tips: Enter all required values in consistent units. The sample size must be at least 2, and standard deviation must be positive.

5. Frequently Asked Questions (FAQ)

Q1: When should I use the t-distribution instead of normal distribution?
A: Use the t-distribution when sample sizes are small (typically <30) or when the population standard deviation is unknown.

Q2: What are degrees of freedom in t-distribution?
A: Degrees of freedom (df) = sample size - 1. It affects the shape of the t-distribution.

Q3: How is the probability calculated?
A: The probability is calculated using the cumulative distribution function (CDF) of the t-distribution with appropriate degrees of freedom.

Q4: What does a high t-value indicate?
A: A high absolute t-value suggests the sample mean is significantly different from the population mean.

Q5: Can this be used for paired or two-sample t-tests?
A: This calculator is for one-sample t-tests. Different formulas are needed for paired or two-sample tests.