1. What is the t-Statistic?

The t-statistic is a ratio of the departure of an estimated parameter from its hypothesized value to its standard error. It's used in hypothesis testing, particularly in t-tests, to determine if there's a significant difference between two sample means.

2. How Does the Calculator Work?

The calculator uses the t-statistic formula:

Where:

• \( \bar{X} \) — Sample mean
• \( \mu_0 \) — Population mean (hypothesized value)
• \( s \) — Sample standard deviation
• \( n \) — Sample size

Explanation: The t-statistic measures how many standard errors the sample mean is from the population mean. Larger absolute values indicate greater deviation.

3. Importance of t-Statistic

Details: The t-statistic is crucial for determining statistical significance in small sample sizes (typically n < 30). It's used in Student's t-tests, confidence interval estimation, and regression analysis.

4. Using the Calculator

Tips: Enter the sample mean, hypothesized population mean, sample standard deviation, and sample size. All values must be valid (n > 0, s ≥ 0).

5. Frequently Asked Questions (FAQ)

Q1: When should I use a t-test instead of a z-test?
A: Use a t-test when the sample size is small (typically < 30) and the population standard deviation is unknown.

Q2: What does a high t-value mean?
A: A high absolute t-value suggests the sample mean is significantly different from the population mean, potentially rejecting the null hypothesis.

Q3: How is the t-statistic related to p-value?
A: The t-statistic is used with degrees of freedom (n-1) to determine the p-value from the t-distribution.

Q4: What are degrees of freedom in t-tests?
A: Degrees of freedom (df = n-1) account for sample size and affect the shape of the t-distribution.

Q5: Can this be used for paired samples?
A: For paired samples, you would calculate differences first and then use a single-sample t-test on the differences.