T-test Distribution Calculator

T-distribution Formula:

\[ CDF = T.DIST(x, df, cumulative) \]

Result:

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1. What is the T-test Distribution?

The T-distribution, also known as Student's t-distribution, is a probability distribution that is used to estimate population parameters when the sample size is small and/or when the population variance is unknown. It is similar to the normal distribution but has heavier tails.

2. How Does the Calculator Work?

The calculator uses the T-distribution formula:

\[ CDF = T.DIST(x, df, cumulative) \]

Where:

\( x \) — The value at which to evaluate the distribution
\( df \) — Degrees of freedom (positive integer)
\( cumulative \) — Logical value that determines the form of the function (TRUE for CDF, FALSE for PDF)

Explanation: The T-distribution approaches the normal distribution as the degrees of freedom increase. For small sample sizes, it provides a more accurate estimation than the normal distribution.

3. Importance of T-distribution

Details: The T-distribution is fundamental in hypothesis testing, confidence interval estimation, and in conducting t-tests. It's particularly important when working with small sample sizes (typically n < 30) where the population standard deviation is unknown.

4. Using the Calculator

Tips: Enter the x value, degrees of freedom (must be a positive integer), and select whether you want the cumulative distribution function (CDF) or probability density function (PDF).

5. Frequently Asked Questions (FAQ)

Q1: What is the difference between CDF and PDF?
A: CDF (Cumulative Distribution Function) gives the probability that a random variable is less than or equal to a certain value. PDF (Probability Density Function) gives the relative likelihood of a random variable taking on a given value.

Q2: How do I determine degrees of freedom?
A: For a single sample t-test, df = n - 1 where n is the sample size. For other tests, the calculation may differ.

Q3: When should I use the T-distribution instead of the Normal distribution?
A: Use the T-distribution when the sample size is small (typically n < 30) and the population standard deviation is unknown.

Q4: What happens as degrees of freedom increase?
A: As degrees of freedom increase, the T-distribution approaches the standard normal distribution.

Q5: Can I use this for one-tailed and two-tailed tests?
A: Yes, the calculator provides the fundamental values that can be used for both one-tailed and two-tailed hypothesis tests.