The t-distribution, also known as Student's t-distribution, is a type of probability distribution that arises when estimating the mean of a normally distributed population in situations where the sample size is small and the population standard deviation is unknown. Here are key aspects of the t-distribution:
History
- The t-distribution was first introduced by William Sealy Gosset in 1908 under the pseudonym "Student." Gosset was a chemist working for the Guinness Brewery in Dublin, Ireland, and he developed this distribution to handle small sample sizes in quality control.
- His work was later refined and published by Ronald A. Fisher, who made the t-distribution widely known in statistical literature.
Characteristics
- The t-distribution is symmetric and bell-shaped like the normal distribution, but it has heavier tails, meaning it accounts for more variability due to small sample sizes.
- It is defined by its degrees of freedom (df), which is related to the sample size. As the degrees of freedom increase, the t-distribution approaches the normal distribution.
- Unlike the normal distribution, the t-distribution changes shape depending on the sample size, becoming more peaked and having fatter tails as the sample size decreases.
Applications
- Confidence Intervals: It's used to construct confidence intervals for the population mean when the population standard deviation is unknown.
- Hypothesis Testing: Particularly in t-tests, which compare means between two groups or assess the significance of regression coefficients.
- Small Sample Sizes: The t-distribution is especially useful when dealing with small sample sizes (typically less than 30), where the central limit theorem might not hold.
Formula
The probability density function for the t-distribution with ν degrees of freedom is:
\[ f(t) = \frac{\Gamma\left(\frac{\nu+1}{2}\right)}{\sqrt{\nu\pi}\,\Gamma\left(\frac{\nu}{2}\right)} \left(1+\frac{t^2}{\nu}\right)^{-\frac{\nu+1}{2}} \]
Comparison with Normal Distribution
External Links
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