Theorem : If and and in addition, and are independent, then
the random variable defined by
has a -distribution with degrees of freedom.

This theorem connects the normal and chi-squared distributions through the -distribution, which is a fundamental concept in statistics, especially in hypothesis testing and confidence intervals for small sample sizes. The derivation stems from combining a standard normal variable with the square root of a chi-squared variable scaled by its degrees of freedom, showcasing that the ratio creates a new distribution. Grab a cup of coffee and think about how often this distribution pops up in real-life scenarios like estimating the mean of a small sample!

In practical applications, the -distribution is crucial when working with small sample sizes, particularly in situations where population variance is unknown. For example, if you’re conducting a survey of 15 students’ test scores and want to determine a confidence interval for the average score, using the -distribution allows you to account for added uncertainty due to the limited data. Remember to use the right degrees of freedom related to your sample size—it’s key to getting accurate results!