Student's t-Distribution

Probability · Axiom Academy

LESSON Student's t-Distribution For inference when the population variance is unknown 1. Construction of t-Distribution The t-distribution arises when we divide a standard normal by the square root of an independent chi-squared variable divided by its degrees of freedom. This exactly describes the distribution of a sample mean when σ is unknown and must be estimated. The t-distribution has thicker tails than the normal distribution, reflecting the added uncertainty from estimating σ. With fewer degrees of freedom, tails are heavier. As df → ∞, the t-distribution converges to N(0,1). The t-distribution is fundamental for confidence intervals and hypothesis tests about means when σ is unknown. The test statistic t = (x̄ - μ)/(s/√n) follows a t-distribution with n-1 degrees of freedom, allowing us to construct t-tests and confidence intervals.

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