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  • STA260 Lecture 04

T Distribution

• Ratio of a random variable from a Standard Normal Distribution and the squared root of another random variable from a Chi-squared Distribution adjusted by its degrees of freedom.
• Both must be independent.
• $T=\frac{Z}{\sqrt{\frac{U}{\nu }}}\sim t_{(\nu )}$
  • $Z\sim N(0,1)$
  • $U\sim {\chi }_{(\nu )}^2$
  • If $Z$ and $U$ are independent, then we can have this $t_{(\nu )}$
  • As $\nu \to \mathrm{\infty }$, then the T Distribution approaches $N(0,1)$
    • $\nu >30$ is a close approximation of the normal.
• Probability Density Function is annoying.
• Example:
  • $T=\frac{\overline{X}-\mu }{\frac{S}{\sqrt{n}}}={\displaystyle \frac{\frac{\overline{X}-\mu }{\frac{\sigma }{\sqrt{n}}}}{\sqrt{\frac{\frac{(n-1)S^2}{{\sigma }^2}}{n-1}}}}=\frac{Z}{\sqrt{\frac{U}{n-1}}}\sim t_{n-1}$
    • We have $n-1$ because we use $S$, $S$ is a model, so we lose a degree of freedom because it's not exactly $\sigma$, the reality.
  • Use $Z$ when $\sigma$ is known, or $T$ when $\sigma$ is unknown.
• Another Example:
  • $X\sim N(\mu ,{\sigma }^2)$
  • $n=6$
  • Find $\overline{X}$ being between $\frac{2S}{\sqrt{n}}$ of population mean $\mu$
  • $P(\frac{-2S}{\sqrt{n}}\le \overline{X}-\mu \le \frac{2S}{\sqrt{n}})$
  • $P(-2<{\displaystyle \frac{\overline{X}-\mu }{\frac{S}{\sqrt{n}}}}<2)$
    • $\sim t_5$
  • $P(-2<t_5<2)$
  • We get from the table, $P(-2.015<t_5<2.015)=0.9$