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STA258 Lecture 06

Lecture

STA258 Lecture 06 Raw

• Scope:: Random sampling from $N(\mu ,{\sigma }^2)$, replacing $\sigma$ with $S$, leading to T Distribution and F Distribution.
• Core pipeline::
  • Start:: $Y_1,\dots ,Y_n\stackrel{iid}{\sim }N(\mu ,{\sigma }^2)$.
  • Known $\sigma$:: Standardize $\overline{Y}$ to get a $Z$-statistic.
  • Unknown $\sigma$:: Replace $\sigma$ by $S$ to get a $T$-statistic.
  • Variance ratios:: Compare $S_1^2$ and $S_2^2$ via an $F$-statistic.
• Atomic notes created from this lecture::
  • Student t statistic (unknown sigma)
  • t distribution construction (Z over sqrt(W over nu))
  • Moments of Student t distribution
  • F distribution construction (ratio of scaled chi-squares)
  • Variance ratio of two normal samples is F
  • F critical value lookup (F_5,9 example)
• Links to prerequisite notes already in vault::
  • Distributions:: Normal Distribution, Chi-squared Distribution, T Distribution, F Distribution.
  • Estimators:: Sample Variance, Marginal Distribution of Sample Variance.
  • Inequalities/tools:: Jensen's Inequality, Gamma Function, Gamma Distribution.

Post-Lecture

• STA258 Post-Lecture 06