Two-Sample Tests

Important

not exam relevant in SS2024

Inference Based on Two Samples

• 2 samples needed
  • comparison between these 2 samples
  • e.g. control group vs medicated group in medical trials
• ${\mu }_1-{\mu }_2$ … difference between means
• ${\sigma }_1$ and ${\sigma }_2$ for deviations

Large Samples

• $z=\frac{({\bar{x}}_1-{\bar{x}}_2)-d_0}{{\sigma }_{{\bar{x}}_1-{\bar{x}}_2}}$
• with $z={\sigma }_{{\bar{x}}_1-{\bar{x}}_2}=\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}$
• two-tailed tests: $H_0={\mu }_1-{\mu }_2=d_0$

Small Samples

• pooled estimator: $s_p^2=\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}$
• t-statistic $t=\frac{({\bar{x}}_1-{\bar{x}}_2)-d_0}{\sqrt{s_p^2(\frac{1}{n_1}+\frac{1}{n_2})}}$
• two-tailed tests: $H_0={\mu }_1-{\mu }_2=d_0$

Paired

• wontfix zoned out from here on