Formulas of HT

2. CI for Mean ($\sigma$ known): $\bar{X}\pm z^{\ast }\cdot \frac{\sigma }{\sqrt{n}}$

3. CI for Mean ($\sigma$ unknown): $\bar{x}\pm t^{\ast }\cdot \frac{s}{\sqrt{n}}$

4. CI for Proportion: $\hat{p}\pm z^{\ast }\cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

5. Sample Mean: $\bar{X}=\frac{1}{n}{\sum }_{i=1}^n{X}_i$

6. Standard Error (SE) for Mean ($\sigma$ known): $SE=\frac{\sigma }{\sqrt{n}}$

7. Standard Error (SE) for Mean ($\sigma$ unknown): $SE=\frac{s}{\sqrt{n}}$ (Used in example, page 8)

8. Z-interval Formula: $[\bar{X}-z_{\alpha /2}\frac{\sigma }{\sqrt{n}},\bar{X}+z_{\alpha /2}\frac{\sigma }{\sqrt{n}}]$

9. Pivotal Quantity (Normal, Known ${\sigma }^2$): $Q=\frac{\bar{X}-\mu }{\sigma /\sqrt{n}}\sim N(0,1)$

10. Pivotal Quantity (Normal, Unknown ${\sigma }^2$): $Q=\frac{\bar{X}-\mu }{s/\sqrt{n}}\sim t_{n-1}$

11. Z-statistic (Single Mean): $Z=\frac{\bar{X}-{\mu }_0}{\sigma /\sqrt{n}}$ (Eq 5, 9)

12. T-statistic (Single Mean): $t=\frac{\bar{X}-{\mu }_0}{s/\sqrt{n}}$ (Eq 6, 12)

13. Z-statistic (Two Means): $Z=\frac{({\bar{X}}_1-{\bar{X}}_2)-{\Delta }_0}{\sqrt{\frac{{\sigma }_1^2}{n_1}+\frac{{\sigma }_2^2}{n_2}}}$ (Eq 22)

14. T-statistic (Two Means, Equal Variance): $t=\frac{({\bar{X}}_1-{\bar{X}}_2)-{\Delta }_0}{\sqrt{s_p^2(\frac{1}{n_1}+\frac{1}{n_2})}}$ (Eq 31)

15. Pooled Variance: $s_p^2=\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}$ (Eq 32)

16. Welch’s T-statistic: $t^{\prime }=\frac{({\bar{X}}_1-{\bar{X}}_2)-{\Delta }_0}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}$ (Eq 46)

17. Welch-Satterthwaite Degrees of Freedom: $df\approx \frac{(\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}{)}^2}{\frac{(s_1^2/n_1{)}^2}{n_1-1}+\frac{(s_2^2/n_2{)}^2}{n_2-1}}$ (Eq 47)