Two sample setting

Setting: two independent samples

$Y_1,\dots ,Y_n$ i.i.d from population with c.d.f $F_Y$, and
$X_1,\dots ,X_m$ i.i.d from population with c.d.f $F_X$

Parameter: now focus on some comparison between the two populations $F_Y$ and $F_X$

Alternative view

Setting: two independent samples

$$\begin{array}{c}(Y_1,G_1),(Y_2,G_2),\dots ,(Y_n,G_n),(Y_{n+1},G_{n+1}),\dots ,(Y_{n+m},G_{n+m})\end{array}$$

where $G$ is a binary grouping variable which indicates which population the observation came from: $$G_i=\{\begin{array}{cc}0,& \text{observation from}Y\\ 1,& \text{observation from}X\end{array}$$

Two views are equivalent

Depending on sampling scheme one view may seem more natural:

• I sample 40 OSU graduate students and 20 OSU undergraduate students:

  • $Y_i$ = graduate student time to complete 1 mile run, $i=1,\dots ,40$
  • $X_i$ = undergraduate student time to complete 1 mile run, $i=1,\dots ,20$

• I sample 60 OSU students and record:

  • $Y_i$ = time to complete 1 mile run, $i=1,\dots ,60$
  • $G_i$ = student’s level (0 = graduate, 1 = undergraduate), $i=1,\dots ,60$

In second view, if we condition on the counts in each group, inference is the same as first view.

Two sample inference for difference in population means

To compare population means: ${\mu }_Y=E\left(Y_i\right)$, ${\mu }_X=E\left(X_i\right)$, we might look at their difference:

$$\delta ={\mu }_Y-{\mu }_X$$

(In alternative view: equivalent to $\delta =E(Y_i|G_i=0)-E(Y_i|G_i=1)$)

• Estimate for $\delta$
• Test for $H_0:\delta ={\delta }_0$
• Confidence interval for $\delta$

Difference in sample means

It seems reasonable to use:

$$\hat{\delta }=\overline{Y}-\overline{X}$$ as a good starting point for inference on $\delta ={\mu }_X-{\mu }_Y$.

Complete worksheet (Charlotte will provide)

Leads to two sample Z-test and intervals

Assume known population variances: $Var\left(Y_i\right)={\sigma }_Y^2$ $Var\left(X_i\right)={\sigma }_X^2$.

$$Z\left({\delta }_0\right)=\frac{(\overline{Y}-\overline{X})-{\delta }_0}{\sqrt{{\sigma }_Y^2/n+{\sigma }_X^2/m}}$$

Reference Distribution: If null hypothesis $H_0:\delta ={\delta }_0$ is true, then $$Z\left({\delta }_0\right)\dot{\sim }N(0,1)$$

Rejection Regions:

• $H_A:\delta >{\delta }_0$, reject $H_0$ for $Z\left({\delta }_0\right)>z_{1-\alpha }$
• $H_A:\delta <{\delta }_0$, reject $H_0$ for $Z\left({\delta }_0\right)<z_{\alpha }$
• $H_A:\delta \ne {\delta }_0$, reject $H_0$ for $|Z\left({\delta }_0\right)|>z_{1-\alpha /2}$

Leads to two sample Z-test and intervals

$(1-\alpha )100$% Confidence interval for $\delta ={\mu }_Y-{\mu }_X$

$$(\overline{Y}-\overline{X})\pm z_{1-\alpha /2}\sqrt{\frac{{\sigma }_Y^2}{n}+\frac{{\sigma }_X^2}{m}}$$

Next time…

What if population variances aren’t known?