STAT 211 Topic 8

Lecture 16 Notes

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Comparing 2 Samples

In topic 7, we did confidence intervals and hypothesis tests for a single sample with the population. Now we're comparing 2 samples with each other.

• Two samples ($X_{1..m}$ and $Y_{1..n}$) must be independent and from separate populations

Look at difference in averages $\overline{X}-\overline{Y}$ for estimate of difference of means ${\mu }_X-{\mu }_Y$ and compare these with the proposed difference ${\mathrm{\Delta }}_0$:

$\begin{array}{rl}E(\overline{X}-\overline{Y})& =E(\overline{X})+E(\overline{Y})={\mu }_X-{\mu }_Y\\ V(\overline{X}-\overline{Y})& =V(\overline{X})+V(\overline{Y})=\frac{{\sigma }_X^2}{m}+\frac{{\sigma }_Y^2}{n}\end{array}$

Cases 1&2: Normal Population or Large Sample

When the sample sizes are large, the distribution of $\overline{X}-\overline{Y}$ is also Normal.

Use the same methods as in STAT 211 Topic 7, substituting $\mu ={\mu }_X-{\mu }_Y$, ${\mu }_0={\mathrm{\Delta }}_0$ and ${\sigma }^2=\frac{{\sigma }_X^2}{m}+\frac{{\sigma }_Y^2}{n}$.

$\begin{array}{rl}\overline{X}-\overline{Y}& \sim \mathrm{N}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}({\mu }_X-{\mu }_Y,\frac{{\sigma }_X^2}{m}+\frac{{\sigma }_Y^2}{n})\\ z& =\frac{\overline{X}-\overline{Y}-{\mathrm{\Delta }}_0}{\sqrt{\frac{{\sigma }_X^2}{m}+\frac{{\sigma }_Y^2}{n}}}\end{array}$

Example

A realtor from the northeast claims that houses are more valuable (higher sales price) than anywhere else in the US.

• ${\mu }_1$: average sales price in Northeast
• ${\mu }_2$: average sales price anywhere but Northeast
• $H_0:{\mu }_1-{\mu }_2=0$
• $H_a:{\mu }_1-{\mu }_2>0$
• $z=\frac{\overline{X}-\overline{Y}-0}{\sqrt{s_X^2/m+s_Y^2/n}}$

Case 3: Small Sample from Normal Population

We do different things depending on the two population variances (not necessarily known):

$\{\begin{array}{ll}{\sigma }_X^2={\sigma }_Y^2& \mathrm{p}\mathrm{o}\mathrm{o}\mathrm{l}\mathrm{e}\mathrm{d}\mathrm{t}\mathrm{-}\mathrm{t}\mathrm{e}\mathrm{s}\mathrm{t}\\ {\sigma }_X^2\ne {\sigma }_Y^2& \mathrm{u}\mathrm{n}\mathrm{p}\mathrm{o}\mathrm{o}\mathrm{l}\mathrm{e}\mathrm{d}\mathrm{t}\mathrm{-}\mathrm{t}\mathrm{e}\mathrm{s}\mathrm{t}\end{array}$

If smaller σ is greater than half of the bigger σ, then we say that they are the same:

${\sigma }_1>\frac{{\sigma }_2}{2}$

Pooled Sample Variance

$s_p^2=\frac{(m-1)s_X^2+(n-1)s_Y^2}{n+m-2}$

Use t-test as normal with df $\nu =n+m-2$

$t=\frac{\overline{X}-\overline{Y}-{\mathrm{\Delta }}_0}{s_p\sqrt{\frac{1}{m}+\frac{1}{n}}}$

Unpooled Sample Variance

t-test would be normal as expected, but degrees of freedom is more complicated (this is why the pooled test is more common):

$\begin{array}{rl}t& =\frac{\overline{X}-\overline{Y}-{\mathrm{\Delta }}_0}{s_p\sqrt{\frac{1}{m}+\frac{1}{n}}}\\ \nu & =\frac{{(\frac{s_X^2}{m}+\frac{s_Y^2}{n})}^2}{\frac{{(s_X^2/m)}^2}{m-1}+\frac{(s_Y^2/n)}{n-1}}\end{array}$

Paired Data

When two samples are related to each other by a third variable (e.g. mother of two children, student who takes two exams, etc.)

Use paired t-test:

1. calculate differences between samples: $D_i=X_i-Y_i$
2. calculate average and standard deviation of the differences
3. use regular t-test on $t=\frac{\overline{D}-{\mathrm{\Delta }}_0}{s_D/\sqrt{n}}$

Confidence Interval

$\overline{D}\pm t_{\alpha /2,n-1}\frac{s_D}{\sqrt{n}}$

Comparing Two Population Proportions

Lecture 17 Notes

Given two sample proportions ${\hat{p}}_X$ and ${\hat{p}}_Y$ from two different populations

We are interested in the difference between the two proportions (Normal distribution). Therefore, we can standardize and perform a z-test with the following parameters:

• H₀: $p_X-p_Y={\mathrm{\Delta }}_0$
• H_a: $\{\begin{array}{l}{p}_X\ne p_Y\\ p_X>p_Y\\ p_X<p_Y\end{array}$
• Test Statistic: $z=\frac{{\hat{p}}_X-{\hat{p}}_Y-{\mathrm{\Delta }}_0}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{m}+\frac{1}{n})}}$

The only problem is that we don't know $p_X$ and $p_Y$ so we estimate it with the second equation.

$\hat{p}=\frac{m}{m+n}{\hat{p}}_1+\frac{n}{m+n}{\hat{p}}_2$

Review

(See STAT 211 Topic 3→)

In general, the exact distribution of our sample proportions are Binomial:

$\begin{array}{rl}{\hat{p}}_1& =\frac{{\sum }_{i=1}^m{X}_i}{m}\\ {\hat{p}}_2& =\frac{{\sum }_{i=1}^n{Y}_i}{n}\end{array}$	$\begin{array}{r}{\sum }_{i=1}^m{X}_i\sim \mathrm{B}\mathrm{i}\mathrm{n}(m,p_1)\\ {\sum }_{i=1}^n{Y}_i\sim \mathrm{B}\mathrm{i}\mathrm{n}(m,p_2)\end{array}$
$\begin{array}{rl}E\left({\sum }_{i=1}^m{X}_i\right)& =m{p}_1\\ E\left({\sum }_{i=1}^n{Y}_i\right)& =m{p}_2\end{array}$	$\begin{array}{rl}V\left({\sum }_{i=1}^m{X}_i\right)& =m{p}_1(1-p_1)\\ V\left({\sum }_{i=1}^n{Y}_i\right)& =m{p}_2(1-p_2)\end{array}$

${\hat{p}}_1-{\hat{p}}_2\sim \mathrm{N}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}(p_1-p_2,\frac{p_1(1-p_1)}{m}+\frac{p_2(1-p_2)}{n})$

Comparing Two Variances

Instead of using ${\chi }^2$ distribution, we use $F$ distribution (F-test).

If we have two samples $X_1,\dots ,X_m$ and $Y_1,\dots ,Y_n$

• H₀: ${\sigma }_X^2={\sigma }_Y^2$
• Test statistic: $F=\frac{s_X^2}{s_Y^2}$

Rejection Range:

Ha	Reject H0 if
${\sigma }_X^2>{\sigma }_Y^2$	$F>F_{a,m-1,n-1}$
${\sigma }_X^2<{\sigma }_Y^2$	$F<F_{a,m-1,n-1}$
${\sigma }_X^2\ne {\sigma }_Y^2$	$F>F_{a/2,m-1,n-1}$ or $F<F_{1-a/2,m-1,n-1}$

Confidence Interval

$(\frac{s_X^2/s_Y^2}{F_{a/2,m-1,n-1}},\frac{s_X^2/s_Y^2}{F_{1-a/2,m-1,n-1}})$