Two Sample T-test for Comparing Two Means

1.Two Sample T-test for Comparing Two Means  
When do we use?
- Suppose we have two random samples which are from population 1(X) and population 2 (Y),
our interest is to compare those two population means. In this case, small sample size will be fine. (When we have large enough sample size, then we use Z-test.)  

Assumption?
- The two samples are identically independently distributed(iid) from approximately Normal populations. And Two samples are independent of each other. 

Hypothesis?
- There is no difference between two population means. $H_{o}=\bar{X}-\bar{Y}=D_{0}=0$

Test statistics?
- $t_{obs}= \frac{(\bar{X}-\bar{Y})-D_{0}}{se(\bar{X}-\bar{Y})}$ Wait!! How do we estimate the standard error of mean difference?


1.1. Pooled Two Sample T-test
When the population variances can be assumed to be equal, we use pooled two sample t-test!
Pooled variance estimate : $S_{p}^2= \frac{ (n_{x}-1)S_{x}^2 + (n_{y}-1)S_{y}^2 }{ n_{x}+n_{y}-2}$

Test statistics?
- $t_{obs}= \frac{(\bar{X}-\bar{Y})-D_{0}}{\sqrt{S_{p}^2( \frac{1}{n_{x}} +\frac{1}{n_{y}})}} \sim t_{n_{x}+n_{y}-2}$, Under the null hypothesis.


1.2. Two Sample T-test using Satterthwaite Approximation  
When the population variances cannot be assumed to be equal, we use Satterthwaite approximation.

Test statistics?
- $t_{obs}= \frac{(\bar{X}-\bar{Y})-D_{0}}{ \sqrt{ \frac{S_{x}^2}{n_{x}}+ \frac{S_{y}^2}{n_{y}}}} \sim t_{\gamma}$, Under the null hypothesis, where  $\gamma = \frac{ (\frac{S_{x}^2}{n_{x}} + \frac{S_{y}^2}{n_{y}})^2}{\frac{(\frac{S_{x}^2}{n_{x}})^2}{n_{x}-1} + \frac{ (\frac{S_{y}^2}{n_{y}})^2}{n_{y}-1} }$ d/f.
Here, this degrees of freedom is calculated by Satterthwaite approximation which rounds down to the nearest integer. 



2. Case Study: The Spock Conspiracy Trial
Description

In 1968, Dr. Benjamin Spock was tried in Boston on charges of conspiring to violate the Selective Service Act by encouraging young men to resist being drafted into military service for Vietnam. The defence in the case challenged the method of jury selection claiming that women were underrepresented. Boston juries are selected in three stages. First 300 names are selected at random from the City Directory, then a venire of 30 or more jurors is selected from the initial list of 300 and finally, an actual jury is selected from the venire in a nonrandom process allowing each side to exclude certain jurors. There was one woman on the venire and no women on the final list. The defence argued that the judge in the trial had a history of venires in which women were systematically underrepresented and compared the judge's recent venires with the venires of six other Boston area district judges.

 

Reference: http://www.inside-r.org/node/159733 

The main question is there is evidence that women underrepresented on Spock judges venire when compared to other judges?

*Two sample t-test in R 
If you are not ready...
> install.packages( "Sleuth3" )
> library(Sleuth3)
> attach(case0502)
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* SLR in R 
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* Two Sample t-test in SAS
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