1. STA 291 Spring 2008
Lecture 22
Dustin Lueker

2. Testing Difference Between Two Population Proportions
Similar to testing one proportion
Hypotheses are set up like two sample mean test
H0:p1-p2=0
Same as H0: p1=p2
Test Statistic
STA 291 Spring 2008 Lecture 21

3. Testing the Difference Between Means from Different Populations
Hypothesis involves 2 parameters from 2 populations
Test statistic is different
Involves 2 large samples (both samples at least 30)
One from each population
H0: μ1-μ2=0
Same as H0: μ1=μ2
Test statistic
STA 291 Spring 2008 Lecture 21

4. Small Sample Tests for Two Means
Used when comparing means of two samples where at least one of them is less than 30
Normal population distribution is assumed for both samples
Equal Variances
Both groups have the same variability
Unequal Variances
Both groups may not have the same variability
STA 291 Spring 2008 Lecture 21

5. Small Sample Test for Two Means, Equal Variances
Test Statistic
Degrees of freedom
n1+n2-2
STA 291 Spring 2008 Lecture 21

6. Small Sample Confidence Interval for Two Means, Equal Variances
Degrees of freedom
n1+n2-2
STA 291 Spring 2008 Lecture 21

7. Small Sample Test for Two Means, Unequal Variances
Test statistic
Degrees of freedom
STA 291 Spring 2008 Lecture 21

8. Small Sample Confidence Interval for Two Means, Unequal Variances
STA 291 Spring 2008 Lecture 21

9. Method 1 (Equal Variances) vs. Method 2 (Unequal Variances)
How to choose between Method 1 and Method 2?
Method 2 is always safer to use
Definitely use Method 2
If one standard deviation is at least twice the other
If the standard deviation is larger for the sample with the smaller sample size
Usually, both methods yield similar conclusions
STA 291 Spring 2008 Lecture 21

10. Comparing Dependent Samples
Comparing dependent means
Example
Special exam preparation for STA 291 students
Choose n=10 pairs of students such that the students matched in any given pair are very similar given previous exam/quiz results
For each pair, one of the students is randomly selected for the special preparation (group 1)
The other student in the pair receives normal instruction (group 2)
STA 291 Spring 2008 Lecture 22

11. Example (cont.) “Matches Pairs” plan
Each sample (group 1 and group 2) has the same number of observations
Each observation in one sample ‘pairs’ with an observation in the other sample
For the ith pair, let
Di = Score of student receiving special preparation – score of student receiving normal instruction
STA 291 Spring 2008 Lecture 22

12. Comparing Dependent Samples
The sample mean of the difference scores is an estimator for the difference between the population means
We can now use exactly the same methods as for one sample
Replace Xi by Di
STA 291 Spring 2008 Lecture 22

13. Comparing Dependent Samples
Small sample confidence interval
Note:
When n is large (greater than 30), we can use the z- scores instead of the t-scores
STA 291 Spring 2008 Lecture 22

14. Comparing Dependent Samples
Small sample test statistic for testing difference in the population means
For small n, use the t-distribution with df=n-1
For large n, use the normal distribution instead (z value)
STA 291 Spring 2008 Lecture 22

15. Example Student 1 2 3 4 5 6 7 8 9 10 Before 60 73 42 88 66 77 90 63 55
Ten college freshman take a math aptitude test both before and after undergoing an intensive training course
Then the scores for each student are paired, as in the following table
Student 1 2 3 4 5 6 7 8 9 10
Before 60 73 42 88 66 77 90 63 55 96
After 70 80 40 94 79 86 93 71 97
STA 291 Spring 2008 Lecture 22

16. Example
STA 291 Spring 2008 Lecture 22

17. Example
Student 1 2 3 4 5 6 7 8 9 10
Before 60 73 42 88 66 77 90 63 55 96
After 70 80 40 94 79 86 93 71 97
Compare the mean scores after and before the training course by
Finding the difference of the sample means
Find the mean of the difference scores
Compare
Calculate and interpret the p-value for testing whether the mean change equals 0
Compare the mean scores before and after the training course by constructing and interpreting a 90% confidence interval for the population mean difference
STA 291 Spring 2008 Lecture 22

18. Example Output from Statistical Software Package SAS N 10 Mean 7
Std Deviation
Tests for Location: Mu0=0
Test Statistic p Value------
Student's t t Pr > |t|
Sign M Pr >= |M|
Signed Rank S Pr >= |S|
STA 291 Spring 2008 Lecture 22

19. Reducing Variability
Variability in the difference scores may be less than the variability in the original scores
This happens when the scores in the two samples are strongly associated
Subjects who score high before the intensive training also dent to score high after the intensive training
Thus these high scores aren’t raising the variability for each individual sample
STA 291 Spring 2008 Lecture 22