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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

- H0:p1-p2=0
- Test Statistic

STA 291 Summer 2010 Lecture 21

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

- Involves 2 large samples (both samples at least 30)

- Test statistic is different
- H0: μ1-μ2=0
- Same as H0: μ1=μ2
- Test statistic

STA 291 Summer 2010 Lecture 21

Comparing Dependent Samples Populations

- 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)

- Example

STA 291 Summer 2010 Lecture 21

Example (cont.) Populations

- “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 Summer 2010 Lecture 21

Comparing Dependent Samples Populations

- 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 Summer 2010 Lecture 21

Comparing Dependent Samples Populations

- 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 Summer 2010 Lecture 21

Comparing Dependent Samples Populations

- 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 Summer 2010 Lecture 21

Example Populations

- 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

STA 291 Summer 2010 Lecture 21

Example Populations

STA 291 Summer 2010 Lecture 21

Example Populations

- 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 Summer 2010 Lecture 21

Reducing Variability Populations

- 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 tend to score high after the intensive training
- Thus these high scores aren’t raising the variability for each individual sample

STA 291 Summer 2010 Lecture 21

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