# STA 291 Summer 2010 - PowerPoint PPT Presentation

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STA 291 Summer 2010. Lecture 21 Dustin Lueker. Testing Difference Between Two Population Proportions. Similar to testing one proportion Hypotheses are set up like two sample mean test H 0 :p 1 -p 2 =0 Same as H 0 : p 1 =p 2 Test Statistic.

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STA 291 Summer 2010

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## STA 291Summer 2010

Lecture 21

Dustin Lueker

### 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 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

• H0: μ1-μ2=0

• Same as H0: μ1=μ2

• Test statistic

STA 291 Summer 2010 Lecture 21

### 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 Summer 2010 Lecture 21

### 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 Summer 2010 Lecture 21

### 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 Summer 2010 Lecture 21

### 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 Summer 2010 Lecture 21

### 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 Summer 2010 Lecture 21

### Example

• 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

STA 291 Summer 2010 Lecture 21

### Example

• 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

• 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