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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

STA 291Summer 2010

Lecture 21

Dustin Lueker


Testing difference between two population proportions
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
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 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)

STA 291 Summer 2010 Lecture 21


Example cont
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 samples1
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 samples2
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 samples3
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
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


Example1
Example Populations

STA 291 Summer 2010 Lecture 21


Example2
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
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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