Sta 291 summer 2010
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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 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

  • 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.)

  • “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

  • 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

  • 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

  • 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

  • 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

STA 291 Summer 2010 Lecture 21


Example2

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

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


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