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STA 291 Fall 2009

STA 291 Fall 2009. Lecture 22 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 Fall 2009

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  1. STA 291Fall 2009 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 Fall 2009 Lecture 22

  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 Fall 2009 Lecture 22

  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 Fall 2009 Lecture 22

  5. Small Sample Test for Two Means, Equal Variances • Test Statistic • Degrees of freedom • n1+n2-2 STA 291 Fall 2009 Lecture 22

  6. Small Sample Confidence Interval for Two Means, Equal Variances • Degrees of freedom • n1+n2-2 STA 291 Fall 2009 Lecture 22

  7. Small Sample Test for Two Means, Unequal Variances • Test statistic • Degrees of freedom STA 291 Fall 2009 Lecture 22

  8. Small Sample Confidence Interval for Two Means, Unequal Variances STA 291 Fall 2009 Lecture 22

  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 Fall 2009 Lecture 22

  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 Fall 2009 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 Fall 2009 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 Fall 2009 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 Fall 2009 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 Fall 2009 Lecture 22

  15. 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 Fall 2009 Lecture 22

  16. Example STA 291 Fall 2009 Lecture 22

  17. 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 Fall 2009 Lecture 22

  18. 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 Fall 2009 Lecture 22

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