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STA 291 Fall 2009

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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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