1. Chapter 7 Introduction to the t Test Part 2: Dependent Samples March 4, 2008

2. t Test for Dependent Means Unknown population mean and variance Two scores for each person –Repeated measures design –aka “Paired Samples t-test” in SPSS Same procedure as t test for single sample, except –Use difference scores –Assume that the comparison mean is 0

3. t Test for Dependent Means Difference scores –For each person, subtract one score from the other –Carry out hypothesis testing with the difference scores Find S 2 for difference scores, Find S M for difference scores Comparison population of difference scores will always have a mean of 0 –That is, the relevant µ for the comparison with M will be 0. –This will always be stated in your null hypothesis for a dependent samples t-test

4. Example #5 in Ch. 7 – program to decrease litter: CityJuly 2001July 2002 Fresno92 Merced104 Bakersfld89 Stockton91 Note: use alpha =.01

5. (cont.) Research hyp: there will be a decrease in litter from time1 to time 2 (  2 0) Null hyp: there will be no difference/effect (  2 =  1, or  1 -  2 = 0) Will need Difference scores for each city, need S 2 and S M based on difference scores S 2 =  (X-M) 2 / N-1 S M = sqrt (S 2 / N)

6. (cont.) CityJuly 01 July 02 Diff (01 – 02) (X-M) 2 Fresno927(7-5) 2 = 4 Merced1046(6-5) 2 = 1 Bksfld89(-1-5) 2 = 36 Stockton918(8-5) 2 = 9 M = 5  (X-M) 2 = 50

7. (cont.) Find S 2 and S M Find observed t from sample: Critical t? Draw distribution… Compare obtained t and critical… Conclusion?

8. Effect Size for t Test for Dependent Means If calculating before data collection,  2 will always be 0,  1 is the expected mean difference in our sample (pre/post-test),  is expected SD of difference scores If calculating after data collection,  2 is still 0,  1 is the actual mean difference (pre/post-test),  is actual SD of difference scores (use S) Use same effect size standards as earlier, small d = |.2|, medium d = |.5|, large d >= |.8|

9. Approximate Power for t Test for Dependent Means (.05 significance level) Note: Table 7-9 shows power in body of table, you need to know N (rows), and effect size (columns)

10. Approximate Sample Size Needed for 80% Power (.05 significance level – Table 7-10) This table shows N needed for 80% power (rule of thumb) given different expected effect sizes.

11. SPSS: Dependent Means t-test Using SATS data, assume ‘sats4’ is pre-semester rating of difficulty of statistics, ‘sats5’ is post- semester rating of difficulty Is there a difference in pre/post semester? –Research hyp: Post should be lower than pre (diff >0) –Null hyp: No difference in pre/post (diff = 0) Analyze  Compare Means  Paired Samples t- test –Pop-up window, under ‘paired variables’, select‘Sats4’ for var1, ‘Sats5’ for var2,  OK

12. (cont.) In output, 1 st section is “Paired Samples Stats”, look for means for ‘sats4’ and ‘sats5’ – this is what we’re comparing In 3 rd section, “Paired Samples Test”, note mean difference score, t observed, df, and ‘sig (2-tail)’. –Mean difference score is compared to 0 –Sig (2 tail) should be compared to alpha level (e.g.,.05). –If ‘sig’ value < alpha  reject Null This example?