1. Introduction to the t Test Part 2: Dependent Samples
Chapter 7
Introduction to the t Test
Part 2: Dependent Samples
Sept. 30, 2014

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 S2 for difference scores, Find SM 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: City July 2001
Fresno 9 2
Merced 10 4
Bakersfld 8
Stockton 1
Note: use alpha
= .01

5. (cont.)
Research hyp: there will be a decrease in litter from time1 to time 2 (2 < 1…or 1 - 2 > 0)
Null hyp: there will be no difference/effect (2 = 1, or 1 - 2 = 0)
Will need Difference scores for each city, need S2 and SM based on difference scores
S2 =  (X-M)2 / N-1
SM = sqrt (S2 / N)

6. (cont.) City July 01 July 02 Diff (01 – 02) (X-M)2 Fresno 9 2 7
(7-5)2 = 4
Merced 10 4 6
(6-5)2 = 1
Bksfld 8 -1
(-1-5)2 = 36
Stockton 1
(8-5)2 = 9
M = 5
 (X-M)2 = 50

7. (cont.) Find S2 and SM 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 (
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 (
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, 1st section is “Paired Samples Stats”, look for means for ‘sats4’ and ‘sats5’ – this is what we’re comparing
In 3rd 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?