Chapter 7
Introduction to the t Test
Part 2: Dependent Samples
March 4, 2008
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
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
Example
• #5 in Ch. 7 – program to decrease litter:
City
July 2001 July 2002
Fresno
9
2
Merced
10
4
Bakersfld 8
9
Stockton
1
9
Note: use alpha
= .01
(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)
(cont.)
(X-M)2
Fresno
July July Diff
01
02
(01 – 02)
9
2
7
Merced
10
4
6
(6-5)2 = 1
Bksfld
8
9
-1
(-1-5)2 = 36
Stockton
9
1
8
(8-5)2 = 9
City
M=5
(7-5)2 = 4
 (X-M)2 = 50
(cont.)
• Find S2 and SM
• Find observed t from sample:
M 
t
SM
• Critical t? Draw distribution…
• Compare obtained t and critical…
• Conclusion?
Effect Size for
t Test for Dependent Means
1   2
d

• 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|
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)
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.
SPSS: Dependent Means t-test
• Using SATS data, assume ‘sats4’ is pre-semester
rating of difficulty of statistics, ‘sats5’ is postsemester 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 ttest
– Pop-up window, under ‘paired variables’, select‘Sats4’ for
var1, ‘Sats5’ for var2,  OK
(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?