Related Samples t-test
Wed, Apr 7th
Related Samples t
 Use when:
– 1 group of subjects is tested more than
once (e.g., pre-test / post-test)…or
– 2 groups of related (or matched)
subjects are measured once (e.g., twins)
 Uses same basic t formula, but focus
now on ‘difference’ scores
– Differences in pre- and post-test or
difference in each partner in a pair
Hypotheses
 Null hyp (Ho) indicates no difference (or no
effect), so we’ll hypothesize the mean
difference in the pop (D) = 0
 Ha indicates there will be a difference (2tailed, D not = 0) or there will be a
difference in a specific direction (1-tailed,
D > 0 or D < 0).
 We’ll reject Ho if | t obs | > | t critical|
(t observed is in critical region)
Ex: What is the effectiveness of a new program
to reduce litter? Measured aver litter in 4 cities in
’01 and ‘02
City
July 2001
July 2002
Fresno
9
2
Merced
10
4
Bakersfld
8
9
Stockton
9
1
Find D (difference score) for each city, and then
the average D (D bar)
City
D
Fresno
July
July
2001 2002
9
2
Merced
10
4
6
Bakersfld
8
9
-1
Stockton
9
1
8
7
D bar =
20/4 = 5
 Next, sum up the deviation scores
(D)
 Then find squared deviation scores for
each city (D2) – add a new column
 Then sum up the squared deviation
scores, (D2)
Here, D = 20, D2 = 150
 Use the T observed formula:
T formula for Related
Samples
Note: D always
 T observed = (Dbar - D)
=0
SDbar
Where SDbar (std error) = sqrt (SD2/ N) and
SD2 = [D2 – (D)2] / N-1
= 150 – (20)2 / 3 = -83.33
SDbar = sqrt (-83.33 / 4) = -4.56
 T obs = (5-0) / -4.56 = -1.09
Finding T critical
 Use the t table as before,
– Need alpha level & 1 or 2-tailed test?
– Need Df = N-1 (here N=pairs of scores or
total # participants if repeated measures)
 Ex) use  = .01, 1-tailed test (expected
litter to decrease), Df= 3
T critical = -4.541
T observed = -1.09
So fail to reject Ho (|t obs| < |t critical|)
There is no difference in litter before &
after new system  not effective
SPSS
 GSS98 dataset example…
 Analyze  Compare Means 
Paired Samples t-test
 Pop-up window, select ‘wife…’ for
var1, ‘children…’ for var2, then click
arrow to put them in “Paired vars”
box  OK
(cont.)
 In output, 1st section is “Paired Samples
Stats”, look for means for ‘wife…” and
‘children’ – 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
 Draw a conclusion about the pre/post
test scores.