TWO-SAMPLE t-TESTS
 Independent versus Related Samples
 Your two samples are independent if you randomly
assign individuals into the two treatment groups.
 Your samples are related if either
▪ Each person in sample A is matched to a partner in
sample B (matched samples) OR
▪ Each person in the study is measured under both
conditions (repeated measures)
INDEPENDENT SAMPLES t-TEST
The steps to conducting an independent samples t-test
are:
 State your research question hypotheses
 Determine your rejection rule
 Calculate the t-statistic
 Use your rejection rule to decide whether you
Reject the null hypothesis
Fail to reject the null hypothesis
INDEPENDENT SAMPLES t-TEST
Two-Tailed Test Hypotheses
 H0: m1 - m2 = 0
 H1: m1 - m2 ≠ 0
INDEPENDENT SAMPLES t-TEST
 Two-tailed Rejection Rule
If you are using the t-table, reject H0

▪
▪
if t(obt) > t(crit, a, n1 + n2 - 2) OR
if t(obt) < -t(crit, a, n1 + n2 - 2)
 If you are using the SPSS printout,
reject H0 if p < .05
INDEPENDENT SAMPLES t-TEST
One-tailed test where you believe the scores in
sample 1 will be greater than the scores in sample 2.
The hypotheses are


H0: m1 - m2 ≤ 0
H1: m1 - m2 > 0
INDEPENDENT SAMPLES t-TEST
One-Tailed Rejection Rule where you believe
the scores in sample 1 will be greater than
the scores in sample 2.
 If using the t-table, reject H0
 if t(obt) > t(crit, a, n1 + n2 - 2)

 If using the SPSS printout, reject Ho if p<
.05.
INDEPENDENT SAMPLES t-TEST
One-tailed test where you believe the scores in
sample 1 will be less than the scores in sample
2. The hypotheses are
H0: m1 - m2 ≥ 0
H1: m1 - m2 < 0
INDEPENDENT SAMPLES t-TEST
 One-tailed Rejection Rule where you believe
the scores in sample 1 will be less than the
scores in sample 2.
 If using the t-table, reject H0
 if t(obt) < -t(crit, a, n1 + n2 - 2)
 If using the SPSS printout, reject Ho if p < .05.
INDEPENDENT SAMPLES t-TEST
Calculating the independent samples t-statistic
1. Calculate the mean for each of the two samples
X1, X 2
2. Calculate the sum of squares for each of the two
samples. What’s a sum of squares? It’s the same as
the formula for the variance, except don’t do the final
step of dividing by n – 1. SS1 for the first sample and
SS2 for the second
2
(
X
)
SS  X 2 n
INDEPENDENT SAMPLES t-TEST
Calculating the independent samples t-statistic:
t obt 
X1 - X 2
 SS1  SS2  1
1 

 

 n1  n2 - 2  n1 n2 
Step 1. NOBODY PANIC!
INDEPENDENT SAMPLES t-TEST
Step 2. Find the difference between the group means.
Note: It doesn’t matter which group you designate as
sample 1 and which as sample 2, AS LONG AS you
take into consideration which group you mean when
you set up your hypotheses and rejection rules.
[Continued on next slide.]
INDEPENDENT SAMPLES t-TEST
Step 3a. Divide the number 1 by the number of
observations in the first sample (n1).
Step 3b. Divide the number 1 by the number of
observations in the second sample (n2).
Step 3c. Add the answers to Step 3a and Step 3b.
Refer to Step 1!
INDEPENDENT SAMPLES t-TEST
Step 4. Add the sum of squares for the first sample (SS1)
to the sum of squares for the second sample (SS2).
See slide #8 for information about calculating the sum
of squares.
Step 5. Find the degrees of freedom by adding the
number of observations in the first sample (n1) to the
number of observations in the second sample (n2)
and then subtracting the number 2. [Continued on
next slide.]
INDEPENDENT SAMPLES t-TEST
Step 6. Divide your answer from Step 4 by the answer
from Step 5.
Step 7. Multiply your answer from Step 6 to the answer
from Step 3c.
Refer to Step 1.
INDEPENDENT SAMPLES t-TEST
Step 8. Take the square root of your answer in Step 7.
Step 9. Divide your answer from Step 2 by the answer
from Step 8.
Compare your obtained t-statistic to the critical t-value
from your rejection rule and decide the appropriate
action.
INDEPENDENT SAMPLES t-TEST
 Find the appropriate t (crit) from the t-table in the back of
the book, using the correct bar at the top depending on a
one-tailed or a two-tailed test, a, and df = n1 + n2 - 2.
 OR use the significance level shown on the SPSS printout.
If it is less than .05, reject Ho.
 Calculate t (obt) using the two independent samples t-test.
 Make your decision based on your rejection rule.
EXAMPLE OF INDEPENDENT SAMPLES t-TEST
Independent Variable is brand of oven (two brands)
Dependent Variable is hours it worked until failure.
Brand A
Brand B
237
208
254
178
246
187
178
146
179
145
183
141
EXAMPLE OF INDEPENDENT SAMPLES t-TEST
Stuff we’ll need
Brand A
X
n
X
X 2
(X ) 2
SS = X n
2
Brand B
1,277
1,005
6
6
1,277 ÷ 6 = 212.833
1,005 ÷ 6 = 167.500
278,415
172,139
(1,277) 2
(1,005) 2
278,415  6,626.833 172,139  3,801.500
6
6
EXAMPLE OF INDEPENDENT SAMPLES t-TEST
Now, for it!
tobt 

X1 - X 2
 SS1  SS2  1
1

  
 n1  n2 - 2  n1 n2 
212.833- 167.500
 6,626.833 3,801.500  1 1 

  
66-2

 6 6 
45.333
45.333


 2.432
(1,042.833)(0.333) 18.635
RELATED SAMPLES t-TEST
Two-Tailed Test
Hypotheses
H0 : mD  0
H1 : m D  0
Rejection Rule--Reject H0
 if t(obt) > t(crit, a, N-1) OR if t(obt) < -t(crit, a, N-1) where N
is the number of differences
 OR if the significance level on the SPSS printout is less than
.05
RELATED SAMPLES t-TEST
One-tailed test where you believe the scores in
sample 1 will be less than the scores in sample 2
(which means that the differences will tend to be
less than 0).
 Hypotheses
H0 : mD  0
H1 : m D  0
 Rejection Rule--Reject H0
 if t(obt) < -t(crit, a, N-1)
 OR if the significance level on the SPSS printout is less than .05
RELATED SAMPLES t-TEST
One-tailed test where you believe the scores in
sample 1 will be greater than the scores in sample
2 (which means that the differences will tend to be
greater than 0).


Hypotheses
H0 : mD  0
H1 : m D  0
Rejection Rule--Reject H0
 if t(obt) > t(crit, a, N-1)
 OR if the significance level on the SPSS printout is less than
.05.
RELATED SAMPLES t-TEST
Calculating the related samples t-statistic
Step 1. Find the difference between each pair of scores.
Again, it doesn’t matter which sample you designate as 1
and which is 2, AS LONG AS you (a) consistently subtract
sample 2 from sample 1, and (b) keep the order in mind
as you set up your hypotheses and rejection rule.
From this point on, ignore the original scores and use only
the difference scores (designed with a subscript D). The
test is conducted pretty much the same as if it were a onesample test. [Continued on next slide.]
RELATED SAMPLES t-TEST
D - mD
SSD
N ( N - 1)
 Step 2. Find the mean of the difference scores.
t
 Step 3. Find the sum of squares of the difference
scores. Be sure to use N as the number of pairs
of scores (or the number of difference scores) to
do the division.
Step 4. Divide your answer from Step 3 by N(N – 1).
Step 5. Take the square root of your answer from Step 4.
Step 6. Divide your answer from Step 2 by the answer from Step 5.
Compare this obtained t-value against the critical t-value from the rejection
rule and decide the appropriate action.
RELATED SAMPLES t-TEST
 Find the appropriate t (crit) from the t-table in the
back of the book, using the correct bar at the top
depending on a one-tailed or a two-tailed test, a, and
df = N - 1.
 OR use the significance level on the SPSS printout.
 Calculate t (obt) using the related samples t-test.
 Make your decision based on your rejection rule.
EXAMPLE OF RELATED-MEASURES t-TEST
Same data as
before
Brand A
Brand B
Difference
Size 1
237
208
29
Size 2
254
178
76
Size 3
246
187
59
Size 4
178
146
32
Size 5
179
145
34
Size 6
183
141
42
EXAMPLE OF RELATED-MEASURES T-TEST
Stuff we’ll need
Differences
D
272
N
6
D
272 ÷ 6 = 45.333
D 2
14,042
2
2
(272)
(
D
)
14,042  1,711.333
SS = D 6
N
2
EXAMPLE OF RELATED-MEASURES t-TEST
And now
t
D - mD
SSD
N ( N - 1)
45.333 - 0 45.333


 6.002
1,711.333 7.553
6(6 - 1)