Section 8.3
Testing the Difference Between
Means (Dependent Samples)
Section 8.3 Objectives
• Perform a t-test to test the mean of the differences for
a population of paired data
t-Test for the Difference Between Means
• To perform a two-sample hypothesis test with
dependent samples, the difference between each data
pair is first found:
 d = x1 – x2 Difference between entries for a data pair
• The test statistic is the mean d of these differences.
 d   d Mean of the differences between paired
n
data entries in the dependent samples
t-Test for the Difference Between Means
Three conditions are required to conduct the test.
1. The samples must be randomly selected.
2. The samples must be dependent (paired).
3. Both populations must be normally distributed.
If these requirements are met, then the sampling
distribution for d is approximated by a t-distribution
with n – 1 degrees of freedom, where n is the number
of data pairs.
-t0
μd
t0
d
Symbols used for the t-Test for μd
Symbol
Description
n
The number of pairs of data
d
The difference between entries for a data pair,
d = x1 – x2
d
The hypothesized mean of the differences of
paired data in the population
Symbols used for the t-Test for μd
Symbol
d
sd
Description
The mean of the differences between the paired
data entries in the dependent samples
d
d 
n
The standard deviation of the differences between
the paired data entries in the dependent samples
2
(

d
)
2
2

d

(d  d )
n
sd 

n 1
n 1
t-Test for the Difference Between Means
• The test statistic is
d 
d
.
n
• The standardized test statistic is
d  d
t
.
sd n
• The degrees of freedom are
d.f. = n – 1.
t-Test for the Difference Between Means
(Dependent Samples)
In Words
1. State the claim mathematically
and verbally. Identify the null
and alternative hypotheses.
In Symbols
State H0 and Ha.
2. Specify the level of significance.
Identify α.
3. Determine the degrees of
freedom.
d.f. = n – 1
4. Determine the critical value(s).
Use Table 5 in
Appendix B if n > 29
use the last row (∞) .
t-Test for the Difference Between Means
(Dependent Samples)
In Words
In Symbols
5. Determine the rejection
region(s).
6. Calculate d and sd .
d  d
n
(d )
2

d

(d  d )
n
sd 

n 1
n 1
2
7. Find the standardized test
statistic.
d  d
t
sd n
2
t-Test for the Difference Between Means
(Dependent Samples)
In Words
8. Make a decision to reject or
fail to reject the null
hypothesis.
9. Interpret the decision in the
context of the original
claim.
In Symbols
If t is in the rejection
region, reject H0.
Otherwise, fail to
reject H0.
Example: t-Test for the Difference
Between Means
A teacher claims that a grammar seminar will help students
reduce the number of grammatical errors made when writing
essays. The table shows the number of errors made before
and after participating in the seminar. Assuming the
populations are normally distributed, is there enough
evidence to support the claim at α = 0.01?
Student
1
2
3
4
5
6
7
Errors (before)
15
10
12
8
5
4
9
Errors (after)
11
9
6
5
1
0
9
Example: t-Test for the Difference
Between Means
Hypotheses:
claim:
mean_before > mean_after
mean_before – mean_after > 0
µbefore - µafter > 0
µ1 - µ 2 > 0
µd > 0
Hypotheses:
H0:µd ≤ 0
Ha: µd > 0 (claim)
Example: t-Test for the Difference
Between Means
Get differences:
Two data sets (samples) become a single set of data (the
set of differences, “d”)
Student
1
2
3
4
5
6
7
Errors
before
15
10
12
8
5
4
9
Errors after
11
9
6
5
1
0
9
d (x1 – x2)
4
1
6
3
4
4
0
22
d2
16
1
36
9
16
16
0
94
So, 𝑑=
𝑑
𝑛
= 22/7 = 3.143 sd 
=
(d  d ) 2

n 1
94−222/7
6
=
sum
(d )
d 
n
n 1
2
94−69.143
6
2
= 4.143=2.035
Example: t-Test for the Difference
Between Means
Get t test statistic:
Two data sets (samples) become a single set of data (the
set of differences)
t=
𝑑 −𝜇𝑑
𝑠𝑑 / 𝑛
=
3.143−0
2.035/ 7
=
3.143
.769
= 4.08
Determine t0  df = n-1=6, α=0.01, one tail (from Ha).
From table t0 = 3.143
Rejection region: t value greater than 3.143
Decision: since t is in rejection region (4.08 > 3.143) we
reject H0.
There is enough evidence at 1% significance level to support
the teacher’s claim that the seminar does reduce the number
Example: t-Test for the Difference
Between Means
A shoe manufacturer claims that athletes can increase their
vertical jump heights using the manufacturer’s new Strength
Shoes®. The vertical jump heights of eight randomly selected
athletes are measured. After the athletes have used the
Strength Shoes® for 8 months, their vertical jump heights are
measured again. The vertical jump heights (in inches) for
each athlete are shown in the table. Assuming the vertical
jump heights are normally distributed, is there enough
evidence to support the manufacturer’s claim at α = 0.10?
Athletes
1
2
3
4
5
6
7
8
Height (old)
24
22
25
28
35
32
30
27
Height (new)
26
25
25
29
33
34
35
30
Example: t-Test for the Difference
Between Means
Hypotheses:
claim:
increases vertical jump
mean_before < mean_after
mean_before – mean_after < 0
µbefore - µafter < 0
µ1 - µ 2 < 0
µd < 0
Hypotheses:
H0:µd ≥ 0
Ha: µd < 0 (claim)
Solution: Two-Sample t-Test for the
Difference Between Means
d = (jump height before shoes) – (jump height after shoes)
•
•
•
•
•
H0: μ d ≥ 0
Ha: μd < 0 (claim)
α = 0.10
d.f. = 8 – 1 = 7
Rejection Region:
Solution: Two-Sample t-Test for the
Difference Between Means
d = (jump height before shoes) – (jump height after shoes)
Before
24
22
After
26
25
d
–2
–3
d2
4
9
25
28
35
25
29
33
0
–1
2
0
1
4
32
30
34
35
27
30
–2
–5
4
25
–3
9
Σ = –14 Σ = 56
 d 14
d 

 1.75
n
8
2 (d )
d 
n
sd 
n 1
(14)
56 
8

8 1
 2.1213
2
2
Solution: Two-Sample t-Test for the
Difference Between Means
d = (jump height before shoes) – (jump height after shoes)
• Test Statistic:
• H0: µd ≥ 0
• Ha: μd < 0 (claim)
d  d
1.75  0
t

 2.333
• α = 0.10
sd n 2.1213 8
• d.f. = 8 – 1 = 7
• Decision: Reject H0.
• Rejection Region:
At the 10% level of significance,
there is enough evidence to
support the shoe manufacturer’s
claim that athletes can increase
their vertical jump heights using
t ≈ –2.333
the new Strength Shoes®.
Section 8.3 Summary
• Performed a t-test to test the mean of the difference
for a population of paired data