𝑑
is the standard deviation of the differences
between the paired or related observations
𝑛
is the number of paired observations

Two Sample Tests: Dependent
Samples
▪
The standard deviation of the differences will be
computed with
?
𝑑
=
Σ
𝑑 −
ҧ
𝑑
2
𝑛 − 1

Example: Dependent Samples
Nickel Savings and Loan wishes to compare the two
companies it uses to appraise the value of residential
homes. Nickel Savings selected a sample of 10
residential properties and scheduled both firms for
an appraisal. The results, reported in $,000 are
At the .05 significance
level, can we conclude that
there is a difference
between the firms’ mean
appraised home values?

Example: Dependent Samples
Step
①
:
State
?
0
and
?
1
Step
②
:
Select
level of significance
𝛼 = 0.05
(given)
Step
③
:
Select
test statistics
?
distribution since
𝜎
is unknown
?
0
:
𝜇
𝑑
= 0
?
1
:
𝜇
𝑑
≠ 0
Two-tailed test

Example: Dependent Samples
Step
④
:
Formulate
decision rule
Two-tailed,
𝛼 = 0.05
,
𝑑𝑓 = 𝑛 − 1 = 10 − 1 = 9
Reject
?
0
if computed
t > 2.262
if computed
t < −2.262
? = 2.262

Example: Dependent Samples
Step
⑤
:
Make a
decision
ҧ
𝑑 =
46
10
= 4.6
?
𝑑
=
174.4
10 − 1
= 4.402
? =
ҧ
𝑑
?
𝑑
/
10
=
4.6
4.402/
10
= 3.305
As computed
? >
critical
?
, reject
?
0
Reject
?
0
if
t > 2.262
if
t < −2.262

Example: Dependent Samples
Step
⑥
:
Interpret
result
There is a difference between the firms’ mean
appraised home values. The largest difference
is $12,000 for Home 3.
What is the
𝑝
-value?
We need the probability value of
?
to be closest to 3.305,
with the
𝑑𝑓
value of 9. This value is between 3.250 and
4.781, thus corresponds to the significance level of .010.
∴ 𝑝
-value
< 0.01
Therefore, there is a small likelihood that the
?
0
is true.