𝑝
2
1
𝑛
1
−
1
𝑛
2

Example: Two-Sample Pooled Test
Owen Lawn Care Inc manufactures and assembles
lawnmowers that are shipped to dealers throughout US and
Canada. Two different procedures have been proposed, the
Welles (W) method and the Atkins(A) method. The
question is: Is there a difference in the mean time to mount
the engines on the frames of the lawnmowers?
To evaluate, a time and motion study
was conducted. A sample of five
employees is timed using the Welles
method and six using the Atkins
method. Is there a difference in the
mean mounting times? Use the .10
significance level.

Example: Two-Sample Pooled Test
Step
①
:
State
?
0
and
?
1
Step
②
:
Select
level of significance
𝛼 = 0.1
(given)
Step
③
:
Select
test statistics
?
distribution since
𝜎
is unknown
?
0
:
𝜇
𝑊
= 𝜇
𝐴
?
1
:
𝜇
𝑊
≠ 𝜇
𝐴
Two-tailed test

Example: Two-Sample Pooled Test
Step
④
:
Formulate
decision rule
Two tailed,
𝛼 =
0.10
𝑑𝑓 = 𝑛
1
+ 𝑛
2
− 2
= 5 + 6 − 2
= 9
From the table, critical
? = ±1.833
Reject
?
0
if computed
t > 1.833
or
if computed
t < −1.8333

Example: Two-Sample Pooled Test
Step
⑤
:
Make a
decision
?
𝑝
2
=
𝑛
𝑊
−1 𝑠
𝑊
2
+
𝑛
𝐴
−1 𝑠
𝐴
2
𝑛
𝑊
+𝑛
𝐴
−2
=
5−1
2.9155
2
−
6−1
2.0976
2
5+6−2
= 6.2222
ҧ𝑥
𝑊
=
20
5
= 4
ҧ𝑥
𝐴
=
30
6
= 5
?
𝑊
=
34
5 − 1
= 2.9155
?
𝐴
=
22
6 − 1
= 2.0976

Example: Two-Sample Pooled Test
Step
⑤
:
Make a
decision
ҧ𝑥
𝑊
=
20
5
= 4
ҧ𝑥
𝐴
=
30
6
= 5
?
𝑊
=
34
5 − 1
= 2.9155
?
𝐴
=
22
6 − 1
= 2.0976
? =
ҧ𝑥
𝑊
−
ҧ𝑥
𝐴
?
𝑝
2
1
𝑛
𝑊
−
1
𝑛
𝐴
=
4 − 5
6.222
1
5
+
1
6
= −0.662
As
−0.662
is between
−1.833
and
1.833
, the
decision is not to reject the null hypothesis.
Reject
?
0
if computed
t > 1.833
or
if computed
t < −1.8333

Example: Two-Sample Pooled Test
Step
⑥
:
Interpret
result
The conclusion is that the sample data failed
to show a difference between the mean
assembly times of the two methods.
What is the
𝑝
-value?
We need the probability value of
?
to be closest to 0.622,
with the
𝑑𝑓
value of 9. This value is 1.383, corresponding
to the significance level of .20.
∴ 𝑝
-value
> 0.20
Therefore, even if the significance level used is 20%, we
would not have rejected the
?
0

The null and alternate hypotheses are:
?
0
: 𝜇
1
= 𝜇
2
?
1
: 𝜇
1
≠ 𝜇
2
A random sample of 15 observations from the first
population revealed a sample mean of 350 and a sample
standard deviation of 12. A random sample of 17
observations from the second population revealed a
sample mean of 342 and a sample standard deviation of
15. At the .10 significance level, is there a difference in
the population means?
Try This
....