d
a
d
H
H
µ
µ
=
≠
0
4,.025
0
.8
0
4.02.
4.02
2.776
/
.447 /
5
We can reject
.
d
d
d
X
t
t
s
n
H
µ
−
−
=
=
≈
>
=
∴
∵
(b)
The distribution of the difference is normal.
2.
Two independent Samples, small samples sizes
1
2
5
n
n
=
=
2
2
1
2
1
2
18,
20,
2.5,
2.5,
2.5
1.58.
p
X
X
s
s
s
=
=
=
=
=
≈
(a)
95% C.I. for
1
2
8,.025
,
2.306
t
µ
µ
=
=
1
2
1
2
1
1
1
1
1
2
8,.025
1
2
8,.025
(
,
)
( 4.306, 0.306).
p
p
n
n
n
n
X
X
t
s
X
X
t
s
−
−
⋅
⋅
+
−
+
⋅
⋅
+
⇒
−
(b)
0
1
2
1
2
:
0 ,
:
0.
a
H
H
µ
µ
µ
µ
−
=
−
≠
1
2
1
2
0
8,.025
1
1
0
0
2.
2
2.306
We can NOT reject
.
p
n
n
X
X
t
t
s
H
−
−
=
= −
−
< −
= −
+
∴
∵
OR
Since
0 is inside the 95% C.I. for
1
2
µ
µ
−
, therefore, we can not reject
H
0
at
0.05
α
=
.
3.
One sample, large sample size, about pop. Mean.
36,
36,
5.
n
X
s
=
=
=
(a)
0
0
:
38 ,
:
38
(
38).
a
H
H
µ
µ
µ
=
<
=
0
0
0
35
38
3.6.
3.6
1.645.
/
5/
36
We can reject
at
.05.
X
z
s
n
H
µ
α
−
−
=
=
≈ −
−
< −
∴
=
∵
(b)
(c)
95% C. I is
0.025
0.025
,
(33.367, 36.633).
s
s
X
z
X
z
n
n
−
⋅
+
⋅
=
AMS 315/576 Test 4
1.
Do fraternities help or hurt your academic progress at college?
To investigate
this question, 5 fraternity members, and 5 non-fraternity members were
randomly selected among the class of 1998. It was found that their GPA are as
follows:
Fraternity members
2
1
2
3
2
Non-fraternity members
3
4
3
3
2
(a)
Please test the research hypothesis at the significance level 0.05
(b)
What assumption(s) did you assume in the above test?
2.
A new weight-reducing diet is currently undergoing tests by the Food and Drug
Administration(FDA).
A typical test is the following:
The weights of a random
sample of five people are recorded before the diet as well as three weeks after
the diet.
The results (in pounds) are as follows:
(a)
Please construct a confidence interval for the difference between the mean
weights before and after the diet is used.
(b)
At the significance level 0.05, can you conclude that the diet is effective?
3.
A local eat-in-pizza restaurant wants to investigate the possibility of starting to
deliver pizzas.
The owner of the store has determined that the home delivery
would be successful if the average time spent on the deliveries does not exceed
38 minutes.
The owner has randomly selected 16 customers and has delivered
pizza to their homes.
He found that the average delivery time was 35 minutes
with a standard deviation of 5 minutes.
(a)
At the significance level 0.05, can you conclude that the average delivery
time would not exceed 38 minutes?
(b)
What is the p-value of your test?
(c)
Please construct a 95% confidence interval for the mean delivery time.
1.
Two independent Samples, Small Sample Sizes.
2
2
1
1
1
1
2
1
2
2
2
2
2,
3,
,
,
.71
p
X
X
S
S
S
=
=
=
=
=
≈
(a)
0
1
2
1
2
:
0 ,
:
0,
0.05.
a
H
H
µ
µ
µ
µ
α
−
=
−
≠
=
1
2
1
2
0
1
1
1
1
5
5
0
0
8,.025
0
0
0
2
3
2.236.
(.71)
Reject
if
2.306 or
2.306.
2.236
2.306.
We can NOT reject
.
p
n
n
X
X
t
s
H
t
t
t
H
−
−
−
=
=
≈ −
+
+
>
=
< −
−
< −
∴
∵
(b)
normal populations, equal population variances
2.
Paired Samples, small sample sizes
3,
4.18,
5.
d
d
X
s
n
=
=
±
(a)
95% C.I. for
,
d
µ
4,.025
4,.025
4,.025
(
,
)
Plug in
2.776;
(-2.19,8.19)
95% C. I.
d
d
d
d
s
s
X
t
X
t
n
n
t
−
⋅
+
⋅
=
←
(b)
0
:
0 ,
:
0.
d
a
d
H
H
µ
µ
=
>
0
4, .05
0
0
3
1.60.
1.87
/
1.60
2.132.
We can NOT reject
.
d
d
X
t
s
n
t
H
−
=
=
≈
>
=
∴
∵
3.
One sample,
population mean. Small sample size.