Example: In a heart study the systolic blood pressure
was measured for 24 men aged 25 and for 30 men
aged 40. Do these data show sufficient evidence to
conclude that the older men have a higher systolic
blood pressure, at the 0.05 level of significance?
Since
The variable concerning systolic blood
pressure is continuous
The sample size of each group is greater
than 10
Systolic blood pressure values in each
group is normally distributed
There are two groups and they are
independent
Independent
samples t-test
is used
Subject
1
2
3
4
5
6
7
8
9
10
11
12
20- year-old
Sbp
Subject
95
13
122
14
130
15
148
16
130
17
150
18
105
19
110
20
130
21
156
22
108
23
124
24
Sbp
132
100
120
125
115
138
100
118
136
110
140
106
Subject
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
40- year-old
Sbp
Subject
150
16
152
17
154
18
160
19
164
20
176
21
108
22
126
23
132
24
142
25
136
26
146
27
114
28
118
29
130
30
Sbp
148
116
128
136
110
126
130
122
140
110
124
136
120
142
114
GROUP
20- year-old
40- year-old
N
24
30
Mean
Std. Deviation
122,8333
16,7790
133,6667
17,3013
24
30
20- year-old
40- year-old
160
Mean  1 SD SBP
150
140
130
120
110
100
N=
GROUP
(1) H0:1=2
Ha: 1<2
(2) Testing the equality of variances
H0:21= 22
Ha: 21 22
S 2max 299.33
F 2 
 1.06  F(30, 24,0.05)  1.94
Smin 281.54
Accept H0. Variances are equal.
(3)
(n1  1) s  (n2  1) s
s 
n1  n2  2
2
1
2
p
2
2
(24  1)281.54 2  (30  1)299.33 2

 291.46
24  30  2
t
( x1  x2 )  (1   2 )
s 2p
n1
(4)

s 2p

n2
t(52,0.05)=1.675< t cal
(122 . 83  133 . 67)  0
291 . 46 291 . 46

24
30
 2.31
 2.31
p<0.05, Reject H0.
(5) The older men have higher systolic blood pressure
Example: A study was conducted to see if a new
therapeutic procedure is more effective than the
standard treatment in improving the digital dexterity of
certain handicapped persons. Twenty-four pairs of
twins were used in the study, one of the twins was
randomly assigned to receive the new treatment, while
the other received the standard therapy. At the end of
the experimental period each individual was given a
digital dexterity test with scores as follows.
Since
The variable concerning digital dexterity
test scores is continuous
The sample size is greater than 10
digital dexterity test score is normally
distributed
There are two groups and they are
dependent
Paired
sample
t-test
New
49
56
70
83
83
68
84
63
67
79
88
48
52
73
52
73
78
64
71
42
51
56
40
81
Standard Difference
54
-5
42
14
63
7
77
6
83
0
51
17
82
2
54
9
62
5
71
8
82
6
50
-2
41
11
67
6
57
-5
70
3
72
6
62
2
64
7
44
-2
44
7
42
14
35
5
73
8
Total
129
Mean 65,46
60,08
5,38
SD
14,38
14,46
5,65
H0: d = 0
Ha: d > 0
di

d
129

 5.38
n
24
2
(
di

d
)

sd2 
 31.90
n 1
d  d
5.38  0
t

sd / n
31.90 / 24
 4.66
t(23,0.05)=1.7139
t
t
Since, calculated
table
reject H0.
We conclude that the new
treatment is effective.
Example: We want to know if children in two
geographic areas differ with respect to the proportion
who are anemic. A sample of one-year-old children
seen in a certain group of county health departments
during a year was selected from each of the
geographic areas composing the departments’
clientele. The followig information regarding anemia
was revealed.
Geographic
Area
1
Number in
sample
450
Number
anemic
105
Proportion
2
375
120
0.32
0.23
H 0 : P2  P1  0
p1  105 / 450  0.23
H a : P2  P1  0
p2  120 / 375  0.32
(450 )(0.23)  (375)(0.32)
p
 0.27
450  375
z
(0.23 - 0.32)  0
 2.78 p  0.0027  0.025
(0.27)(0.73) (0.27)(0.73)

450
375
Reject H0.
We concluded that the proportion of anemia is different
in two geographic areas.
Example: To test the median level of energy intake of
2 year old children as 1280 kcal reported in another
study, energy intakes of 10 children are calculated.
Energy intakes of 10 children are as follows:
Child
Energy
Intake
1
2
3
4
5
6
7
8
9
10
1500
825
1300
1700
970
1200
1110
1270
1460
1090
Since
The variable concerning energy intake is
continuous
The sample size is not greater than 10
Energy intake is not normally distributed
There is only one group
Sign
test
H0: The population median is 1280.
HA: The population median is not 1280.
Child
Energy
İntake
1
2
3
4
5
6
7
8
9
10
1500
825
1300
1700
970
1200
1110
1270
1460
1090
+
-
+
+
-
-
-
-
+
-
Number of (-) signs = 6
and number of (+) signs = 4
For k=4 and n=10
From the sign test table p=0.377
Since p > 0.05 we accept H0
We conclude that the median energy intake level
in 2 year old children is 1280 kcal.
Example: Cryosurgery is a commonly used therapy
for treatment of cervical intraepithelial neoplasia
(CIN). The procedure is associated with pain and
uterine cramping. Within 10 min of completing the
cryosurgical procedure, the intensity of pain and
cramping were assessed on a 100-mm visual analog
scale (VAS), in which 0 represent no pain or
cramping and 100 represent the most severe pain
and cramping. The purpose of study was to compare
the perceptions of both pain and cramping in women
undergoing the procedure with and without
paracervical block.
5 women were selected randomly in each
groups and their scores are as follows:
Group
Women without
a block
Score
14
88
37
27
0
50
Women with a
paracervical
block
70
37
66
75
Since
The variable concerning pain/cramping
score is continuous
The sample size is less than 10
There are two groups and they are
independent
Mann
Whitney
U test
H 0 : M I  M II
H A : M I  M II
Group
I
I
I
I
II
II
II
II
II
I
Score
0
14
27
37
37
50
66
70
75
88
R1= 1+2+3+4.5+10 = 20.5
Rank
1
2
3
4.5
4.5
6
7
8
9
10
n1 (n1  1)
U1  n1n2 
 R1
2
5(5  1)
 55 
 20.5  19.5
2
U 2  n1n2  U1  5  5 19.5  5.5
U  19.5
From the table, critical value is 21
19.5 < 21 accept H0
We conclude that the median pain/
cramping scores are same in two groups.
Example: A study was conducted to analyze the
relation between coronary heart disease (CHD) and
cigarette smoking. 40 patients with CHD and 50
control subjects were randomly selected from the
records and smoking habits of these subjects were
examined. Observed values are as follows:
Smoking
CHD
Total
+
-
10
30
40
No
4
46
50
Total
14
76
90
Yes
Observed and expected frequencies
CHD
Smoking
+
-
10
6.2
30 33.8
40
No
4
7.8
46 42.2
50
Total
14
76
90
Yes
2
Total
2
χ 2  
(O ij  E ij ) 2
E ij
i 1 j1

10  6.2

2
6.2

30  33.8

2
33.8

4  7.8

2
7.8

46  42.2

2
42.2
 4.95
df = (r-1)(c-1)=(2-1)(2-1)=1
2
2 =4. 95 >  (1,0.05)=3.845
reject H0
Conclusion: There is a relation between CHD and
cigarette smoking.
Example:To test whether the weight-reducing diet is effective
9 persons were selected. These persons stayed on a diet for
two months and their weights were measured before and
after diet. The following are the weights in kg:
Weights
Subject
Since
Before After
1
85
82
The variable concerning
weight is continous.
2
91
92
3
68
62
The sample size is less than 10
4
5
6
76
82
87
73
81
83
7
8
105
93
85
88
9
98
90
There are two groups and they
are dependent
Wicoxon signed
ranks test
Subject
1
2
3
4
5
6
7
8
9
Weights
Before
After
85
82
91
92
68
62
76
73
82
81
87
83
105
85
93
88
98
90
Difference
Di
Sorted
Di
Rank
Signed
Rank
3
-1
6
3
1
4
20
5
8
-1
1
3
3
4
5
6
8
20
1.5
1.5
3.5
3.5
5
6
7
8
9
-1.5
1.5
3.5
3.5
5
6
7
8
9
T = 1.5
T = 1.5 < T(n=9,a =0.05) = 6
reject H0, p<0.05
T = 1.5 < T(n=9,a =0.01) = 2
reject H0, p<0.01
We conclude 99% cinfident that diet is effective.