1. Significance Test: One sample t- Test (Single Sample t- Test)
Dr. Amjad El-Shanti
2012

2. One sample t- test
By this test we will study whether the sample mean consistent with a certain hypothesized value for the population mean.
The method used is known as significance testing based on the normal distribution if the sample size large and the t-distribution for small sample size.

3. One sample t- test & Paired t- test
The paired t-test is a special case of the one sample t-test whether a sample mean is different from some specific value µ, which need no be zero.

4. One-sample t-test Formula
The general formula:
SE= S/√n

5. S = ∑Xi2 – (∑Xi)2 n √ n-1

6. Example (1):
The following are the heights in centimeters of 24 two –year old boys with homozygous sickle cell disease (SS):
Height and weight standard for the UK give a reference height for two-Year old males of 86.5 cm. Does the above sample suggest that two-year old males SS children differ in height from the standards?
Answer
H0: µ= 86.5
Ha: µ = 86.5
84.4
89.9 89
81.9 87
78.5
84.1
86.3
80.6 80
81.3
86.6
84.3
89.8
85.4 85
82.5
80.7
85.5

7. Answer example (1): cont….
X-x
(X-x )2
84.4
0.3
0.09
89.9
5.8
33.64 89
4.9
24.01
81.9
-2.2
4.84 87
3.4
11.56
78.7
-5.6
31.36
84.1
86.3
2.2
80.6
-3.5
12.25 80
-4.1
16.81
81.3
-2.8
7.84
86.8
2.7
7.29
83.4
-0.7
0.49
89.8
5.7
32.49
85.4
1.3
1.96 85
0.9
0.81
82.5
-1.6
2.56
80.7
-3.4
84.3
0.2
0.04
1.69
85.5
1.4
∑x=
∑(x-x)2=225.9
X= ∑X/n= /24= 84.1cm
S=√ ∑(x-x)2/n-1=
= √ 225.9/23 = 3.13
Se= s/√n = 3.13/ √ 24= 0.63
X= 84.1, µ= 86.5
Se= 0.63
n=24
d . f.= 23
t test = x-µ/se= /0.63
t test = -3.81
α=0.05 , d . f. = 23
t table=
T test > t table
3.81 > 2.06
P<α p< 0.05
Reject Ho
There is significant difference in heights of SS children and height of Two –year children

8. Example (2):
A sample of size n =100 has a mean age x= 54.85, Test whether this sample mean comes from a population whose mean age is 53, given that =5.5?
Answer
H0: µ= 53
Ha: µ = 53
α= 0.05
Z test = (x - µ)/ (/√ n)= ( )/ (5.5/√100)= 3.36
Z table (α=0.05) = 1.96
Z test > Z table ………………… >1.96
Reject Ho …………………… There is significant difference between sample mean age and population mean age.