1. University of Palestine
Significance Test (Independent Sample t-Test)
Dr. Amjad El-Shanti
MD, PMH,Dr PH
University of Palestine
2016

2. Steps in Test of Hypothesis
Formulate H0 and Ha
Determine the appropriate test
Establish the level of significance:α
Calculate the test statistic
Determine the degree of freedom
Compare observed test statistic against a Theoretical value

3. 1. Determine the Appropriate Test
If comparing a sample to a population, use one sample tests.
If comparing two samples in order to draw inferences about group differences in the population use two sample t-test.
Here the test statistic is based on a theoretical sampling distribution known as sampling distribution of the difference between two means.
Mdiff =
The standard deviation of such a sampling distribution is referred to as the standard error of the difference.

4. 1. Determine the Appropriate Test
Assumptions and Requirements for the two sample test (comparing groups means) are:
Independent variable consists of two levels of a nominal-level variable (when there are two and only two groups).
Dependent variable approximates quantitative (Numerical)
Normal distribution or large enough sample size to assume normality due to the central limit theorem.

5. 1. Determine the Appropriate Test
If the two groups are independent of each other uses independent group t-test (Pooled Test).
If the two groups are not independent of each other use dependent group t-test also known as paired t-test.
This lecture focuses on
independent sample t-test
which is a parametric test

6. 2. Establish Level of Significance
α is a predetermined value
The convention
α = .05
α = .01
α = .001

7. 3. Calculating Test Statistics
For the independent groups t-test the formula is:
The numerator is the difference in means between the two samples, and the denominator is the estimated standard error of the difference. 7

8. 3. Calculating Test Statistics
The estimated standard error of the difference is estimated on the basis of variances of the two samples (Pooled Variance t-test).
Where
S21= variance of Group 1
S22 = variance of Group 2
n 1= number of cases in Group 1
n 2= number of cases in Group 2 8

9. 4. Determine Degrees of Freedom
Degrees of freedom, df, is value indicating the number of independent pieces of information a sample can provide for purposes of statistical inference.
Df = Sample size – Number of parameters estimated
Df is n1 +n2 -2 for two sample test of means because the population variance is estimated from the sample

10. 5. Compare the Computed Test Statistic Against a Tabled Value
If |tc| > |tα| Reject H0
If p value < α Reject H0

11. Example of Independent Groups t-tests
Suppose that we plan to conduct a study to alleviate the distress of preschool children who are about to undergo the finger-stick procedure for a hematocrit (Hct) determination.
Note: Hct = % of volume of a blood sample occupied by cells.

12. Example of Independent Groups t-tests, Continued
Twenty subjects will be used to examine the effectiveness of the special treatment.
10 subjects randomly assigned to treatment group.
10 assigned to a control group that receives no special preparation.

13. 1. Determine the Appropriate Test
Testing hypothesis about two independent means (t-test)
Dependent variable = the child’s pulse rate just prior to the finger-stick
Independent variable or grouping variable = treatment conditions (2 levels)
H0: There is no difference in the mean of pulse of group under treatment and the mean of pulse of non treated group (control group).
MT = MNT . MT - MNT =0
Ha: There is difference in the mean of pulse of group under treatment and the mean of pulse of non treated group (control group).

14. 1. Determine the Appropriate Test
Two samples are independent.
The variables are:
*pulse: Numerical
*groups of treatment (Group A: Treated
(Group B: Non Treated)
So, the test is T test??
Pooled t Test: The difference of means between two samples which are independent (Subjects in group A rather than subjects in group B).
Can’t reject the hypothesis that the two variances are equal

15. 2. Establish Level of Significance
The convention
α = .05
α = .01
α = .001
In this example, assume α = 0.05

16. 3. Calculating Test Statistics
Group A
Group B 1
100
105 2 86 95 3
112
120 4 80 85 5
115
110 6 83 7 90 8 94 93 9
107 10 16

17. Rearrange the Data Treated Group (A) Non treated Group (B) 1 100 2 863
112 4 80 5
115 6 83 7 90 8 94 9 85 10
105 11 12 95 13
120 14 15
110 16 17 18 93 19
107 20
Treated
Group (A)
Non treated
Group (B)
Rearrange the Data

18. Group A sn XA X-X  (x-x)2 1 100 5 25 2 86 -9 81 3 112 17 289 4 80 -15
225
115 20
400 6 83
-12
144 7 90 -5 8 94 -1 9 85
-10 10
105
sum
950
1390
*X= 950/10=95
*Variance (s2)=  (x-x)2/n-1
= 1390/9= 154.4
*Standard deviation(s)=
 variance =  154.4= 12.42

19. Group B sn XB X-X  (x-x)2 1 105 2 95 -10 100 3 120 15 225 4 85 -202 95
-10
100 3
120 15
225 4 85
-20
400 5
110 25 6 -5 7
115 10 8 93
-12
144 9
107
sum
1050
1248
*X= 1050/10=105
*Variance (s2)=  (x-x)2/n-1
= 1248/9= 138.6
*Standard deviation(s)=
 variance =  138.6= 11.77

20. 3. Calculating Test Statistics (continued)
Group A (Treated) Group B (Control) (Non Treated)
X X2

21. 3. Calculating Test Statistics (continued)

22. 6. Compare the Computed Test Statistic Against a Tabled Value

23. 6. Compare the Computed Test Statistic Against a Tabled Value
If we had chosen a one tail test:
H0 : µ1 = µ2
Ha : µ1 < µ2
1.73
The null hypothesis can be rejected

24. T or z distribution??
If n of both samples are >25 and standard deviation of both populations are unknown use t distribution .
If n of both samples are large <=25 use z distribution.
z= x1-x2/ s12+ s22
n n2