This document provides an overview and examples of conducting independent samples t-tests. It begins with explaining the assumptions and steps to conduct an independent samples t-test, including defining the null and alternative hypotheses, significance level, test statistic, critical region, and conclusion. It then provides three examples demonstrating how to apply these steps to compare the mean serum creatinine levels between different groups of patients. The document concludes with explaining how to compute a confidence interval for the difference between two independent means.

2. Shakir Rahman
BScN, MScN, MSc Applied Psychology,
PhD Nursing (Candidate)
University of Minnesota USA.
Principal & Assistant Professor
Ayub International College of Nursing &
AHS Peshawar
Visiting Faculty
Swabi College of Nursing & Health
Sciences Swabi
Nowshera College of Nursing & Health
Sciences Nowshera
T –test for two
Independent
Samples

3. Example 1
– Q1.The mean serum creatinine level measured in 36 patients
after they received a newly proposed antibiotic was 1.2 mg/dl
and the standard deviation was 0.4 mg/dl . A physician claims that
the mean serum creatinine in the general population is 1 mg/dl.
– Using a significance level of 0.05, test if the mean serum
creatinine level in the general population is greater than 1
mg/dl.

4. Z test

5. Example 2
Q2.The mean serum creatinine level measured in 25
patients, after they received a newly proposed
antibiotic, was 1.1 mg/dl and the sample standard
deviation is 0.6 mg/dl. A physician claims that the
mean serum creatinine in the general population is
0.9 mg/dl.
Using a significance level of 0.05, test if the mean
serum creatinine level in the general population is
different from 0.9 mg/dl.

6. t test for Onesample

7. Example 3
Q3. The mean serum creatinine level is measured in 13 white
women who received a newly proposed antibiotic was 1.2
mg/dl and the standard deviation was 0.6 mg/ dl.
Another sample of 12 black women who have receivedan
old antibiotic have mean serum creatinine level of 1.0
mg/dl with standard deviation of 0.4 mg/dl.
Using a significance level of 0.05, test if the mean serum
creatinine level in white women is different than the black
women.

8. What is Different in the 3rd Example?
• Two groups
• Two means
• Two SDs
Comparison of two
means By
t- test

9. Decide 2 samples are independent or dependent on each
other.
Independent:
If there is no connection between two samples then
Perform t- test for two
independent samples

10. t Test for Two Independent Samples
• Compare means of two groups
– Experimental—treatment versus control
– Existing groups—males versus females
9

11. Underlying Assumptions for Independent Samples t-test
1. The samples have been randomly selected from
normally distributed populations
2. The samples are independent of each other
3. Population variances 1
2 and 2
2 are unknown

12. Contd…
4. In order to estimate the unknown population
variances, there are two ways:
– If the two population variances are assumed to be
equal i.e. 1
2 = 2
2 , a pooled variance is calculated as
an estimate of the unknown population variances
– If the two population variances are assumed to be
unequal i.e. 1
2 ≠ 2
2, a different formula isused
2
2
2
1 1 2
1 2
n  n  2
n  1s  n  1s
s2
p 

13.  1 2 
p
n n
 1 1 
S2   
X1  X2  μ1 μ2 
t 
12
1
2
3
4
5
State null
and
Alternative
hypotheses
Level of
significance
= α
Test
Statistics
Critical
region/
P value
Conclusion
Follow the same steps of
Hypothesis Testing
t
TEST

14. Hypotheses to test two independent
means
STEP I:
Lower tail test:
H0: μ1  μ2
Ha: μ1 <μ2
Upper tail test:
H : μ ≤ μ
0 1 2
Ha: μ1 >μ2
Two-tailedtest:
H0: μ1 =μ2
Ha: μ1 ≠μ2

15. Notations to indicate two Groups
Group I Group II
n1
s1
X1
n2
s2
X2

16. STEP II:
Level of Significance α =0.05

17. STEP III:
Test Statistic =
1 2 
t 
p
n n
 1 
S2 1
X1  X2  μ1 μ2 

18. Now since 2 is unknown, we need to find an
estimate for it, using the variances from the
1 2.
two samples S2 & S2 This estimate is called
the pooled variance, Sp2
2
2
2
1 1 2
1 2
n  n  2
 1
n  1s  n s
s2
p 

19. STEP IV:
Critical region: for significance level α and
df = n1+n2-2
Reject H0 if
tcal > ttab
or
tcal <- ttab
STEP V: Conclusion

20. Example
A researcher wishes to determine whether the
salaries of professional nurses employed by
private hospitals are higher than those of nurses
employed by government hospitals. She selects a
sample of nurses from each type of hospital and
calculates the means and standard deviations of
their salaries.
At α = 0.01, can she conclude that the private
hospitals pay more than the government
hospitals? Assume that the populations are
approximately normally distributed.

21. Data:
Private Government
X1 = $ 26,800
s1 = $ 600
n1 = 10
X2= $ 25, 400
s2 = $ 450
n2 = 8

22. Step I:
Ho : µ1 ≤ µ2
Ha: µ1 > µ2
Step II:
α = 0.01
t tabα, d.f. = t tab 0.01, 16 =2.583

24. Step III:
t statistic = 5.47
1 2 
t 
p
 n n
 1 
X1  X2  μ1 μ2 
S2 1
2
2
2
1 2
1
n1  n2  2
s
n  1s  n  1
2
sp 

25. Step IV:
Critical region:
Reject Ho if t cal> t tab
5.47 > 2.583

26. Step V:
– Since our t cal is greater than our t tab (5.47 > 2.583) and falls under
critical region, so we reject our Ho at 1 % level of significance and we
have sufficient evidence to conclude that the salaries paid to nurses
employed by private hospitals are higher than those paid to nurses
employed by government hospitals.

27. Computing Confidence Interval for the
difference in means of 2 groups


1 2
(x  x )  t ,df
2
2  1 1 
n2 
 n1
s p

29. Bluman (2012). Elementary Statistics: A Step by StepApproach
(8th.). McGraw Hill.
Daniel (2014). Biostatistics: Basic Concepts and Methodology
for the Health Sciences. New York: John Wiley &Sons.

30. Acknowledgements
Dr Tazeen Saeed Ali
RM, RM, BScN, MSc ( Epidemiology & Biostatistics), Ph.D. (Medical Sciences),
Post Doctorate (Health Policy & Planning)
Associate Dean School of Nursing & Midwifery
The Aga Khan University Karachi.
Kiran Ramzan Ali Lalani
BScN, MSc Epidemiology & Biostatistics (Candidate) Registered Nurse (NICU)
Aga Khan University Hospital