OBJECTIVES: Run the test of hypothesis for mean difference using paired samples. Construct a confidence interval for the difference in population means using paired samples. Observation of interest will be the difference in the readings before and after intervention called paired difference observation. Paired t test: A paired t-test is used to compare two means where you have two samples in which observations in one sample can be paired with observations in the other sample. Examples of where this might occur are: Before-and-after observations on the same subjects (e.g. students’ test results before and after a particular module or course). A comparison of two different methods of measurement or two different treatments where the measurements/treatments are applied to the same subjects (e.g. blood pressure measurements using a sphygmomanometer and a dynamap). When there is a relationship between the groups, such as identical twins. This test is concerned with the pair-wise differences between sets of data. This means that each data point in one group has a related data point in the other group (groups always have equal numbers). ASSUMPTIONS: The sample or samples are randomly selected The sample data are dependent The distribution of differences is approximately normally distributed. Note: The under root is onto the entire numerator and denominator, so you should take the root after solving it entirely where “t” has (n-1) degrees of freedom and “n” is the total number of pairs.

2. PAIREDT-TEST
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

3. OBJECTIVE
S
By the end of this session the students should be
able to:
• Run the test of hypothesis for mean difference
using paired samples
• Construct a confidence interval for the
difference in population means usingpaired
samples.

4. T-test for onesample
s n
X 
t 


5. T-test for two dependent sample
(Paired sample)

6. POPULATION
Sample
Before
Intervention
Intervention
Sample
AfterIntervention
Observation of interest will be the difference in the readings
before and after intervention called paired difference observation

7. WHATISPAIREDT-TEST?
• A paired t-test is used to compare two means where you have two
samples in which observations in one sample can be paired with
observations in the other sample.
Examples of where this might occur are:
– Before-and-after observations on the same subjects (e.g. students’ test
results before and after a particular module or course).
– A comparison of two different methods of measurement or two different
treatments where the measurements/treatments are applied to the same
subjects (e.g. blood pressure measurements using a sphygmomanometer and
a dynamap).
– When there is a relationship between the groups, such as identical twins.

8. 8
PAIRED DIFFERENCE SAMPLES “ T TEST” AN
EXPLANATION
• This test is concerned with the pair-wise differences
between sets of data.
• This means that each data point in one group has a
related data point in the other group (groups always
have equal numbers).

9. ASSUMPTIONS
• The sample or samples are randomly selected
• The sample data are dependent
• The distribution of differences is approximately normally
distributed.

10. 10
Reading1
(Before
intervention/Ca
se)
Reading 2 (After
intervention/M
atched control)
Means of
Differences
(XD)
Difference
d= X1-X2
D2 OR
(X1-X2)2
X1 X2 X1+X2/2 D1=X1-X2
2
(X1- X2)
X1 X2 X1+X2/2 D2=X1-X2
2
(X1-X2)
X1 X2 X1+X2/2 D3=X1-X2 .
. . X1+X2/2 . .
. . . .
X1 X2 Di=X1-X2 .
. . . .
. . . .
X1 X2 Dn=X1-X2 .
∑X1 ∑X2 Xd ∑d
2
∑d
What is a Paired Difference Observation?

11. 11
X D = Mean of Differences
SD = Standard deviation of Differences
n= number ofpairs
D0 = Difference between the population means
Where D0 is mostly zero
 S D 
 n 
t 
x D
T
E
S
TST
A
TISTICANDASSUMPTIONSFORP
AIREDDIFFERENCE
METHOD
- In this method, the test statistic becomes:
 D 0

12. FORMULAFORSTANDARD
DEVIATION
Sd =√ n∑d2 –(∑d)2
n (n-1)
Note: The under root is onto the entirenumerator
and denominator, so you should take the root
after solving it entirely

13. 13
(1-)% CONFIDENCEINTERV
ALFORP
AIREDMEAN DIFFERENCE
(D)
where “t” has (n-1) degrees of freedom and “n” is
the total number of pairs.
Formula:
n
SD
D ,df
2
x  t

14. 14
HYPOTHESIST
E
S
TFORPAIREDMEANDIFFERENCE(D)
• One-tailed Test:
– H0: µD ≤0 Ha: µD > 0 (Righttailed)
– Ho: μD ≥0 Ha: μD < 0 (Lefttailed)
• Two-tailed Test:
– H0: µD =0 Ha: μD 0
– Step Two: α =0.05

15. • Step 3 Test
statistic:
Sd = √ n∑d2 –(∑d)2
n (n-1)
where “t” has (n-1) degrees of freedom (df) and “n” is the total
number of
pairs.
• Step 4: CriticalRegion:
Reject Ho if:
t cal > t tab OR t cal < -
t tab
• Step 5: Conclusion

 n 
t  D
 s D 
x  D 0

16. EXAMPLE
BLOOD SAMPLES FROM 10 PATIENTS WERE SENT TO EACH OF
TWO LABS FOR CHOLESTEROL DETERMINATION.
MEASUREMENTS WERE AS FOLLOWS:
Participants Lab 1 Lab 2
1 296 318
2 268 287
3 244 260
4 272 279
5 240 245
6 244 249
7 282 294
8 254 271
9 244 262
10 262 285

17. QUE
STION
- Is there a statistically significant difference at α
= .05 in the cholesterol levels reported by lab1
and lab 2.

19. SOLUTION
Step1:
H0: µD = 0
Ha: µD ≠0
Step 2:
α= .05
d.f. = n-1, 10-1 =9
Critical value = +2.26
Step 3:
t = -14.4-0 = -6.73
6.77/√10
Sd = √10x 2486-(- 144)2
10x 9
= 6.77
-6.73 -2.26 2.26

20. SOLUTION
Step 4:
Reject Ho if: t cal > t tab OR if t cal < - t tab
Since in this case, t cal < -t tab = -6.73 < -2.26, so
we reject Ho
Step 5 (Conclusion):
Reject Ho at 5 % level of significance, and we have
sufficient evidence to conclude that results from
the both labs are differentthan each other.

21. CONFIDENCE INTERVAL
• 14.4+3.25 X6.77/√10
• INTERPRETATION: WE ARE9 9 % CONFIDENT THAT THE
MEAN DIFFERENCE WILL FALL BETWEEN LOWER AND UPPERL
IMITS
99%C.I
n
SD
D ,df
2
x  t

23. 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