STATISTICAL INFERENCE ABOUT
MEANS AND PROPORTIONS WITH
TWO POPULATIONS
STATISTICS IN PRACTICE
 The testing process in the pharmaceutical industry
usually consists of three stages :
 1.preclinical testing
 2.testing for long-term usage and safety
 3.clinical efficacy testing
10.1 ESTIMATION OF THE DIFFERENCE
BETWEEN THE MEANS OF TWO
POPULATINS:INDEPENDENT SAMPLES
 Point Estimator of the Difference Between the
Means of populations

x1  x 2
(10.1)
 Sampling Distribution of x1  x2
 Expected Distribution of


E( x1  x2 )  1   2
(10.2)
 x  x   12 / n1   22 / n2
1
2
x1  x 2
(10.3)
Interval Estimate of
1
 2
:large-sample case
 Interval Estimate of the Difference Between the Means of
Two Populations: Large-Sample Case (n1  30, n2  30)

x1  x 2  z a / 2 x
 Point Estimator of

(10.4)
1  x2
sx x 
1
2

x1  x2
s12 s 22

n1 n 2
(10.5)
Interval Estimate of the Difference Between
the Means of Two Population: Large-Sample
Case (n1  30, n2  30) With  1 and  2 Estimated by s1
and s2
x1  x2  z a / 2 s x  x
1
2
(10.6)
1   2
Interval Estimate of
:Small-Sample Case
 We begin by making the following assumption:




1.Both populations have normal distributions
2
2
2
2.The variances of the populations are equal ( 1   2   )
 x x
1
2
1 1
  / n1   / n2   (  )
n1 n2 (10.7)
2
1
2
2
2
2
Pooled estimator of
2
2
(
n

1
)
s

(
n

1
)
s
1
2
2
s2  1
n1  n2  2
Point Estimate of
sx x
1
 x x
1
1
 s (  )
n1 n2
1
2
When
2
2
(10.9)
(10.8)
 12   22   2
Interval Estimate of the Difference
Between the Means of Two Populations:
Small-Sample Case (n1  30 and / or n2  30)
With  1 and  2Estimated by s1 and s2
x1  x 2  t a / 2 s x  x
1
(10.10)
2
10.2HYPOTHESIS TESTS ABOUT THE
DIFFERENCE BETWEEN THE MEANS OF
TWO POPULATIONS :INDIFFERENT
SAMPLE
 Large-sample Case

z
( x1  x 2 )  ( 1   2 )
 12 / n1   22 / n2
(10.11)
Small-Sample Case
t 
where
( x1  x 2 )  ( 1   2 )
s (1 / n1  1 / n 2 )
2
(10.12)
(n1  1) s12  (n2  1)s22
s 
n1  n2  2
2
10.3 INFERENCES ABOUT THE
DIFFERENCE BETWEEN THE MEANS OF
TWO POPULATIONS:MATCHD SAMPLE
 In choosing the sampling procedure that
will be used to collect production time data
and test the hypotheses ,we consider two
alternative designs:
 1.Independent sample design
 2.Matched sample design

t 
d 
sd /
d
n
(10.13)
10.4 INFERENCES ABOUT THE
DIFFERENCE BETWEEN THE
PROPORTIONS OF TWO POPULATIONS
 Point Estimator of the difference Between
the Proportions of Two Populations

p1  p2
(10.14)
Sampling Distribution of
 Expected Value:
E( p1  p2 )  p1  p2 (10.15)
 Standard deviation:
 p p 
1
2
p1  p2
p1 (1  p1 ) p2 (1  p2 )

n1
n2
(10.16)
Interval Estimation of
 Point Estimator of  p  p
1

s p1  p2 
p1 (1  p1 ) p2 (1  p2 )

n1
n2
p1  p2
2
(10.17)
 Interval Estimate of the Difference Between
the Two Populations: Large-Sample Case
With n1 p1 , n1 (1  p1 ), n2 p2 , and n2 (1  p2 )  5

p1  p 2  z a / 2 s p  p
1
(10.18)
2
Hypothesis Tests About

z
 p p
1
2
n1 p1  n2 p2
p
n1  n2


( p1  p 2 )  ( p1  p 2 )
sp p 
1
2
p(1  p)(
1 1
 )
n1 n2
(10.19)
(10.20)
(10.21)
p1  p2