Sampling Distribution of the Sample Proportion:
Suppose that the size of a population is N. Each element of the
population can be classified as type A or non-type A. Let p be
the proportion of elements of type A in the population. A random
sample of size n is drawn from this population. Let p̂ be the
proportion of elements of type A in the sample.
Let X = no. of elements of type A in the sample
p =Population Proportion
no. of elements of type A in the population

N
p̂ = Sample Proportion
no . of elements of type A in the sample X


n
n
Result:
(1)
X ~ Binomial (n, p)
(2)
E( p̂ )= E( X )= p
n
(3)
X
Var( p̂ ) = Var(
)=
(4)
For large n, we have
n
p̂ ~ N(p,
pq )
n
pq
n
; q 1 p
(Approximately)
pˆ  p
Z
~ N(0,1) (Approximately)
pq
n
Sampling Distribution of the Difference between Two
Proportions:
Suppose that we have two populations:
· p1 = proportion of the 1-st population.
· P2 = proportion of the 2-nd population.
·
We are interested in comparing p1 and p2, or equivalently,
making inferences about p1 p2.
·
We independently select a random sample of size n1 from
the 1-st population and another random sample of size n2 from
the 2-nd population:
· Let X1 = no. of elements of type A in the 1-st sample.
· Let X2 = no. of elements of type A in the 2-nd sample.
X1
· pˆ1 
= proportion of the 1-st sample
n1
X2
· pˆ 2 
= proportion of the 2-nd sample
n2
· The sampling distribution of pˆ 1  pˆ 2 is used to make
inferences about p1 p2.
Result:
(1) E ( pˆ 1  pˆ 2 )  p1  p2
(2) Var( pˆ1  pˆ 2 )  p1 q1  p2 q2 ; q1  1  p1 , q2  1  p2
n1
n2
(3)
For large n and n , we have
1
2
p1 q1 p2 q2
pˆ 1  pˆ 2 ~ N ( p1  p2 ,

)
n1
n2
(p̂1  p̂ 2 )  (p1  p 2 )
Z
~ N(0,1)
p1 q1 p 2 q 2

n1
n2
(Approximately)
(Approximately)
Critical Values of the t-distribution (t )
Critical Values of the t-distribution (t )