Page 352 - DMTH404_STATISTICS

Statistics

Notes
1 é 2 ù
= ê ns + å å Cov ( X X, j )ú
i
2
n ê ë i j ú û

Case I. If the sample is drawn with replacement, then X , X ...... X are independent random
1 2 n
variates and hence, Cov(X , X) = 0. Thus, we have
i j
ns 2 s 2
Var X 2 = .
( ) =
n n
Case II. If the sample is drawn without replacement, then

(
Cov X X, j ) = - s 2 , therefore,
i
N 1
-
1 é s 2 ù N n s 2
-
2
Var X 2 ê ns - ( n n 1- ) ú = ×
( ) =
-
-
n ë N 1 û N 1 n
-
N n
We note that if N   (i.e., population becomes large),  1 and therefore, in this case
-
N 1
2
s
( )
also, Var X = .
n
Remarks:
1. The standard deviation of a statistic is termed as standard error. The standard error of X ,
s
to be written in abbreviated form as S.E. X d i, is equal to , when sampling is with
n
-
s N n
replacement and it is equal to × , when sampling is without replacement.
n N 1
-
2. S.E. X d i is inversely related to the sample size.
N n
-
3. The term is termed as finite population correction (fpc). We note that fpc tends to
N 1
-
become closer and closer to unity as population size becomes larger and larger.
4. As a general rule, fpc may be taken to be equal to unity when sample size is less than 5%
of population size, i.e., n < 0.05N.

Example 1: Construct a sampling distribution of the sample mean for the following
population when random samples of size 2 are taken from it (a) with replacement and (b)
without replacement. Also find the mean and standard error of the distribution in each case.

Population Unit : 1 2 3 4
Observation : 22 24 26 28

Solution.

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