Page 355 - DMTH404_STATISTICS

Unit 24: Sampling Distributions

Notes
s æ N n s 2 ö
-
with mean m and standard error ç or × ÷ as n becomes larger and larger.
-
n ç è N 1 n ÷ ø
Remarks: As a general rule, when n ³ 30, the sampling distribution of X is taken to be normal for
practical purposes.

Application of the Sampling Distribution

Decisions by various government and non-government agencies are made on the basis of sample
results. For example, a sales manager may take a sample of quantities purchased of its product
to predict sales. A government agency may take a sample of residents to assess the effect of a
certain welfare program etc. Thus, in order to draw reliable conclusions, we must have a sound
knowledge regarding the sample. An extremely common and quite useful knowledge about the
sample is given by the sampling distribution of the relevant statistic.
An important application of sampling distribution is to determine the probability of the statistic
lying in a given interval.

24.2.2 Sampling Distribution of the Difference Between two Sample
Means

Let there be two populations of sizes N and N , means m and m and standard deviations s and
1 2 1 2 1
s respectively. Let X be the mean of the random sample of size n obtained from the first
1
2 1
population and X be the mean of the random sample of size n obtained from the second
2
2
population. Thus, we can regard X and X as two independent random variables with means
1
2
m and m and standard errors as
1 2
s 1 æ N - n s 1 2 ö s 2 æ N - n 2 s 2 2 ö
1
2
1
ç or × ÷ and ç or × ÷ respectively.
n ç N - 1 n ÷ n ç N - 1 n ÷
2 è
1 è
1 ø
2 ø
1
2
Further, their difference, X - X , will also be a random variable with mean
d
1
2
=E X - X i =E X d i- E X d i =m - m and standard error
1 2 1 2 1 2
( ) Var X+
= Variance ( X - X 2 ) = Var X 1 ( )
2
1
2 2
s 1 s 2
= + when both the samples are drawn using srswr) or
n n
1 2
N - n s 1 2 N - n 2 s 2 2
1
2
1
= × + × (when both the samples are drawn using srswor).
N - 1 n 1 N - 1 n 2
1
2
Remarks:
1. When both the populations are normal, then X - X will be distributed normally with
1
2
2 2
s 1 s 2
mean m - m and standard error + .
1 2
n n
1 2
LOVELY PROFESSIONAL UNIVERSITY 347