Page 354 - DMTH404_STATISTICS

Statistics

Notes shown below:

Sample No. Sample Values X
1 22,24 23
2 22,26 24
3 22,28 25
4 24,26 25
5 24,28 26
6 26,28 27
1
Since all the samples are equally likely, the probability of each value of X is . Thus, we
6
can write the sampling distribution of X as
X 23 24 25 26 27 Total
p X d i 1 1 2 1 1 1
6 6 6 6 6

1
( ) =
Further, m = E X [23 24 25 2 26 27+ + ´ + + ] 25.=
X 6
2
To find S.E. X d i, we first find E X e j given by
1 2 2 2 2 2 3760
2
E X é 23 + 24 + 2 25 + 26 + 27 ù = = 626.67.
( ) =
´
6 ë û 6
2
Thus, s = 626.67 25 = 1.67 = 1.292.
-
X
2
-
-
N n s 4 2 5
Alternatively, s = × = ´ = 1.67 = 1.292.
X
-
N 1 n 3 2
24.2.1 Nature of the Sampling Distribution of Mean
It can be deduced that when a random sample X , X ...... X is obtained from a normal population
1 2 n
with mean m and standard deviation s, then each of the X 's are also distributed normally with
i
mean m and standard deviation s.
By the use of additive (or reproductive) property of normal distribution, it follows that the
distribution of X , a linear combination of X , X ...... X , will also be normal. As shown in the
1 2 n
s
previous section, the mean and standard error of the distribution would be m and respectively.
n
Remarks: Since normal population is often a large population, the fpc is always taken equal to
unity.
The nature of the sampling distribution of X , when parent population is not normal, is provided
by Central Limit Theorem. This theorem states that:

If X , X ...... X is a random sample of size n from a non-normal population of size N with mean
1 2 n
m and standard deviation s, then the sampling distribution of X will approach normal distribution

346 LOVELY PROFESSIONAL UNIVERSITY