Page 402 - DMTH404_STATISTICS

Statistics

Notes 2. For a given level of confidence and  , n is inversely related to Î , the square of the
2
2
Î
maximum error of estimation. This implies that to reduce Î to , the size of the sample
k
must be k times the original sample size.
2
3. The lesser the magnitude of Î, the more precise will be the interval estimate.

Example 9: What should be the sample size for estimating mean of a normal population
if the probability that sample mean differs from population mean by not more than 30% of
standard deviation is 0.99.
Solution.
Let n be the size of the sample. It is given that

P ( X -  0.30 ) 0.99= .... (1)


Assuming that the sampling distribution of X is normal with mean m and . .S E = , we can
X
n
write
æ  ö
-
P X   2.58 ÷ = 0.99 (from table of areas) .... (2)
ç
è n ø
Comparing (1) and (2), we get
 æ 2.58ö 2
0.30 = 2.58 Þ n = ç ÷ = 73.96 or 74 (approx.)
n è 0.30 ø

Example 10: A survey of middle class families of Delhi is proposed to be conducted for
the estimation of average monthly consumption (in Rs) per family. What should be the size of
the sample so that the average consumption is estimated within a range of Rs 300 with 95% level
of confidence. It is known that the standard deviation of the consumption in population is Rs
1,600.
Solution.
Let n denote the size of the sample to be drawn. With usual notations, we want to find n such that

P ( X -  300 ) = 0.95 .... (1)


Assuming that the sampling distribution of X is normal with mean m and . .S E = , we can
X
n
æ  ö
write P X -  1.96 ÷ = 0.95
ç
è n ø
´
æ 1.96 1600ö
or P X   ÷ = 0.95 .... (2)
-
ç
è n ø

394 LOVELY PROFESSIONAL UNIVERSITY