Page 374 - DECO504_STATISTICAL_METHODS_IN_ECONOMICS_ENGLISH

Unit 29: Methods of Point Estimation and Interval Estimation

and almost sure limits for μ are: Notes
σ
x ± 3
n
In fact, using values from the normal probability integral table (showing areas under standard normal
curve), confidence limits corresponding to any specified percentage can be obtained. These are exact
confidence limits.
In some cases, the population may not be truely a normal distribution, but the sampling distributions
of statistics based on large samples are approximately normal. For example, the sample mean ( ) x based
on a large random sample drawn (with or without replacement) from any population is approximately
normally distributed. Similarly, the sample proportion (p) calculated from a large random sample
has approximately a normal distribution. It is therefore possible to utilise the properties relating to
the percentage of area under the standard normal curve to find approximate confidence limits for the
population mean μ and the population proportion P, provided the sample size n is large.
Approximate Confidence Limits (large samples) any Distribution

(1) for Mean μ :

95% confidence limits = ± 1.96 S.E. of x ( ) x

99% confidence limits = ± 2.58 S.E. of x ( ) x
x
Almost sure limits = ± 3 (S.E. of x ) ... (5)
(2) for Proportion P:
95% confidence limits = ± 1.96 S.E. of p ( ) p

99% confidence limits = ± 2.58 S.E. of p ( ) p

Almost sure limits = ± 3 S.E. of p ( ) p ... (6)

(3) for Difference of Means ( 1 μ − 2 )μ :
95% confidence limits = ( − x 2 ) 1 ± x 1.96 (S.E. of x 1 − x 2 )

99% confidence limits = ( − x 2 ) 1 ± x 2.58 (S.E. of x 1 − x 2 )

Almost sure limits = ( − x 2 ± x 3S.E. of 1 − x x 2 ) ... (7)
(
) 1
(4) for Difference of Proportions( 1 2 ) :
P– P
95% confidence limits = ( 1 − p 2 ) ± p 1.96 (S.E. of p 1 − p 2 )

99% confidence limits = ( 1 − p 2 ) ± p 2.58 (S.E. of p 1 − p 2 )

Almost sure limits = ( 1 − p 2 ) ± p ( 3S.E. of p 1 − p 2 ) ... (8)

LOVELY PROFESSIONAL UNIVERSITY 369