Page 366 - DECO504_STATISTICAL_METHODS_IN_ECONOMICS_ENGLISH

Unit 28: Theory of Estimation: Point Estimation, Unbiasedness, Consistency, Efficiency and Sufficiency

Solution: A random sampling is one where each sample has an equal chance of being selected. Notes
We draw a random sample of size ‘n’.
Then,
x + ⎡ + x + ... x ⎤ 1
E ( ) x = ⎢ E 2 n ⎥ Where x is the sample observation.
⎣ n ⎦ 1

1
= ⎣ ( ⋅ ) 1 ( + ⎡ ) 2 + E x ( ...E x ) n ⎤E x ⎦
n
Now the expected values of x (a member of the population) is population mean μ .
i
∴ ( ) x = [ E 1 μ + + μ + ... ]μ ⎡ ⎣ ( QE x ) 1 ( =E x ) 2 =...E x ) n = μ⎤ ⎦
(
n
1
n
= [ ⋅ n ] μ = μ [ Q 1 +C = C C 2 + ...C = C ] Σ
n
n
Thus, sample mean X is an unbiased estimate of population mean.
Self-Assessment

1. Fill in the Blanks:
(i) The two types of estimates are ............ and ............ .
(ii) The numerical value of a sample mean is said to be an estimate of the population ............
figure.
(iii) A point estimate is a single number which is used as an estimate of the unknown ............
parameter.
(iv) Point estimate provides one single value of the ............ .
(v) Parameter of a sample denoted by ............ .
28.4 Summary

• The topic of estimation in Statistics deals with estimation of population parameters like mean
of a statistical distribution. It is assumed, that the concerned variable of the population follows
a certain distribution with some parameter(s). For instance, it may be assumed that the life of
the electric bulbs follows a normal distribution which has two parameters viz. mean (m) and
σ
standard deviation () . While one of the parameters, say, standard deviation is known to be
equal to 200 hours from past experience, the other parameter, viz. the mean life of the bulbs, is
not known, and which we wish to estimate.
• An example of point and interval estimation could be provided from our day-to-day conversation
when we talk about commuting time to office. We do make statements like “It takes about 45
minutes ranging from 40 to 50 minutes depending on the traffic conditions.” The statistical
details of these two types of estimation are described below.
• A point estimate is a single value, like 10, analogous to a point in a geometrical sense. It is used
to estimate a population parameter, like mean, with the help of a sample of observations.
• It may be noted that the observations x , x , x , ... x are random variables, and therefore, any
1 2 3 n
function of these observations will also be a random variable. Any function of the sample
observations is called a Statistic.
• the standard deviation of the sample mean is known as the standard error of the mean. It is a
measure of the extent to which sample means could be expected to vary from sample to sample.

LOVELY PROFESSIONAL UNIVERSITY 361