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Statistical Methods in Economics

Notes
σ 2
( )
Var X n 2 14 7 ⎡ 22 ⎤
Efficiency = = πσ 2 = = = = 0.64 ⎢ Q π = ⎥
( Var )M π 22 11 ⎣ 7 ⎦
2n

() = 0.64 Var (M)
∴ Var X
Therefore, sample mean X is 64% more efficiency than the sample median.
Hence, the sample mean is more efficient estimator of the population mean as
compared to sample median.
(4) Sufficient Estimator: The last property that a good estimator should possess is sufficiency. An
ˆ
estimator θ is said to be a ‘sufficient estimator’ of a parameter θ if it contains all the informations
in the sample regarding the parameter. In other words, a sufficient estimator utilises all
informations that the given sample can furnish about the population. Sample means X is said
to be a sufficient estimator of the population mean.

28.3 Application of Point Estimation

The applications relating to point estimation are studied under two headings:
(1) Point Estimation in case of Single Sampling
(2) Point Estimation in case of Repeated Sampling.
(1) Point Estimation in case of Single Sampling: When a single independent random sample is
drawn from the unknown population, the point estimate of the population parameter can be
illustrated by the following examples:
Example 7: A sample of 10 measurements of the diameter of a sphere gave a mean X = 4.38
inches and a standard deviation = .06 inches. Determine the unbiased and efficient
estimates of (a) the true mean (i.e., population mean) and (b) the true variance (i.e.,
population variance).
Solution: We are given: n = 10, X = 4.38, s = .06

(a) The unbiased and efficient estimate of the true mean μ is given by:
X = 4.38
(b) The unbiased and efficient estimate of the true variance σ 2 is:

n 2
ˆ s 2 = ⋅s
n − 1
Putting the values, we get

10
ˆ s 2 = 10 − 1 ×.06 = 1.11 × 0.06 = .066

2
Thus, μ = 4.38, σ = 0.666
Example 8: The following five observations constitute a random sample from an unknown
population:

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