Page 362 - DECO504_STATISTICAL_METHODS_IN_ECONOMICS_ENGLISH

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

6.33, 6.37, 6.36, 6.32 and 6.37 centimeters. Notes
Find out unbiased and efficient estimates of (a) true mean, and (b) true variance.
Solution: (a) The unbiased and efficient estimate of the true mean (i.e. population mean) is
given by the value of

+
ΣX + 6.33 6.37 + + 6.36 6.32 6.37 31.75
X = n = 5 = 5 = 6.35
(b) The unbiased and efficient estimate of the true variance (i.e., population variance)
is:

Σ ( ) − XX 2
ˆ s 2 =
n − 1

where, ˆ s 2 = modified sample variance.

( )6.33 6.35 2 + ( − )6.37 6.35 2 + ( − − )6.36 6.35 2
=
−
51
( ) + 2 + ( − 6.32 6.35 − )6.37 6.35 2

.0022
= = .00055 cm 2
4
Example 9: The following data relate to a random sample of 100 students in Kurukshetra
University classified by their weights (kg):
Weight (kg): 60—62 63—65 66—68 69—71 72—74

No. of Students: 5 18 42 27 8

Determine unbiased and efficient estimates of (a) population mean and (b) population
variance.
Solution:

Calculation of Mean and Variance
Weight No. of Students (f) M.V. (m) A = 67 d = m – A d’ = d/3 fd’ fd’ 2

60—62 5 61 – 6 – 2 – 10 20
63—65 18 64 – 3 – 1 – 18 18
66—68 42 67 0 0 0 0
69—71 27 70 + 3 + 1 + 27 27
72—74 8 73 + 6 + 2 + 16 32

2
n = 100 Σ '= 15 Σ ' = 97
f
d
d
f
(a) The unbiased and efficient estimate of the population mean is given by the value:
Σ '
fd
X = Α+ n × i

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