Page 365 - DECO504_STATISTICAL_METHODS_IN_ECONOMICS_ENGLISH

Statistical Methods in Economics

Notes 1 1
) +
7. (3, 1) ( 31 = 2.0 ⋅ ⎡ ( ) − 32 2 + ( ) − 1 2 2 ⎤ = 1.00 2.00
2 2 ⎣ ⎦
1 1 ⎡ 2 2 ⎤
) +
8. (3, 2) ( 32 = 2.5 ⎣ ( 3 ) − + ( 2.5 2 2.5 ) − ⎦ = 0.25 0.50
2 2
1 1 ⎡ 2 2 ⎤
) +
9. (3, 3) ( 33 = 3.0 ⎣ ( 3 ) − + ( 3 3 3 ) − ⎦ = 0.00 0.00
2 2
Total k = 9 ΣX = 18 Σˆ s 2 = 6

ΣX 18
(a) Mean of Sampling Distribution of Means = μ = k = 9 = 2. Here, K = No. of
x
samples.
++ 3
12
Population Mean μ = = 2.
3
Since, μ = μ , sample mean X is an unbiased estimate of the population mean
x
μ .

Σs 2 3 1
(b) Mean of the Sampling Distribution of Variance = μ 2 = k = 9 = 3
s
( 12 ) − 2 + ( ) − 2 ( 2 2 3 2 ) − 2 2
Population Variance σ 2 = =
3 3

2
≠ σ , sample variance s is not an unbiased estimate of the population
2
Since, μ 2
s
2
σ
variance ( ) .
n 2
But the modified sample variance defined as ˆ s 2 = s will be unbiased estimate
n − 1
of the population variance σ 2 because:
μ 2 = Σˆ s 2 = 6 = 2
ˆ s k 9 3

2
σ 2 = 3

∴ μ ˆ s 2 = σ 2

Since μ ˆ s 2 = σ 2 , the modified sample variation is an unbiased estimate of the
population variance.

X
Example 12: Show that the sample mean () is an unbiased estimate of the population mean.
or
An independent random sample x , x , x , ..., x is drawn from a population with
1 2 3 n
mean μ . Prove that the expected value of the sample mean X equals the population
mean μ .

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