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Statistical Methods in Economics

Notes
σ 2 1
(10) for Variance-Ratio 2 : (means unknown)
σ 2

n
n
If two independent random samples of sizes 1 and 2 are drawn from two Normal populations
s 2 / σ 2
2
2
with unknown means μ μ, 2 and variances σ , σ respectively, then s 1 2 2 / σ 1 2 2 follows F
1
1
2
distribution with degrees of freedom ( n − 2 − 1,n 1 . If F .975 and F .025 denote the lower and the
) 1
upper 2.5% points of F distribution, we have with probability 95% the following inequalities
s 2 / σ 2
F .975 ≤ 1 1 ≤ F .025 ... (15)
s 2 2 / σ 2 2
2
2
The 95% confidence interval for σ / σ can be obtained from this as
1
2
1 ⋅ s 1 2 ≤ σ 1 2 ≤ 1 ⋅ s 1 2
F .025 s 2 2 σ 2 2 F .975 s 2 2 ... (16)
2
2
2
2
where s and s denote unbiased estimators of σ , σ respectively from the two samples.
2
1
2
1
Self-Assessment
1. Tick the Correct Answer
(i) What is the best description of a point estimate ?
(a) any value from the sample used to estimate a parameter.
(b) a sample statistic used to estimate a parameter.
(c) the margin of error used to estimate a parameter.
(ii) Which best describes the lower endpoint of a confidence interval ?
(a) point estimate
(b) point estimate minus margin of error
(c) point estimate plus margin of error
(iii) Which best describes the upper endpoint of a confidence interval ?
(a) point estimate
(b) point estimate minus margin of error
(c) point estimate plus margin of error
(iv) Which value will be at the centre of a confidence interval
(a) population parameter
(b) point estimate
(c) margin of error

(v) What is the relationship between a 95% confidence interval and a 99% confidence interval
from the same sample ?
(a) the 95% interval will be wider
(b) the 99% interval will be wider
(c) both intervals have the some width

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