Page 373 - DECO504_STATISTICAL_METHODS_IN_ECONOMICS_ENGLISH

Statistical Methods in Economics

Notes Such an interval, when it exists, is called a Confidence Interval for θ . The two quantities t and t
1 2
which serve as the lower and upper limits of the interval are known as Confidence Limits. The probability
(c) with which the confidence interval will include the true value of the parameter is known as
Confidence Coefficient of the interval.
The significance of confidence limits is that if many independent random samples are drawn from
the same population and the confidence interval is calculated from each sample, then the parameter
will actually be included in the intervals in c proportion of cases in the long run. Thus the estimate of
the parameter is stated as an interval with a specified degree of confidence.
The calculation of confidence limits is based on the knowledge of sampling distribution of an
appropriate statistic. Suppose, we have a random sample of size n from a Normal population
N ( μ σ, 2 ) , where the variance σ 2 is known. It is required to find 95% confidence limits for the

unknown parameter μ . We know that the sample mean ( ) x follows normal distribution with mean

μ and variance σ /n , and so
2
x − μ
z = σ
n
has a standard normal distribution. Since 95% of the area under the standard normal curve lies
between the ordinates at z = ± 1.96, we have

⎡ ⎤
⎢ x − μ ⎥
− P ≤ ⎢ ≤ 1.96 ⎥ 1.96
⎢ σ ⎥ = 0.95
⎢ n ⎦ ⎥ ⎣

i.e. in 95% of cases the following inequalities hold
x − μ
− ≤1.96 ≤ 1.96
σ
n

Separating out μ we get
σ σ
x − 1.96 ≤ μ ≤ +1.96x
n n

⎡ σ σ ⎤
The interval ⎢ x − , 1.96 x + 1.96 ⎥ is known as the 95% confidence interval for μ , and the 95%
⎣ n n ⎦
confidence limits are
σ
x ± 1.96
n

Again, 99% of area under the standard normal curve lies between the ordinates at z = ± 2.58, and
99.73% (i.e. almost whole) of the area lies between z = ± 3. Hence proceeding exactly in the same
manner, the 99% confidence limits for μ are
σ
x ± 2.58
n

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