Mathematical Methods for Physics and Engineering : A Comprehensive Guide

STATISTICS

31.3.5 Confidence limits for a Gaussian sampling distribution

An important special case occurs when the sampling distribution is Gaussian; if

the mean isaand the standard deviation isσˆathen

P(aˆ|a, σaˆ)=

1

√

2 πσ^2 aˆ

exp

[

−

(aˆ−a)^2

2 σ^2 aˆ

]

. (31.30)

For almost any (consistent) estimatoraˆ, the sampling distribution will tend to

this form in the large-sample limitN→∞, as a consequence of the central limit

theorem. For a sampling distribution of the form (31.30), the above procedure

for determining confidence intervals becomes straightforward. Suppose, from our

sample, we obtain the valueˆaobsfor our estimator. In this case, equations (31.28)

and (31.29) become

Φ

(

aˆobs−a+

σˆa

)

=α,

1 −Φ

(

aˆobs−a−

σˆa

)

=β,

where Φ(z) is the cumulative probability function for the standard Gaussian distri-

bution, discussed in subsection 30.9.1. Solving these equations fora−anda+gives

a−=aˆobs−σˆaΦ−^1 (1−β), (31.31)

a+=aˆobs+σˆaΦ−^1 (1−α); (31.32)

we have used the fact that Φ−^1 (α)=−Φ−^1 (1−α) to make the equations symmetric.

The value of the inverse function Φ−^1 (z) can be read off directly from table 30.3,

given in subsection 30.9.1. For the normally used central confidence interval one

hasα=β. In this case, we see that quoting a result using the standard error, as

a=aˆ±σˆa, (31.33)

is equivalent to taking Φ−^1 (1−α) = 1. From table 30.3, we findα=1− 0 .8413 =

0 .1587, and so this corresponds to a confidence level of 1−2(0.1587)≈ 0 .683.

Thus, the standard error limits give the 68.3% central confidence interval.

Ten independent sample valuesxi,i=1, 2 ,..., 10 , are drawn at random from a Gaussian

distribution with standard deviationσ=1. The sample values are as follows (to two decimal

places):

2 .22 2.56 1.07 0.24 0.18 0.95 0. 73 − 0 .79 2.09 1. 81

Find the90%central confidence interval on the population meanμ.

Our estimatorμˆis the sample mean ̄x. As shown towards the end of section 31.3, the
sampling distribution of ̄xis Gaussian with meanE[ ̄x] and varianceV[x ̄]=σ^2 /N.Since
σ= 1 in this case, the standard error is given byσxˆ=σ/

√

N=0.32. Moreover, in
subsection 31.3.3, we found the mean of the above sample to be ̄x=1.11.