546 Chapter 9. Essential Statistics for Data Analysis
Withf=kx,weget
(
∂f
∂k) 2
= x^2⇒
1
f(
∂f
∂k) 2
=
x
k⇒∫ 1
01
f(
∂f
∂k) 2
dx =1
2 kSubstituting this in the above relation forNgives
N=
1
( k)^21
2 kBut we havek=0.21 and k=0. 05 k=0.0105, which gives
N =
1
(0.0105)^2
1
(2)(0.21)
=2. 1 × 104.
9.4 Confidence Intervals
Suppose we have a rough idea of the activity level in a radiation environment and
we use this information to estimate the dose that we expect a radiation worker to
receive while working there for some specific period of time. The problem with this
scheme is that there are a number of uncertainties involved in the computations,
such as level of actual activity, its space dependence (the radiation sources might
not be isotropic), and its variation with time. In such a situation what can be
done is to define a confidence interval within which the value isexpectedto lie with
a certainprobability. For example, we can say that there is 90% probability that
the person will receive a dose of somewhere between 10-20mrem.Herewehave
two parameters that we are reporting: the confidence interval and its associated
probability. The choice of a confidence interval is more or less arbitrary, although
generally it is based on some rationale, such as our rough estimation based on
some known parameters (in our example, we might have gotten a value of 15mrem
and then decided to give ourselves a leverage of± 5 mremto compensate for any
uncertainty in the calculations.). The probability, on the other hand, depends on
the confidence interval and the probability distribution.
If the probability distribution of a variablex(such as dose) is given byL(x), then
the probability thatxlies betweenx 1 andx 2 is given by
P(x 1 <x<x 2 )=∫x 2
∫x^1 L(x)dx
∞
−∞L(x)dx