Материал: Крючков Фундаменталс оф Нуцлеар Материалс Пхысицал Протецтион 2011

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			− μ	.
t =		X			(4.54)


	σ n

If the hypothesis is just, then the quantity t obeys to Student’s distribution with the (n– 1) degree of freedom.

Example. Let there be a group of people growing as follows: 160, 160, 167, 170, 173, 176, 178, 178, 181 and 181.

Let hypothesis Н0 consist in that these values are distributed by normal law with the average value of 167 cm and the hypothesis Н1 in that `Х > 167.

Solution:

∑xi

n=1

= 172.4 cm;

∑(xi −172.4)2

σ 2

i=1

= 62.9;

σ = 7.93 cm;

62.9

= 2.51 cm;

t =

172.4 −167.0

= 2.15 ( p = 0.90, n −1 = 9);

2.51

t p = 1.83.

So the hypothesis is rejected.

Fisher criterion

The

Fisher criterion is used

to check if variances for two normal

samples differ significantly.

When two variances are compared, it is convenient to check the relation:

Fv ,v	2	= σ12	,	(4.55)
1	2	σ 22
		σ 22

and compare this quantity with that in the table for the number of the degrees of freedom ν1 and ν2 and the significance level р.

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Example. Let there be data from three laboratories (Table 4.4):

∑ ∑xij ,

(4.56)

30 i=1 j=1

i =

∑xij .

(4.57)

10 j=1

Table 4.4

Measurement data from three laboratories

Meas.

Laboratory number

number

Х1.1

Х2.1

Х3.1

Х1.2

Х2.2

Х3.2

Х1.3

Х2.3

Х3.3

…….

Х1.10

Х2.10

Х3.10

`Xi

∑x1, j

∑x2, j

∑x3, j

10 j =1

The criterion is

F =

σ between groups

< F p,ν inside ,ν between ,

(4.58)

σ inside groups

where ν inside = ∑ni

− k = 27;ν between = k −1 = 2,

ni is the

number of

i=1

measurements in the i laboratory, and k is the number of laboratories.

The formulas to calculate the number of the degrees of freedom and the variances are given in Table 4.5.

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Table 4.5

Basic formulas to calculate the numbers of degrees of freedom and variances

Parameter

Total

Inside groups

Between groups

Number of

degrees of

N– 1

∑ni − k

k– 1

freedom

i=1

k ni

∑∑(xij −

i )

k ni

Variance

∑∑(xij

− X )

i=1 j=1

∑ni (Xi − X )2

i=1 j =1

i=1

N − 1

∑ni − k

k − 1

i=1

Sampling studies

Sampling studies are used in cases where complete enumeration cannot be done or makes no economic sense. This particularly concerns quality checks of some product or material types requiring destruction (destructive analyses for NM).

The specimen or some of the items taken for the study is called a sample (or a sample set), while the whole set of items sampled is called a universe.

A sampling check is done to decide on the nature of a set based on the set’s representative sample.

Observations and measurements will be done only for a sample taken from a universe. The quality of the sampling study results is sure to depend primarily on to what extent the sample composition is representative of the universe or, in other words, on to what extent the sample is representative.

To be representative, a sample needs items and specimens to be randomly selected. Randomness suggests that no factor other than chance determines if the sample should include an item or not.

There are different ways to form samples (sample sets) depending on the characteristics thereof and the characteristics of the universe [6].

Set characteristics

Quantitative characteristics:

∙total quantity of nuclear material;

∙average level of enrichment in 235U. Qualitative characteristics:

198

∙proper use of tamper indicating devices;

∙the numbers on tested items which should coincide with database.

To have a representative sample, one should be aware of how the characteristic of interest is distributed inside the set:

∙ uniformly throughout the set; ∙ uniformly within clusters;

∙ uniformly inside layers; ∙ not known.

Each sampling test program includes the following items. 1. Purpose of sampling test.

2. Analysis of the set to be tested and characteristics thereof. 3. Analysis of constraints and statistical concerns.

4. Analysis of constraints and nonstatistical concerns.

5. Calculation of sample size.

6. Selection of sampling strategy.

Let us take a closer look at the above items.

Purpose of sampling test:

∙check of earlier measurements;

∙determination of the NM quantity as described in the given inventory listing;

∙check of database records and the inventory listing for conformity;

∙finding out if tamper indication devices are properly applied.

Identification of the set to be tested. When analyzed, the set should

have its sample-affecting characteristics identified, so one shall find out:

∙if the set is homogeneous in terms of the given characteristic;

∙if not, if the set layers are homogeneous;

∙how the items of interest are stored;

∙if radiation safety concerns exist.

Statistical characteristics. Qualitative (discrete) characteristics:

∙set size;

∙distribution of the given characteristic inside the set;

∙required significance level;

∙admissible number of defects.

So, for example, a study is required to secure with a 95% probability that not more than 1% of the set elements are defective.

Quantitative (continuous) characteristics:

∙what loss, if any, should be regarded significant;

∙estimate uncertainty;

199

∙ permissible number of type I and II errors (type I error is a false loss detection (the true hypothesis is rejected), type II error is omission of a real loss (a false hypothesis is accepted)).

The foregoing determines to some extent the size of the desired sample.

Nonstatistical constraints:

a)regulatory requirements;

b)safety and security requirements;

c)constraints determined by limited physical resources, finding included;

d)time constraints.

The sample size is determined by the following parameters:

∙size of the set to be tested;

∙maximum permissible number of defects;

∙significance level.

An inspection to check the system of tamper indicating devices (TID) suggests a check for two reliability conditions.

Condition 1. The recording system should show the exact location and identification of at least 99% of the TIDs.

Condition 2. The TIDs should be properly applied in not less than 95% of cases.

For a sampling test, the confidential probability is taken as 95%. Example. Let us consider a set of 2000 items. We shall try to determine

the minimum sample size at which one can check with a probability of 95% (i.e. α = 0.05) the second condition: not more than 5% of the elements (i.e. not more than 100) have defects.

The function of distributing the enumeration probability for х defective elements in a sample of n elements from a set consisting of N items and having only D defective items is described by a hypergeometric distribution (4.21):

D N

p(x) =	x n
	N

	n
where, in our case, х = 0, 1, 2,…, min(	D,n).

It is obvious that the smallest sample defective items are tolerable, so then:

− D −

x ,

will be for the case where 0

200

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