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E.1 Sign Test

Contents:
E.1.1 Sample sizes for Sign Test;
E.1.2 Critical values for Sign Test;
E.1.3 Power of the Sign Test;

E.1.1 Sample sizes for Sign Test

Table E.1 Sample sizes for Sign Test.
(Number of measurements to be performed in each survey unit)

(α,β) or (β,α)
0.01 0.01 0.01 0.01 0.01 0.025 0.025 0.025 0.025 0.05 0.05 0.05 0.1 0.1 0.25
Δ/σ 0.01 0.025 0.05 0.1 0.25 0.025 0.05 0.1 0.25 0.05 0.1 0.25 0.1 0.25 0.25
0.1 4095 3476 2984 2463 1704 2907 2459 1989 1313 2048 1620 1018 1244 725 345
0.2 1035 879 754 623 431 735 622 503 333 518 410 258 315 184 88
0.3 468 398 341 282 195 333 281 227 150 234 185 117 143 83 40
0.4 270 230 197 162 113 192 162 131 87 136 107 68 82 48 23
0.5 178 152 130 107 75 126 107 87 58 89 71 45 54 33 16
0.6 129 110 94 77 54 92 77 63 42 65 52 33 40 23 11
0.7 99 83 72 59 41 70 59 48 33 50 40 26 30 18 9
0.8 80 68 58 48 34 57 48 39 26 40 32 21 24 15 8
0.9 66 57 48 40 28 47 40 33 22 34 27 17 21 12 6
1 57 48 41 34 24 40 34 28 18 29 23 15 18 11 5
1.1 50 42 36 30 21 35 30 24 17 26 21 14 16 10 5
1.2 45 38 33 27 20 32 27 22 15 23 18 12 15 9 5
1.3 41 35 30 26 17 29 24 21 14 21 17 11 14 8 4
1.4 38 33 28 23 16 27 23 18 12 20 16 10 12 8 4
1.5 35 30 27 22 15 26 22 17 12 18 15 10 11 8 4
1.6 34 29 24 21 15 24 21 17 11 17 14 9 11 6 4
1.7 33 28 24 20 14 23 20 16 11 17 14 9 10 6 4
1.8 32 27 23 20 14 22 20 16 11 16 12 9 10 6 4
1.9 30 26 22 18 14 22 18 15 10 16 12 9 10 6 4
2 29 26 22 18 12 21 18 15 10 15 12 8 10 6 3
2.5 28 23 21 17 12 20 17 14 10 15 11 8 9 5 3
3 27 23 20 17 12 20 17 14 9 14 11 8 9 5 3

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E.1.2 Critical values for the Sign Test

Table E.2 Critical values for the Sign test statistic S+

Alpha
N 0.005 0.01 0.025 0.05 0.1 0.2 0.3 0.4 0.5
4 4 4 4 4 3 3 3 2 2
5 5 5 5 4 4 3 3 3 2
6 6 6 5 5 5 4 4 3 3
7 7 6 6 6 5 5 4 4 3
8 7 7 7 6 6 5 5 4 4
9 8 8 7 7 6 6 5 5 4
10 9 9 8 8 7 6 6 5 5
11 10 9 9 8 8 7 6 6 5
12 10 10 9 9 8 7 7 6 6
13 11 11 10 9 9 8 7 7 6
14 12 11 11 10 9 9 8 7 7
15 12 12 11 11 10 9 9 8 7
16 13 13 12 11 11 10 9 9 8
17 14 13 12 12 11 10 10 9 8
18 14 14 13 12 12 11 10 10 9
19 15 14 14 13 12 11 11 10 9
20 16 15 14 14 13 12 11 11 10
21 16 16 15 14 13 12 12 11 10
22 17 16 16 15 14 13 12 12 11
23 18 17 16 15 15 14 13 12 11
24 18 18 17 16 15 14 13 13 12
25 19 18 17 17 16 15 14 13 12
26 19 19 18 17 16 15 14 14 13
27 20 19 19 18 17 16 15 14 13
28 21 20 19 18 17 16 15 15 14
29 21 21 20 19 18 17 16 15 14
30 22 21 20 19 19 17 16 16 15
31 23 22 21 20 19 18 17 16 15
32 23 23 22 21 20 18 17 17 16
33 24 23 22 21 20 19 18 17 16
34 24 24 23 22 21 19 19 18 17
35 25 24 23 22 21 20 19 18 17
36 26 25 24 23 22 21 20 19 18
37 26 26 24 23 22 21 20 19 18
38 27 26 25 24 23 22 21 20 19
39 27 27 26 25 23 22 21 20 19
40 28 27 26 25 24 23 22 21 20
41 29 28 27 26 25 23 22 21 20
42 29 28 27 26 25 24 23 22 21
43 30 29 28 27 26 24 23 22 21
44 30 30 28 27 26 25 24 23 22
45 31 30 29 28 27 25 24 23 22
46 32 31 30 29 27 26 25 24 23
47 32 31 30 29 28 26 25 24 23
48 33 32 31 30 28 27 26 25 24
49 33 33 31 30 29 27 26 25 24
50 34 33 32 31 30 28 27 26 25

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For N greater than 50, the table (critical) value can be calculated from N/2 + z/2*√(N), where z is the (1-α) percentile of a standard normal distribution, which can be found in Table E.3 or in Table 3.33.

E.1.3 Power of the Sign Test

The power of the Sign test for detecting residual radioactivity at the concentration level LBGR = DGCL – Δ, may be found using equation E-1.

1 – β = 1 – Σki=0 (iN)(q*)i.(1 – q*)N-i
1 – β = 1 – Φ((k – Nq*)/√(Nq*(1-q*))) …………………………………………………. (E.1)

with

q* = Φ(Δ/σ) …………………………………………………. (E-2)

The function Φ(z) is the standard normal cumulative distribution function tabulated in Table E.1. Note that if Δ/σ is large, q* approaches one, and the power also approaches one. This calculation can be performed for other values, Δ*, in order to construct a power curve for the test. These calculations can also be performed using the standard deviation of the actual measurement data, s, in order to construct a retrospective power curve for the test. This is an important step when the null hypothesis is not rejected, since it demonstrates whether the DQOs have been met.

The retrospective power curve for the Sign test can be constructed using Equations E-1 and E-2, together with the actual number of concentration measurements obtained, N. The power as a function of Δ/σ is calculated. The values of Δ/σ are converted to concentration using:

Concentration = DCGLW – (Δ/σ)(observed standard deviation)

The results for the Class 3 exterior survey unit example of Section 3.10.3.5 are plotted in Figure E.1. This figure shows the probability that the survey unit would have passed the release criterion using the Sign test versus concentration of residual radioactivity. This curve shows that the data quality objectives were met, despite the fact that the actual standard deviation was larger than that used in designing the survey. This is primarily due to the additional 20% that was added to the sample size, and also that sample sizes were always rounded up. The curve shows that a survey unit with less than 135 Bq/kg would almost always pass, and that a survey unit with more than 145 Bq/kg would almost always fail.

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Figure D.1 Retrospective power curve for Class 3 exterior survey unit
Figure E.1 Retrospective power curve for Class 3 exterior survey unit.

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