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 – Σ^{k}_{i=0} (i^{N})(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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