Figure 7-5 shows a 7-well design used for production units with moderate connectivity. The 7-well design contains 25 well zones with an average of approximately 7 wells in each zone. As shown  in Table 7-10, 17 monitoring wells are required to achieve 90% confidence that at least P=95% of the well zones are in compliance. To achieve 95% confidence that at least 95% of the well zones are in compliance requires 20 monitoring wells.

Figure 7-6 and Figure 7-7 show the 11-well and 20-well design options used for production units with good or high connectivity, respectively. The 11-well design contains 16 well zones with an average of approximately 11 wells in each zone. The 20-well design contains 9 well zones with an average of approximately 20 wells in each zone. Based on Table 7-10, 15 monitoring wells are required in the 11-well design to achieve 90% confidence that at least 95% of the well zones are in compliance. In the 20-well design, all 9 production units must be monitored to achieve the 90% confidence level. To achieve 95% confidence that at least 95% of the well zones are in compliance, all well zones must be monitored when using the 11-well or 20-well designs.

Figure 7-8 through Figure 7-11 show 4 sampling design options for 7-spot production units for units with minimal, low, good and high connectivity. Figure 7-8 shows a 2-well design with 72 well zones and an average of 2.25 wells per zone. Figure 7-9 describes a 4-well design with 36 well zones and an average of 4.5 wells per zone. Figure 7-10 shows a 10-well design with 16 well zones and an average of 10 wells per zone. Figure 7-11 shows an 18-well design with 9 well zones and an average of 18 wells per zone.

Table 7-11 shows the value of N required to achieve 90% confidence that at least P percent of the well zones are in compliance when using a 5-spot pattern. Table 7-13 is used for a 90% confidence level with 7-spot patterns.
Table 7-10 and Table 7-11 are useful only for production units with 181 wells, and Table 7-12 and Table 7-13 for 162 wells. For larger or smaller production units with different shapes, the number of well zones determined using a k-well design will vary. Table 7-14 and Table 7-15 contain the information required to construct Table 7-10 through Table 7-12 with design options for production units with a different number of well zones and different geometries. Geometries with disjointed clusters of wells require a separate well zone for each cluster.

To compare the number of monitoring wells required in this section with connectivity of varying degrees and in the previous section with no assumed connectivity, consider again the 5-spot unit with 181 wells. Table 7-11 shows that 35, 27, 19, 16, or 9 monitoring wells are required for a posterior probability of 95% that at least 95% of the wells are expected to be in compliance using a 2-, 5-, 7-, 11- or 20-well design, respectively. In comparison, 45 wells are required in Table 7-7 for a posterior probability of 95% that the percent of wells expected to not be in compliance is less than 5% in a unit with no connectivity. With no connectivity, 45 wells are designated as monitoring wells. If minimal connectivity is assumed and a 2-well design adopted, the required number of monitoring wells is reduced to 35. If a 5-well design is adopted, the required number of monitoring wells is reduced further to 27. Additional reductions in the required number of monitoring wells are obtained if a higher degree of connectivity is demonstrated to support a 7-well or 11-well design.

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Production Unit with 5-spot pattern, 100 injection wells (●) and 81 production wells (۞).
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Figure 7-3.	2-Well Design for Production Unit with Minimal Connectivity

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Production Unit with 5-spot pattern, 100 injection wells (●) and 81 production wells (۞).
    Unit has 36 well zones (shading) and an average zone contains 5 wells.
Figure 7-4.	5-Well Design for Production Unit with Low Connectivity
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Production Unit with 5-spot pattern, 100 injection wells (●) and 81 production wells (۞).
    Unit has 25 well zones (shading) and an average zone contains 7 wells.
Figure 7-5.	7-Well Design for Production Unit with Moderate Connectivity

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Production Unit with 5-spot pattern, 100 injection wells (●) and 81 production wells (۞).
    Unit has 16 well zones (shading) and an average zone contains 11 wells.
Figure 7-6.	11-Well Design for Production Unit with Good Connectivity
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Production Unit with 5-spot pattern, 100 injection wells (●) and 81 production wells (۞).
    Unit has 9 well zones (shading) and an average zone contains 20 wells.
Figure 7-7.	20-Well Design for Production Unit with High Connectivity

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Production Unit with 7-spot pattern, 126 injection wells (●) and 36 production wells (۞).
    Unit has 72 well zones (shading) and an average zone contains 2 wells.
Figure 7-8.	2-Well Design for Production Unit with Minimal Connectivity

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Production Unit with 7-spot pattern, 126 injection wells (●) and 36 production wells (۞).
  Each production well defines a well zone. Unit has 36 well zones (shading).
Figure 7-9.	4-Well Design for Production Unit with Low Connectivity

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Production Unit with 7-spot pattern, 126 injection wells (●) and 36 production wells (۞).
    Unit has 16 well zones (shading) and an average zone contains 10 wells.
Figure 7-10.	10-Well Design for Production Unit with Good Connectivity

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Production Unit with 7-spot pattern, 126 injection wells (●) and 36 production wells (۞).
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Figure 7-11.	18-Well Design for Production Unit with High Connectivity

Table 7-9.	Ratio of Monitor Wells to Production Wells
                                     Site
                                Production Unit
                   Actual Number of Ore Zone Monitor Wells
                          Number of Extraction Wells*
                          Number of Injection Wells*
                             Total Number of Wells
Highland Uranium Project
A-Wellfield
                                       5
                                                                             38
                                                                             48
                                                                             86
Irigary
Units 1 to 9
                                      45
                                                                            453
                                                                            565
                                                                          1,018
Christensen Ranch
Units 2 to 5
                                      78
                                                                            663
                                                                            838
                                                                          1,501

                                                                          44.2%
                                                                          55.8%

Crow Butte
Unit 1
                                      12
                                                                             52
                                                                             64
                                                                            116
Totals

                                      140
                                                                          1,206
                                                                          1,515
                                                                          2,721
Averages

                                      9.3
                                                                           80.4
                                                                            101
                                                                          181.4
* Other sites estimated using Christensen Ranch proportion of injection/extraction wells

Table 7-10.	Number of Monitoring Wells Required for Five Design Options for a Production Unit with 181 Wells in 5-spot Pattern (Posterior Probability=90%)
                           Posterior Probability=90%
              Percent of Well Zones Demonstrating Compliance (P)
Design Option
                             Number of Well Zones
                                     P=70
                                      75
                                      80
                                      85
                                      90
                                      91
                                      92
                                      93
                                      94
                                      95
                                      96
                                      97
                                      98
                                      99
                                 2-Well Design
                                      81
                                       6
                                       7
                                       9
                                      12
                                      17
                                      19
                                      21
                                      25
                                      29
                                      29
                                      34
                                      42
                                      55
                                      72
                                 5-Well Design
                                      36
                                       5
                                       7
                                       8
                                      10
                                      15
                                      15
                                      18
                                      18
                                      18
                                      24
                                      24
                                      24
                                      32
                                      32
                                 7-Well Design
                                      25
                                       5
                                       6
                                       7
                                      10
                                      13
                                      13
                                      13
                                      17
                                      17
                                      17
                                      17
                                      23
                                      23
                                      23
                                11-Well Design
                                      16
                                       5
                                       5
                                       6
                                       8
                                      11
                                      11
                                      11
                                      11
                                      15
                                      15
                                      15
                                      15
                                      15
                                      15
                                20-Well Design
                                       9
                                       4
                                       4
                                       6
                                       6
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                                       -
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                                       -
                                       -
                                       -
                                       -
                                       -
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Note: All clusters must be monitored.

Table 7-11.	Number of Monitoring Wells Required for Five Design Options for a Production Unit with 181 Wells in 5-spot Pattern (Posterior Probability=95%)
                           Posterior Probability=95%
              Percent of Well Zones Demonstrating Compliance (P)
Design Option
                             Number of Well Zones
                                     P=70
                                      75
                                      80
                                      85
                                      90
                                      91
                                      92
                                      93
                                      94
                                      95
                                      96
                                      97
                                      98
                                      99
                                 2-Well Design
                                      81
                                       7
                                       9
                                      11
                                      15
                                      21
                                      24
                                      27
                                      30
                                      35
                                      35
                                      41
                                      50
                                      62
                                      76
                                 5-Well Design
                                      36
                                       7
                                       8
                                      10
                                      13
                                      18
                                      18
                                      22
                                      22
                                      22
                                      27
                                      27
                                      27
                                      34
                                      34
                                 7-Well Design
                                      25
                                       6
                                       7
                                       9
                                      12
                                      15
                                      15
                                      15
                                      19
                                      19
                                      19
                                      19
                                      24
                                      24
                                      24
                                11-Well Design
                                      16
                                       6
                                       6
                                       8
                                      10
                                      12
                                      12
                                      12
                                      12
                                       -
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                                       -
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                                       -
                                20-Well Design
                                       9
                                       5
                                       5
                                       7
                                       7
                                       -
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                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
Note: All clusters must be monitored.
Table 7-12.	Number of Monitoring Wells Required for Four Design Options for a Production Unit with 162 Wells in 7-spot Pattern (Posterior Probability=90%)
                           Posterior Probability=90%
              Percent of Well Zones Demonstrating Compliance (P)
Design Option
                             Number of Well Zones
                                     P=70
                                      75
                                      80
                                      85
                                      90
                                      91
                                      92
                                      93
                                      94
                                      95
                                      96
                                      97
                                      98
                                      99
                                 2-Well Design
                                      72
                                       5
                                       7
                                       9
                                      12
                                      16
                                      19
                                      21
                                      25
                                      25
                                      30
                                      37
                                      37
                                      48
                                      63
                                 4-Well Design
                                      36
                                       5
                                       7
                                       8
                                      10
                                      15
                                      15
                                      18
                                      18
                                      18
                                      24
                                      24
                                      24
                                      32
                                      32
                                10-Well Design
                                      16
                                       5
                                       5
                                       6
                                       8
                                      11
                                      11
                                      11
                                      11
                                      15
                                      15
                                      15
                                      15
                                      15
                                      15
                                18-Well Design
                                       9
                                       4
                                       4
                                       6
                                       6
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
Note: All clusters must be monitored.

Table 7-13.	Number of Monitoring Wells Required for Four Design Options for a Production Unit with 162 Wells in 7-spot Pattern (Posterior Probability=95%)
                           Posterior Probability=95%
              Percent of Well Zones Demonstrating Compliance (P)
Design Option
                             Number of Well Zones
                                     P=70
                                      75
                                      80
                                      85
                                      90
                                      91
                                      92
                                      93
                                      94
                                      95
                                      96
                                      97
                                      98
                                      99
                                 2-Well Design
                                      72
                                       7
                                       9
                                      11
                                      15
                                      21
                                      23
                                      26
                                      31
                                      31
                                      36
                                      44
                                      44
                                      54
                                      67
                                 4-Well Design
                                      36
                                       7
                                       8
                                      10
                                      13
                                      18
                                      18
                                      22
                                      22
                                      22
                                      27
                                      27
                                      27
                                      34
                                      34
                                10-Well Design
                                      16
                                       6
                                       6
                                       8
                                      10
                                      12
                                      12
                                      12
                                      12
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                18-Well Design
                                       9
                                       5
                                       5
                                       7
                                       7
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
Note: All clusters must be monitored.

Table 7-14.	Number of Monitoring Wells Required for Posterior Probability of 90% that at Least P% of the Well Zones Demonstrate Compliance
                             Number of Well Zones
              Percent of Well Zones Demonstrating Compliance (P)

                                     P=70
                                      75
                                      80
                                      85
                                      90
                                      91
                                      92
                                      93
                                      94
                                      95
                                      96
                                      97
                                      98
                                      99
                                       5
                                       3
                                       3
                                       3
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       6
                                       4
                                       4
                                       4
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       7
                                       3
                                       5
                                       5
                                       5
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       8
                                       4
                                       4
                                       5
                                       5
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       9
                                       4
                                       4
                                       6
                                       6
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                      10
                                       4
                                       5
                                       5
                                       7
                                       7
                                       9
                                       9
                                       9
                                       9
                                       9
                                       9
                                       9
                                       9
                                       9
                                      11
                                       4
                                       5
                                       5
                                       7
                                       7
                                      10
                                      10
                                      10
                                      10
                                      10
                                      10
                                      10
                                      10
                                      10
                                      12
                                       4
                                       4
                                       6
                                       8
                                       8
                                       8
                                      11
                                      11
                                      11
                                      11
                                      11
                                      11
                                      11
                                      11
                                      13
                                       5
                                       5
                                       6
                                       9
                                       9
                                       9
                                       9
                                      12
                                      12
                                      12
                                      12
                                      12
                                      12
                                      12
                                      14
                                       4
                                       5
                                       7
                                       7
                                       9
                                       9
                                       9
                                      13
                                      13
                                      13
                                      13
                                      13
                                      13
                                      13
                                      15
                                       5
                                       6
                                       6
                                       8
                                      10
                                      10
                                      10
                                      10
                                      14
                                      14
                                      14
                                      14
                                      14
                                      14
                                      16
                                       5
                                       5
                                       6
                                       8
                                      11
                                      11
                                      11
                                      11
                                      15
                                      15
                                      15
                                      15
                                      15
                                      15
                                      17
                                       4
                                       5
                                       7
                                       9
                                      11
                                      11
                                      11
                                      11
                                      11
                                      16
                                      16
                                      16
                                      16
                                      16
                                      18
                                       5
                                       6
                                       7
                                       9
                                      12
                                      12
                                      12
                                      12
                                      12
                                      17
                                      17
                                      17
                                      17
                                      17
                                      19
                                       5
                                       6
                                       8
                                      10
                                      13
                                      13
                                      13
                                      13
                                      13
                                      18
                                      18
                                      18
                                      18
                                      18
                                      20
                                       4
                                       5
                                       6
                                       8
                                      10
                                      14
                                      14
                                      14
                                      14
                                      14
                                      18
                                      18
                                      18
                                      18
                                      25
                                       5
                                       6
                                       7
                                      10
                                      13
                                      13
                                      13
                                      17
                                      17
                                      17
                                      17
                                      23
                                      23
                                      23
                                      30
                                       5
                                       6
                                       7
                                      10
                                      12
                                      16
                                      16
                                      16
                                      20
                                      20
                                      20
                                      27
                                      27
                                      27
                                      35
                                       5
                                       7
                                       8
                                      10
                                      15
                                      15
                                      18
                                      18
                                      18
                                      24
                                      24
                                      24
                                      32
                                      32
                                      40
                                       5
                                       6
                                       8
                                      10
                                      14
                                      17
                                      17
                                      21
                                      21
                                      21
                                      27
                                      27
                                      36
                                      36
                                      45
                                       5
                                       7
                                       8
                                      12
                                      16
                                      16
                                      19
                                      19
                                      24
                                      24
                                      31
                                      31
                                      41
                                      41
                                      50
                                       5
                                       7
                                       8
                                      11
                                      15
                                      18
                                      18
                                      21
                                      21
                                      26
                                      26
                                      34
                                      34
                                      45
                                      55
                                       6
                                       7
                                       8
                                      11
                                      17
                                      19
                                      19
                                      23
                                      23
                                      29
                                      29
                                      37
                                      37
                                      50
                                      60
                                       5
                                       7
                                       8
                                      11
                                      16
                                      18
                                      21
                                      21
                                      26
                                      26
                                      32
                                      41
                                      41
                                      54
                                      65
                                       6
                                       7
                                       8
                                      12
                                      17
                                      20
                                      20
                                      23
                                      28
                                      28
                                      34
                                      44
                                      44
                                      59
                                      70
                                       5
                                       7
                                       9
                                      12
                                      16
                                      19
                                      21
                                      25
                                      25
                                      30
                                      37
                                      37
                                      48
                                      63
                                      75
                                       6
                                       7
                                       9
                                      12
                                      18
                                      20
                                      20
                                      23
                                      27
                                      32
                                      32
                                      40
                                      51
                                      68
                                      80
                                       6
                                       7
                                       9
                                      12
                                      17
                                      19
                                      21
                                      25
                                      29
                                      29
                                      34
                                      42
                                      55
                                      72
                                      85
                                       6
                                       7
                                       9
                                      12
                                      18
                                      20
                                      23
                                      26
                                      26
                                      30
                                      36
                                      45
                                      58
                                      77
                                      90
                                       6
                                       7
                                       9
                                      12
                                      17
                                      19
                                      21
                                      24
                                      28
                                      32
                                      39
                                      48
                                      61
                                      81
                                      95
                                       6
                                       7
                                       9
                                      12
                                      18
                                      20
                                      23
                                      26
                                      29
                                      34
                                      41
                                      50
                                      65
                                      86
                                      100
                                       6
                                       7
                                       9
                                      12
                                      18
                                      19
                                      21
                                      27
                                      27
                                      31
                                      36
                                      43
                                      53
                                      68
                                      110
                                       6
                                       7
                                       9
                                      13
                                      18
                                      21
                                      24
                                      26
                                      30
                                      34
                                      40
                                      47
                                      58
                                      75
                                      120
                                       6
                                       7
                                       9
                                      12
                                      18
                                      21
                                      23
                                      26
                                      29
                                      33
                                      43
                                      52
                                      64
                                      82
                                      130
                                       6
                                       7
                                       9
                                      13
                                      18
                                      21
                                      23
                                      26
                                      31
                                      35
                                      40
                                      56
                                      69
                                      89
                                      140
                                       6
                                       7
                                       9
                                      12
                                      19
                                      21
                                      23
                                      28
                                      30
                                      34
                                      44
                                      51
                                      75
                                      96
                                      150
                                       6
                                       7
                                       9
                                      13
                                      19
                                      21
                                      23
                                      27
                                      30
                                      36
                                      41
                                      54
                                      65
                                      102
                                      160
                                       6
                                       7
                                       9
                                      13
                                      19
                                      21
                                      25
                                      27
                                      32
                                      35
                                      44
                                      58
                                      69
                                      109
                                      170
                                       6
                                       7
                                       9
                                      13
                                      19
                                      21
                                      24
                                      28
                                      31
                                      37
                                      47
                                      53
                                      74
                                      116
                                      180
                                       6
                                       7
                                       9
                                      13
                                      19
                                      21
                                      24
                                      28
                                      33
                                      36
                                      44
                                      56
                                      78
                                      123
                                      190
                                       6
                                       7
                                       9
                                      13
                                      19
                                      21
                                      24
                                      27
                                      32
                                      38
                                      46
                                      60
                                      82
                                      130
                                      200
                                       6
                                       7
                                       9
                                      13
                                      19
                                      21
                                      24
                                      29
                                      31
                                      37
                                      44
                                      55
                                      73
                                      107
                                      225
                                       6
                                       7
                                       9
                                      13
                                      20
                                      22
                                      24
                                      29
                                      33
                                      38
                                      45
                                      62
                                      82
                                      120
                                      250
                                       6
                                       7
                                       9
                                      13
                                      20
                                      22
                                      25
                                      29
                                      32
                                      39
                                      46
                                      61
                                      79
                                      133
                                      300
                                       6
                                       7
                                       9
                                      13
                                      20
                                      22
                                      25
                                      30
                                      33
                                      39
                                      47
                                      60
                                      83
                                      131
                                      350
                                       6
                                       7
                                      10
                                      13
                                      20
                                      23
                                      25
                                      29
                                      33
                                      41
                                      48
                                      65
                                      86
                                      152
                                      400
                                       6
                                       7
                                      10
                                      13
                                      20
                                      23
                                      25
                                      30
                                      34
                                      40
                                      49
                                      64
                                      89
                                      147
                                      450
                                       6
                                       7
                                      10
                                      13
                                      20
                                      23
                                      26
                                      30
                                      34
                                      41
                                      50
                                      67
                                      91
                                      165
                                      500
                                       6
                                       7
                                      10
                                      13
                                      20
                                      23
                                      26
                                      30
                                      34
                                      41
                                      50
                                      66
                                      93
                                      158
                                      600
                                       6
                                       7
                                      10
                                      13
                                      21
                                      23
                                      26
                                      30
                                      35
                                      41
                                      51
                                      67
                                      96
                                      167
                                      700
                                       6
                                       7
                                      10
                                      13
                                      21
                                      23
                                      26
                                      31
                                      35
                                      42
                                      52
                                      68
                                      98
                                      174
                                      800
                                       6
                                       7
                                      10
                                      13
                                      21
                                      23
                                      26
                                      31
                                      35
                                      42
                                      52
                                      69
                                      100
                                      179
                                      900
                                       6
                                       7
                                      10
                                      13
                                      21
                                      23
                                      26
                                      31
                                      35
                                      42
                                      53
                                      70
                                      101
                                      184
                                     1000
                                       6
                                       7
                                      10
                                      13
                                      21
                                      23
                                      26
                                      31
                                      35
                                      43
                                      53
                                      70
                                      102
                                      188
Note: All clusters must be monitored.

Table 7-15.	Number of Monitoring Wells Required for Posterior Probability of 95% that at Least P% of the Well Zones Demonstrate Compliance
                             Number of Well Zones
              Percent of Well Zones Demonstrating Compliance (P)

                                     P=70
                                      75
                                      80
                                      85
                                      90
                                      91
                                      92
                                      93
                                      94
                                      95
                                      96
                                      97
                                      98
                                      99
                                       5
                                       4
                                       4
                                       4
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       6
                                       4
                                       4
                                       4
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       7
                                       4
                                       5
                                       5
                                       5
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       8
                                       4
                                       4
                                       6
                                       6
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       9
                                       5
                                       5
                                       7
                                       7
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                      10
                                       4
                                       6
                                       6
                                       8
                                       8
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                      11
                                       5
                                       6
                                       6
                                       8
                                       8
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                      12
                                       5
                                       5
                                       7
                                       9
                                       9
                                       9
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                      13
                                       6
                                       6
                                       8
                                      10
                                      10
                                      10
                                      10
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                      14
                                       5
                                       7
                                       8
                                       8
                                      11
                                      11
                                      11
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                      15
                                       6
                                       7
                                       7
                                       9
                                      12
                                      12
                                      12
                                      12
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                      16
                                       6
                                       6
                                       8
                                      10
                                      12
                                      12
                                      12
                                      12
                                       -
                                       -
                                       -
                                       -
                                       -
                                       -
                                      17
                                       5
                                       7
                                       8
                                      10
                                      13
                                      13
                                      13
                                      13
                                      13
                                       -
                                       -
                                       -
                                       -
                                       -
                                      18
                                       6
                                       7
                                       9
                                      11
                                      14
                                      14
                                      14
                                      14
                                      14
                                       -
                                       -
                                       -
                                       -
                                       -
                                      19
                                       6
                                       8
                                       9
                                      11
                                      15
                                      15
                                      15
                                      15
                                      15
                                       -
                                       -
                                       -
                                       -
                                       -
                                      20
                                       6
                                       7
                                       8
                                      10
                                      12
                                      15
                                      15
                                      15
                                      15
                                      15
                                      19
                                      19
                                      19
                                      19
                                      25
                                       6
                                       7
                                       9
                                      12
                                      15
                                      15
                                      15
                                      19
                                      19
                                      19
                                      19
                                      24
                                      24
                                      24
                                      30
                                       6
                                       8
                                       9
                                      13
                                      15
                                      18
                                      18
                                      18
                                      23
                                      23
                                      23
                                      29
                                      29
                                      29
                                      35
                                       7
                                       8
                                      10
                                      13
                                      18
                                      18
                                      22
                                      22
                                      22
                                      27
                                      27
                                      27
                                      34
                                      34
                                      40
                                       7
                                       8
                                      10
                                      13
                                      17
                                      20
                                      20
                                      25
                                      25
                                      25
                                      31
                                      31
                                      38
                                      38
                                      45
                                       7
                                       8
                                      10
                                      14
                                      19
                                      19
                                      23
                                      23
                                      28
                                      28
                                      35
                                      35
                                      43
                                      43
                                      50
                                       7
                                       9
                                      10
                                      14
                                      19
                                      22
                                      22
                                      26
                                      26
                                      31
                                      31
                                      39
                                      39
                                      48
                                      55
                                       7
                                       9
                                      11
                                      14
                                      20
                                      24
                                      24
                                      28
                                      28
                                      34
                                      34
                                      43
                                      43
                                      53
                                      60
                                       7
                                       9
                                      11
                                      14
                                      20
                                      22
                                      26
                                      26
                                      31
                                      31
                                      37
                                      46
                                      46
                                      57
                                      65
                                       7
                                       9
                                      11
                                      15
                                      21
                                      24
                                      24
                                      28
                                      33
                                      33
                                      41
                                      50
                                      50
                                      62
                                      70
                                       7
                                       9
                                      11
                                      15
                                      21
                                      23
                                      26
                                      31
                                      31
                                      36
                                      44
                                      44
                                      54
                                      67
                                      75
                                       7
                                       9
                                      11
                                      15
                                      22
                                      25
                                      25
                                      28
                                      33
                                      39
                                      39
                                      47
                                      58
                                      72
                                      80
                                       7
                                       9
                                      11
                                      15
                                      21
                                      24
                                      27
                                      30
                                      35
                                      35
                                      41
                                      50
                                      62
                                      76
                                      85
                                       7
                                       9
                                      11
                                      16
                                      23
                                      25
                                      28
                                      32
                                      32
                                      37
                                      44
                                      53
                                      66
                                      81
                                      90
                                       7
                                       9
                                      11
                                      16
                                      22
                                      24
                                      27
                                      30
                                      34
                                      40
                                      47
                                      56
                                      70
                                      86
                                      95
                                       7
                                       9
                                      11
                                      16
                                      23
                                      26
                                      28
                                      32
                                      36
                                      42
                                      49
                                      59
                                      74
                                      91
                                      100
                                       7
                                       9
                                      12
                                      15
                                      22
                                      24
                                      27
                                      34
                                      34
                                      38
                                      44
                                      52
                                      63
                                      78
                                      110
                                       7
                                       9
                                      12
                                      16
                                      23
                                      27
                                      30
                                      33
                                      37
                                      42
                                      49
                                      57
                                      69
                                      85
                                      120
                                       7
                                       9
                                      12
                                      16
                                      23
                                      27
                                      30
                                      33
                                      36
                                      41
                                      53
                                      62
                                      75
                                      93
                                      130
                                       7
                                       9
                                      12
                                      16
                                      23
                                      27
                                      30
                                      32
                                      39
                                      44
                                      50
                                      68
                                      82
                                      101
                                      140
                                       7
                                       9
                                      12
                                      16
                                      24
                                      27
                                      29
                                      35
                                      38
                                      42
                                      54
                                      62
                                      88
                                      109
                                      150
                                       8
                                       9
                                      12
                                      17
                                      24
                                      27
                                      29
                                      34
                                      37
                                      46
                                      51
                                      67
                                      78
                                      116
                                      160
                                       8
                                       9
                                      12
                                      16
                                      24
                                      27
                                      31
                                      34
                                      40
                                      44
                                      54
                                      71
                                      84
                                      124
                                      170
                                       8
                                      10
                                      12
                                      17
                                      24
                                      27
                                      31
                                      36
                                      39
                                      47
                                      58
                                      66
                                      89
                                      132
                                      180
                                       8
                                       9
                                      12
                                      16
                                      25
                                      27
                                      31
                                      35
                                      41
                                      45
                                      55
                                      70
                                      94
                                      140
                                      190
                                       8
                                      10
                                      12
                                      17
                                      25
                                      27
                                      31
                                      35
                                      40
                                      48
                                      58
                                      74
                                      99
                                      147
                                      200
                                       8
                                       9
                                      12
                                      17
                                      25
                                      27
                                      31
                                      37
                                      40
                                      46
                                      55
                                      68
                                      89
                                      126
                                      225
                                       8
                                      10
                                      12
                                      17
                                      26
                                      28
                                      31
                                      37
                                      42
                                      48
                                      57
                                      77
                                      100
                                      142
                                      250
                                       8
                                      10
                                      12
                                      17
                                      25
                                      29
                                      32
                                      37
                                      41
                                      50
                                      58
                                      77
                                      97
                                      157
                                      300
                                       8
                                      10
                                      12
                                      17
                                      26
                                      29
                                      32
                                      38
                                      42
                                      50
                                      60
                                      76
                                      103
                                      157
                                      350
                                       8
                                      10
                                      12
                                      17
                                      26
                                      29
                                      33
                                      38
                                      43
                                      52
                                      62
                                      82
                                      108
                                      184
                                      400
                                       8
                                      10
                                      13
                                      17
                                      26
                                      29
                                      33
                                      39
                                      43
                                      51
                                      63
                                      81
                                      112
                                      179
                                      450
                                       8
                                      10
                                      13
                                      17
                                      26
                                      30
                                      33
                                      38
                                      44
                                      53
                                      64
                                      85
                                      115
                                      202
                                      500
                                       8
                                      10
                                      13
                                      17
                                      27
                                      30
                                      33
                                      39
                                      44
                                      53
                                      65
                                      84
                                      118
                                      195
                                      600
                                       8
                                      10
                                      13
                                      17
                                      27
                                      30
                                      34
                                      39
                                      45
                                      53
                                      66
                                      86
                                      122
                                      208
                                      700
                                       8
                                      10
                                      13
                                      18
                                      27
                                      30
                                      34
                                      40
                                      45
                                      54
                                      67
                                      87
                                      125
                                      217
                                      800
                                       8
                                      10
                                      13
                                      18
                                      27
                                      30
                                      34
                                      40
                                      46
                                      55
                                      68
                                      89
                                      128
                                      225
                                      900
                                       8
                                      10
                                      13
                                      18
                                      27
                                      30
                                      34
                                      40
                                      46
                                      55
                                      68
                                      90
                                      130
                                      232
                                     1000
                                       8
                                      10
                                      13
                                      18
                                      27
                                      30
                                      34
                                      40
                                      46
                                      55
                                      69
                                      90
                                      131
                                      237
Note: All clusters must be monitored.

Determining Connectivity of the Wellfield

Connectivity of the wellfield may be measured using both physical and statistical methods. An example of a physical measure of connectivity is the hydraulic conductivity of the ground water. Connectivity may also be demonstrated based on an analysis of the spatial correlation between measurements in nearby wells. Spatial autocorrelation is a generalization of one-dimensional temporal autocorrelation. Spatial correlation may be defined in two, three or more dimensions. In this application two-dimensional spatial autocorrelation is of interest. In two or more dimensions autocorrelation is multi-directional.

Two- and three-dimensional data with spatial correlation are best analyzed using geostatistical methods. Geostatistics is capable of using the information revealed by a correlation analysis of the data to estimate concentrations at unsampled locations and in areas where data are sparse. Although application of these methods often requires specialized software and skilled practitioners, operators are encouraged to perform geostatistical analysis to obtain a better understanding of the site. Geostatistical analysis may reveal anomalous baseline conditions in certain areas of the site or areas where remediation has not been successful. A full discussion of geostatistics is beyond the scope of this document. Some examples of the application of geostatistics in the analysis of ground water contamination can be found in Chapter 6 of EPA 2000b.

Moran's I statistic is one measure of spatial association that does not require specialized software. This statistic is a global measure of spatial autocorrelation characterized by a correlation among nearby wells. In addition to the data from each well, Moran's I statistic requires matrix of spatial weights. In the most elementary case, the weight matrix may be derived from a distance matrix. The distance matrix has elements that represent the "distance" between well i and well j. The distance matrix may be expressed in physical units of length, or may contain other measures of distance, such as the travel time between wells.

The weight matrix for the I statistic has elements equal to the inverse of the elements of the distance matrix. The weight matrix measures the "nearness" of each well to the other wells. The weighting entries for pairs of wells that are close together are higher than for pairs of wells that are far apart. The weight matrix has off-diagonal entries wi, j equal to 1 divided by the distance between well i and well j. The diagonal entries of the weight matrix are set to 0. This is one of several ways to calculate an inverse distance matrix. For example, the inverse distance matrix may have off-diagonal entries equal to 1 divided by 1 plus the distance between well i and well j.

Moran's I is calculated using the formula:

where N is the number of spatial units indexed by i, j=1, ..., N; X is the analyte of interest;is the mean of X; and wi,j is an element of the spatial weight matrix.

The expected value of Moran's I statistic under a hypothesis of no spatial autocorrelation is . Positive (negative) values of I indicate positive (negative) spatial autocorrelation. Values of I range from −1, indicating perfect dispersion, to +1, indicating perfect correlation between the wells. A zero value indicates a random spatial pattern. For statistical hypothesis testing, Moran's I values can be transformed to normal scores in which values greater than 1.96 (or smaller than −1.96) indicate spatial autocorrelation that is significant at the 5% level.

Hypothesis Testing and Data Quality Objectives

Hypothesis testing is a statistical tool for deciding when the ground water has reached steady state and for comparing post-restoration conditions with baseline conditions. The hypothesis tests are conducted for individual wells and, when wells exhibit homogeneous dynamics, for all wells combined.
The first step in developing a hypothesis test is to transform the problem into statistical terminology by formulating a null hypothesis and an alternative hypothesis. These hypotheses form the two alternative decisions that the hypothesis test will evaluate. When a well is compared with the baseline, the unknown parameter of interest (δ) is the amount by which the post-restoration distribution exceeds the baseline distribution (See Box 7-1). Delta (δ) is an unknown value, and statistical tests may be used to evaluate hypotheses relating to its possible values. A hypothesis test is designed to reject or not reject hypotheses about δ based on test statistics computed from the sample data.

At its core, this is another example of the "How clean is clean?" problem. The action level for baseline comparisons is the largest difference in the two distributions that is acceptable to the decision maker. In this report, the action level for this difference is defined as a substantial difference (Δ), which may be zero or a positive value based on the risk assessment, an applicable regulation, a screening level, or guidance.
This document does not establish a specific value for a substantial difference Δ, as the value will vary from parameter to parameter and from site to site. Therefore, specific values for Δ should be considered on a case-by-case basis. In many cases, the minimum feasible value of Δ is determined by the usual variability in that parameter during the pre-leaching phase (Phase 1 in Figure 3-1). Appendix A to EPA 2002a discusses further the selection of a value for Δ. The determination of Δ for each parameter of interest should be considered during the development of a quality assurance project plan as part of the planning process for the site evaluation.

Hypothesis testing is a quantitative method to determine whether a specific statement concerning the unknown difference δ (a statement known as the "null hypothesis") can be rejected based on the data at hand. Decisions concerning the true value of δ (e.g., is δ > 0?) reduce to a choice of "yes" or "no." When viewed in this way, two types of incorrect decisions, or decision errors, may occur:

   * Incorrectly deciding the answer is "yes" when the true answer is "no"
   * Incorrectly deciding the answer is "no" when the true answer is "yes"

While the possibility of decision errors can never be totally eliminated, it can be controlled to acceptable levels. To control decision errors, it is necessary to control the uncertainty in the estimate of δ. Uncertainty arises from three sources:

   * Sampling error
   * Measurement error
   * Natural variability

The decisionmaker has some control over the first two sources of uncertainty. For example, a larger number of samples may lead to fewer decision errors because the probability of a decision error decreases as the number of samples increases. Use of more precise measurement techniques or duplicate measurements can reduce measurement error, thus minimizing the likelihood of a decision error. The third source of uncertainty is more difficult to control. Natural variability arises from the uneven distribution of chemical concentrations and conditions at the site.

Natural variability includes both spatial and temporal variability. When measurements are made in different wells over a relatively short period of time, spatial variability is measured by the standard deviation (σ) of the measurements around their mean value. A large value of σ indicates that a large number of measurements will be needed to achieve a desired limit on decision errors. Baseline samples may have a different variability than post-restoration samples. As post-restoration variability is usually higher than in the baseline, post-restoration data ideally would be used to estimate σ.

Temporal variability measures the variability of concentration in a well over time. Short-term variability may be random or have a seasonality pattern. Long-term temporal variability may appear as a trend. In the case of a trend, temporal variability is defined in terms of the variation of the measurements from the trendline.

It is advisable to overestimate σ rather than underestimate the true variability. A very crude approximation for σ may be made by dividing the anticipated range (maximum minus minimum) by 6 (EPA 2002a, Section 3.1). It is important that overly optimistic estimates for σ be avoided, because this may result in a sample size that fails to generate sufficient data to distiguish between the null and alternative hypotheses. In statistical terms, the test will lack sufficient power if the sample size is too small. The power of a statistical hypothesis test is discussed in the following section.

The minimum detectable difference (MDD) for a statistical test indicates that differences smaller than the MDD cannot be detected reliably. If the test is used to decide if post-restoration concentrations exceed the baseline concentrations by more than Δ, it is necessary to ensure that the MDD for the test is less than Δ. In the planning stage, this requirement is met by designing a sampling plan with sufficient power to detect differences as small as Δ (MDD <= Δ). If data were collected without the benefit of a sampling plan, retrospective calculation of the power of the test may be necessary before making a decision.

In the planning stage, the absolute size of the MDD is of less importance than the ratio of the MDD to the natural variability of the post-restoration concentrations. This ratio is termed the relative difference and is defined as MDD/σ, where σ is an estimate of the standard deviation of the post-restoration distribution. The relative difference expresses the power of resolution of the statistical test (MDD) in units of uncertainty (σ). Relative differences much less than 1 standard deviation (MDD/σ << 1) are more difficult to resolve unless a larger number of measurements are available. Relative differences of more than 3 standard deviations (MDD/σ > 3) are easier to resolve.

Decision Errors and Confidence Levels

A key step in developing a sampling and analysis plan is to establish the level of precision required of the data used for decision making. These requirements will determine the required sample size. An increased number of samples generally increases the level of precision. Due to the uncertainties that result from sampling variation, decisions will be subject to errors. There are two ways to err when analyzing data (Table 7-16):

   * Type I Error: Based on the observed data, the test may reject the null hypothesis when, in fact, the null hypothesis is true (a false positive). This is a Type I error. The probability of making a Type I error is α (alpha).
   * Type II Error: On the other hand, the test may fail to reject the null hypothesis when the null hypothesis is, in fact, false (a false negative). This is a Type II error. The probability of making a Type II error is β (beta).

Table 7-16.	Hypothesis Testing: Type I and Type II Errors
                         Decision Based on Sample Data
                             Actual Site Condition

                                  H0 Is True
                                H0 Is Not True
                              H0 is not rejected
                          Correct Decision: (1 - α)
                      Type II Error: False Negative (β)

                                H0 is rejected
                       Type I Error: False Positive (α)
                          Correct Decision: (1 - β)

		Note:  H is the null hypothesis.

The acceptable level of decision error associated with hypothesis testing is defined by two key parameters: confidence level and power. These parameters are closely related to the two error probabilities, α and β.
   * Confidence level: 100(1 - α)%. As the confidence level is lowered (or alternatively, as α is increased), the likelihood of committing a Type I error increases.
   * Power: 100(1 - β)%. As the power is lowered (i.e., as β is increased), the likelihood of committing a Type II error increases.

The selection of appropriate levels for decision errors and the resulting number of samples is a critical component of the DQO process that should concern all stakeholders.

Because there is an inherent tradeoff between the probability of committing a Type I or Type II error, a simultaneous reduction in both types can occur only by increasing the number of samples. If the probability of committing a false positive is reduced by increasing the level of confidence of the test (in other words, by decreasing α), the probability of committing a false negative is increased, because the power of the test is reduced (increasing β).

When the site is sampled for a number of species, the selection of appropriate DQOs for each contaminant will be influenced by the relative health risks and costs of control for each species. If a single contaminant is the major focus of concern, the DQOs (α and β) may be based on this species. If more than one species is a matter of concern, the Bonferroni correction is a simple approach for addressing the problem. If the species are of equal concern, the nominal significance level for each test (α) is divided by the number of contaminants that are to be tested. Similarly, when the species have different levels of concern, adjustments may be made using a different α for each species. However, the sample size calculations (described below) show that this reduction in the significance level requires a major increase in the amount of data to be collected. The issue of multiple comparisons is beyond the intended scope of this document. A complete discussion of Bonferroni confidence intervals and newer alternative approaches to the multiple-comparison problem is presented in Bickel and Doksum (2006). In terms of not exceeding the regulatory limits, we believe the confidence level for decisions involving the listed radionuclides should be equivalent. The limits for other analytes may be less stringent but still  consistent with the value of the parameter in making a safety case.

Hypothesis Tests for Comparisons with Baseline

Statistical hypothesis tests are used for comparing post-restoration conditions with baseline conditions and for demonstrating stability of the site after restoration. Three statistical methods are presented for the comparison with baseline conditions. The two-sample t-test and prediction limits for a future mean are two parametric methods used in the comparison with baseline conditions. Prediction limits (PLs) are designed to provide an upper bound for the mean of a future sample with a specified probability equal to (1 - α), known as the confidence level of the PL. It represents the chance that the PL will contain the mean of a future (post-operations) sample from the monitoring wells. The nonparametric WRS test is also used to compare post-restoration well conditions with baseline values. This test compares the relative ranks of the two data sets. The two-sample t-test and the WRS test are recommended for comparing baseline and post-remedial wells (EPA 2006a). The RCRA Unified Guidance (EPA 2009) recommends the PL method.

The three different statistical tests for comparisons with baseline conditions are used in Phase 4 to determine if the restoration goals have been met. A choice of one of the three different statistical tests should be made before post-restoration data are collected. The selected test procedure is then applied for all wells. The prior selection of a single testing approach is necessary for comparability of results across wells and to avoid the possibility of running all three tests, then selecting the test with most favorable results.

Two forms of hypothesis tests are used for comparisons with baseline conditions. The null hypothesis in the first form of the test (Test Form 1) states that there is no statistically significant difference between the means of the baseline and post-restoration concentration distributions. The null hypothesis in the second form of the test (Test Form 2) states that the post-restoration mean exceeds the baseline mean by more than a substantial difference Δ. Either test form may be used with the two-sample t-test and the WRS test. Test Form 1 is used for PLs.

Test Form 1

The null hypothesis for Test Form 1 is formulated for the express purpose of being rejected when restoration has not been successful. If the null hypothesis is rejected, the alternative hypothesis is accepted, indicating that baseline conditions have not been achieved.

   * The null hypothesis (H0): The mean of the post-restoration distribution does not exceed the baseline mean. Symbolically, the null hypothesis for Test Form 1 is H0: δ <= 0.
   * The alternative hypothesis (HA): The post-restoration distribution mean exceeds the baseline mean: (HA: δ > 0).

When using Test Form 1, the null hypothesis is presumed true until it is rejected. A problem may arise in the use of Test Form 1 when insufficient data are available for the decision. If insufficient data are available, Test Form 1 may lead to inconclusive results due to a lack of power. Figure 7-12 illustrates an example of a test performance plot for Test Form 1. At the origin of the plot, the true difference between the means of the two distributions is zero. Positive values of the difference are plotted on the horizontal axis to the right of the origin, negative values to the left. The vertical axis shows the probability of rejecting the null hypothesis, which for this test form is the probability of deciding the post-restoration mean concentration exceeds the mean baseline concentration. This probability ranges from 0 to 1.0 (0% to 100%).

The gray region for the test extends from 0 up to the MDD. It is necessary to specify a gray region for the test, because the decision may be "too close to call" due to uncertainty. This may occur when the difference in means is small compared to the MDD for the test. To the left or right of the gray region, the test outcome is easy to determine. In the gray region, the statistical test has difficulty deciding between the two alternatives. To the left of the gray region, the test performance curve is no greater than α. In the gray region, the test performance curve increases as the difference between the means increases. The number of samples and the standard deviation, σ, determine the rate of increase. The right edge of the gray region is located at the MDD. The MDD measures the width of the gray region for the test. When the difference between the means is equal to the MDD, the probability of deciding that the post-restoration mean exceeds the baseline mean is equal to the power of the test (1  - β).
Figure 7-12 also shows a hypothetical value of a substantial difference for this analyte of Δ = 100. This value is used in the DQO process as an upper limit for the width of the gray region (MDD). In this example, an MDD less than Δ was selected for the test. If the MDD is selected to be smaller than Δ, then differences from baseline smaller than Δ can be detected by the test with a high probability.

Figure 7-12.	Test Performance Plot with Parameter Definitions for Test Form 1

Test Form 2

With Test Form 2, the null hypothesis is that the post-restoration samples exceed the baseline by a substantial difference. The null hypothesis is formulated for the express purpose of being rejected if the data support the alternative:

   * The null hypothesis (H0): The post-restoration distribution mean exceeds the baseline mean by more than Δ. Symbolically, the null hypothesis is written as H0: δ > Δ.
   * The alternative hypothesis (HA): The post-restoration distribution mean does not exceed the baseline mean by more than Δ (HA: δ <= Δ).

In Test Form 2, the hypothesis test is structured so that the post-restoration data must provide evidence that the site is within acceptable limits. The null hypothesis is assumed to be true unless the statistical test indicates that it should be rejected in favor of the alternative. Figure 7-13 illustrates an example of a test performance plot for Test Form 2. The horizontal and vertical axes for this plot are the same as in Figure 7-12. The gray region for Test Form 2 again has a width equal to the MDD. To the left of the gray region, the test performance curve is no greater than β. In the gray region, the test performance curve increases as the difference between the means increases. The right edge of the gray region is located at Δ, the value selected as a substantial difference. When the difference between the means is equal to Δ, the probability of deciding that the post-restoration mean exceeds the baseline mean is equal to (1 - α).

An MDD less than Δ was selected for the test. If the MDD is selected to be smaller than Δ, then the test will have greater power to accept sites when the difference in means is less than Δ.

Figure 7-13.	Test Performance Plot with Parameter Definitions for Test Form 2

Selecting a Test Form

When comparing Test Forms 1 and 2, it is important to distinguish between the selection of the null hypothesis, which is a burden-of-proof issue, and the selection of the substantial difference, which involves determination of an action level.

Test Form 1 uses a conservative level of 0 as the maximum acceptable difference, but relaxes the burden of proof by selecting a null hypothesis that the post-restoration mean is not statistically different from the baseline mean. Test Form 2 requires a stricter burden of proof, but relaxes the action level from 0 to Δ. See Box 7.2 for more information on the parameters of Test Form 1 and Test Form 2. EPA2002a and MARSSIM (EPA 2000a) include further discussion of how to choose between Test Forms 1 and 2 (MARSSIM uses the terms Scenario A and Scenario B for Test Forms 2 and 1, respectively), and provide additional guidance for setting up the hypotheses.

Regardless of the choice of hypothesis for the comparison with baseline conditions, an incorrect conclusion could be drawn from the data analysis using either form of test. To account for this inherent uncertainty, one should specify the limits on the decision errors. This task was described in Section 7.3.1.

The selection of a null hypothesis depends on what the "working assumption" is for each monitoring phase and, perhaps more generally, what has already occurred. For post-restoration, one "assumes the worst," i.e., that post-restoration values exceed the baseline by more than Δ. Compliance can be demonstrated only by collecting sufficient data to reject the null hypothesis (i.e., "proving" the alternative). For this phase, a regulator would be primarily concerned with the occurrence of Type I errors (showing compliance when not justified). However, when using a trend test to demonstrate that the site is stable, the null hypothesis would be a statement of no trend. The null hypothesis would stand until sufficient data show evidence of a trend.

A Type 2 error occurs if the null hypothesis is accepted when it is not true. With Test Form 1, a Type 1 error occurs when the site is incorrectly determined to require further restoration. A Type 2 error means that the site was incorrectly determined to be in compliance. From a human health perspective, a Type 2 error is more serious than a Type 1 error. Hence, it is reasonable that the Type 2 error rate (β) should be smaller than the Type 1 error rate (α). Using Test Form 1, the regulator would be concerned with Type 2 errors, and the specification of MDDs and minimum sample sizes will be of particular importance to the regulator.

With Test Form 2, a Type 1 error occurs when the site is incorrectly determined to be in compliance. A Type 2 error means that the site was incorrectly determined to require further restoration. From a human health perspective, a Type 1 error is more serious than a Type 2 error. In this case, it is reasonable that the Type 1 error rate (α) should be smaller than the Type 2 error rate (β). Once α is selected, a higher value of β will reduce the required number of samples, but there will be a greater likelihood that the site is incorrectly determined to be out of compliance. In this case, the site operator faces a trade-off and may select to reduce the value of β (at the expense of a greater number of samples) and increase the power of the test in order to avoid the possibility of Type 2 errors.

For the purposes of this report, the minimum recommended performance measures are:

Test Form 1:
      Confidence level at least 80% (α <= 0.20) and power at least 90% (β <= 0.10)

Test Form 2:
      Confidence level at least 90% (α <= 0.10) and power at least 80% (β <= 0.20). (EPA 2002a, Section 3.2)

Box 7-2 describes these performance standards in more detail.

--------------------------------------------------------------------------------
   Box 7-1. Definitions
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   δ (delta): The true difference between the post-restoration distribution and the baseline distribution of parameter X. Delta is an unknown value that describes the true state of nature. Hypotheses about its value are evaluated using statistical hypothesis tests. In principle, we can select any specific value for δ and then test if this difference is statistically significant or not with a given confidence and power.
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   Δ (a substantial difference): A difference between the two distributions that is sufficiently large to warrant additional interest based on health or ecological information. Δ is the investigation level. If δ exceeds Δ, the difference in concentrations is judged to be sufficiently large to be of concern for the purpose of the analysis. A hypothesis test uses baseline and post-restoration measurements to determine if δ exceeds Δ.
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   MDD (minimum detectable difference): The smallest difference that the statistical test can resolve. The MDD depends on sample-to-sample variability, the number of samples, and the power of the statistical test. The MDD is a property of the survey design.

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   Box 7-2. Interpretation of the Statistical Measures for Test Forms 1 and 2
--------------------------------------------------------------------------------

--------------------------------------------------------------------------------
   Test Form 1
--------------------------------------------------------------------------------
   Confidence level = 80%: On average, in 80 out of 100 cases, post-restoration concentrations are correctly identified as not exceeding baseline concentrations by more than Δ, while in 20 out of 100 cases, post-restoration concentrations will be incorrectly identified as exceeding baseline concentrations by more than Δ when, in fact, they do not. For a confidence level of 80%, choose α=0.20.
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   Power = 90%: On average, in 90 out of 100 cases, post-restoration concentrations will be correctly identified as exceeding baseline concentrations by more than Δ, while in 10 out of 100 cases, post-restoration concentrations will be incorrectly identified as not exceeding baseline concentrations by more than Δ when, in fact, they do. For power of 90%, choose β=0.10.
--------------------------------------------------------------------------------

--------------------------------------------------------------------------------
   Test Form 2
--------------------------------------------------------------------------------
   Confidence level = 90%: On average, in 90 out of 100 cases, post-restoration concentrations are correctly identified as exceeding baseline concentrations by more than Δ, while in 10 out of 100 cases, post-restoration concentrations will be incorrectly identified as not exceeding baseline concentrations by more than Δ when, in fact, they do. For a confidence level of 90%, choose α=0.10.
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   Power = 80%: On average, in 80 out of 100 cases, post-restoration concentrations will be correctly identified as not exceeding baseline concentrations by more than Δ, while in 20 out of 100 cases, post-restoration concentrations will be incorrectly identified as exceeding baseline concentrations by more than Δ when, in fact, they do not. For power of 80%, choose β=0.20.

Hypothesis Tests for Detecting Trends