Question: (Requires knowledge of trigonometry.) In Example, a systematic sampling scheme was proposed for detecting hazardous wastes in landfills. How far apait should sampling points be placed? Suppose there is a leakage and it spreads to a circular region with radius R. Let 21) be the distance between adjacent sampling points;: in the same row or column.

a. Calculate the probability with which a contaminant will be detected.

b. Propose a sampling design that gives a higher probability that a contaminant wil be detected than the square grid, but does not increase the number of sampling points.

Example: Many dumps and landfills in the United States contain toxic materials. These materials may have been sealed in containers when deposited but may now be suspected of leaking. But we no longer know where the materials were deposited-containers of hazardous waste may be randomly distributed throughout the landfill, or they may be concentrated in one area, or there may be none at all. A common practice is to take a systematic sample of grid points and to take soil samples from each to look for evidence of contamination. Choose a point at random in the area, then construct a grid containing that point so that grid points are an equal distance apart. One such grid is shown in Figure 5.8. The advantages of taking a systematic sample rather than an SRS are that the systematic sample forces an even coverage of the region and is easier to implement in the field. If you are not worried about periodic patterns in the distribution of toxic materials and you have little prior knowledge where the toxic materials might be, a systematic sample is a good design. With any grid in systematic sampling, you need to worry if the toxic materials are regularly placed so that the grid may miss all of them, as shown in Figure 5.9. If this is a concern, you would be better off taking a stratified sample. Lay out the grid but select a point at random in each square at which to take the soil sample.