Question

2. (25 pts.) The reflow from fracture-based NG drilling (spent fracking water) contains substantial dissolved salt content. One quick method of determining total dissolved salt concentration (nominally TDS is total dissolved solids – a well-known water quality metric, which in this case, equates to total dissolved salt) across a wide spectrum of is to measure the fluid conductivity. A sample from each of the n = 5 wells is taken, and repeated over the course of four days, yielding m = 4 aliquots. Suppose the cost of collecting a sample is c₁ = $500, the cost of measuring an aliquot is c₂ = $100, and the cost function C=c₁n+c₂nm applies. Estimate the optimum number of aliquots to measure for specific conductance, using the data in Table 2, below. Compute the mean, standard error of the mean, and the optimum m and n if c₁=$500, c₂=$100, and C=$2000.

Well	Aliquot
1	3110	3180	3180	3190
2	3400	3400	3400	3400
3	3090	3100	3080	3040
4	3750	3750	3700	3750
5	3100	3100	3140	3100

                                        Table 2.

Answer 1

Well	Aliquot					sh	whsh	opt nh
1	3110	3180	3180	3190	3165	36.968	7.394	7
2	3400	3400	3400	3400	3400	0.000	0.000	0
3	3090	3100	3080	3040	3077.5	26.300	5.260	5
4	3750	3750	3700	3750	3737.5	25.000	5.000	5
5	3100	3100	3140	3100	3110	20.000	4.000	4
Total							21.654

Since, n_h cannot be a fraction.

Cost:	2000= 500n+100nm
	Therefore,
	n=n1=7
	nm=n2=0	i.e. m=0

Theory: