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Lecture

5sampling statistics.pdf

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Department
Chemistry
Course
ENCH 213
Professor
Diane Beauchemin
Semester
Fall

Description
Sampling statistics Sampling considerations • Lot: total material • Gross or bulk sample: representative of the lot • Laboratory sample: with exact same composition as the bulk sample • Test portions Select the one best answer: aAbulk sample is taken from a lot. b Abulk sample is taken from a laboratory sample. cAbulk sample is also called a gross sample. d a and c Statistics of sampling 2 2 2 • s o s +ss a ◦ s = overall variance ◦ s = variance of the sampling process 2 ◦ s = aariance of the analytical procedure • if sa< s s3 ◦ loss of time to try to reduce s (maan source of error is then the sampling error) ◦ s obaained from replicate measurements of a standard ◦ s oboained from replicate measurements of the samples Gross sample size depends on: • Uncertainty between the composition of the gross sample and that of the lot • Heterogeneity of the sample • Particle size distribution • Analyte concentration • Homogeneous solutions of liquids and gases: ◦ heterogeneity and particle size at the molecular level ◦ sample size can be relatively small ◦ depends on final use Particulate solids • Simplify by treating the sample as 2-component mixtures with a single particle size ◦ soil mixture= 367 Smartees + 554 Reeces pieces nA nB • p = probability of drawing a Smartee = n /(n +A ) =A0.3B8 • q = probability of drawing a Reeces piece = n /(n +nB) =A0.6B2 = 1-p Small sample size • small sample size ≈ 15 particles ◦ expected number of Smartees = np = 15 x 0.398 = 6.0 ◦ expected number of Reeces pieces = nq = 15 x 0.602 = 9.0 • standard deviation in sampling operation (i.e., of many drawings) = s = (npq)½ =n1.9 ◦ applies to both kinds of particles relative errors ◦ 32% of expected number of Smartees ◦ 21% of expected number of Reeces pieces Large sample size • large sample size ≈ 70 particles ◦ expected number of Smartees = np = 70 x 0.398 = 28 ◦ expected number of Reeces pieces = nq = 70 x 0.602 = 42 • standard deviation in sampling operation (i.e., f many drawings) = s = (npq)½ n 4.1 ◦ applies to both kinds of particles ◦ 15% of expected number of Smartees ◦ 10% of expected number of Reeces pieces How many particles for representative sampling with σ ≤ 4%? r • relative standard deviation σ = sr/n =n(npq) /n = (pq/n) ½ ½ ◦ small sample: (0.398 x 0.602/15) = 0.13 or 13% ◦ large sample: (0.398 x 0.602/70) = 0.059 or 5.9% • to decrease it to 4% ◦ σ = r /n n (npq) /n = (pq/n) ½ 1/2 ◦ 0.04 = (0.398 x 0.602 /n) → solve for n ◦ n = 0.398 x 0.602 / (0.04) = 1502 Establishing and using a sampling constant • because sample mass (m) α n, and n σ = pq: r 2 ◦ mσ = K r sampsing constant = mass of sample producing a relative sampling standard
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