CHEM 524 -- Course Outline (Part 10)

VII. Chapter 6 and Appendix A (Read--continued)

F. Statistical Sampling error (random error) yields to error evaluation,

1. Systematic error more difficult- Calibration, Matrix, Sampling errors

2. Random distribution of error is Gaussian -- z test, large set

--- m = true mean; s = true S.D.

--- -probability in interval

and P(z < z_a) = 1 - a, P(|z| < z_a) = 1 - 2a

--- values from table of z and a (Table A1)

3. Student t-test (Table A-2, s is unknown) -- measure first (n < 30)

-- for a small number of data, the error (uncertainly) increases

--- s and s differ -- need table for a depend on n

--- t = (E - m)/(s/ n^1/2) E - average of n samples table gives t (a, n)

same form P(t < tⁿ_a) = 1 - a, P(t > tⁿ_a) = a,

also P(|t |< tⁿ_a) = 1 - 2a, and P(0 < t < tⁿ_a) = 0.5 - a,

4. Hypothesis testing -- is difference between E and m significant?

--- test confidence interval m = E ± zs/ n^1/2 (or m = E ± ts/ n^1/2)

-- confidence (or probability) that an interval (error range) encloses the true mean

-- as confidence increases, interval must increase, as n increases, interval decrease

-- example problem

G. Concentration Sensitivity

1. Calibration curve gives S = f(c), sensitivity: m = dE/dc

-- Concentration Confidence interval: m_c = c ± ts_c/ n^1/2 s_c = s/m

-- Actual conficence (error) also affected by calibration error

-- Analytical senstivity: g = m/s = 1/s_c corrects for gain, etc.

2. Detection Limit

-- DL = k^.s_bk / m s_bk -- S.D. of blank, k -- confidence factor, t = k / 2^1/2

(goal make measurements at >10*DL)