Mathematical Methods for Physics and Engineering - Matematica.NET

STATISTICS31.16 The function y(x) is known to be a quadratic function of x. The following tablegives the measured values and uncorrelated standard errors of y measured atvarious values of x (in which there is negligible error):x 1 2 3 4 5y(x) 3.5 ± 0.5 2.0 ± 0.5 3.0 ± 0.5 6.5 ± 1.0 10.5 ± 1.0Construct the response matrix R using as basis functions 1, x,x 2 . Calculate thematrix R T N −1 R and show that its inverse, the covariance matrix V, hastheform⎛⎞V = 1 12 592 −9708 1580⎝−9708 8413 −1461⎠ .9184 1580 −1461 269Use this matrix to find the best values, and their uncertainties, for the coefficientsof the quadratic form for y(x).31.17 The following are the values and standard errors of a physical quantity f(θ)measured at various values of θ (in which there is negligible error):θ 0 π/6 π/4 π/3f(θ) 3.72 ± 0.2 1.98 ± 0.1 −0.06 ± 0.1 −2.05 ± 0.1θ π/2 2π/3 3π/4 πf(θ) −2.83 ± 0.2 1.15 ± 0.1 3.99 ± 0.2 9.71 ± 0.4Theory suggests that f should be of the form a 1 + a 2 cos θ + a 3 cos 2θ. Show thatthe normal equations for the coefficients a i are481.3a 1 + 158.4a 2 − 43.8a 3 = 284.7,158.4a 1 + 218.8a 2 +62.1a 3 = −31.1,−43.8a 1 +62.1a 2 + 131.3a 3 = 368.4.(a) If you have matrix inversion routines available on a computer, determine thebest values and variances for the coefficients a i and the correlation betweenthe coefficients a 1 and a 2 .(b) If you have only a calculator available, solve for the values using a Gauss–Seidel iteration and start from the approximate solution a 1 =2,a 2 = −2,a 3 =4.31.18 Prove that the expression given for the Student’s t-distribution in equation (31.118)is correctly normalised.31.19 Verify that the F-distribution P (F) given explicitly in equation (31.126) is symmetricbetween the two data samples, i.e. that it retains the same form but with N 1and N 2 interchanged, if F is replaced by F ′ = F −1 . Symbolically, if P ′ (F ′ )isthedistribution of F ′ and P (F) =η(F,N 1 ,N 2 ), then P ′ (F ′ )=η(F ′ ,N 2 ,N 1 ).31.20 It is claimed that the two following sets of values were obtained (a) by randomlydrawing from a normal distribution that is N(0, 1) and then (b) randomlyassigning each reading to one of two sets A and B:Set A: −0.314 0.603 −0.551 −0.537 −0.160 −1.635 0.7190.610 0.482 −1.757 0.058Set B: −0.691 1.515 −1.642 −1.736 1.224 1.423 1.165Make tests, including t- andF-tests, to establish whether there is any evidencethat either claims is, or both claims are, false.1302