Mathematical Methods for Physics and Engineering : A Comprehensive Guide

STATISTICS

31.16 The functiony(x) is known to be a quadratic function ofx. The following table
gives the measured values and uncorrelated standard errors ofymeasured at
various values ofx(in which there is negligible error):

x 1234 5

y(x)3. 5 ± 0. 52. 0 ± 0. 53. 0 ± 0. 56. 5 ± 1. 010. 5 ± 1. 0

Construct the response matrixRusing as basis functions 1,x,x^2. Calculate the

matrixRTN−^1 Rand show that its inverse, the covariance matrixV,hastheform

V=

1

9184





12 592 − 9708 1580

− 9708 8413 − 1461

1580 − 1461 269



.

Use this matrix to find the best values, and their uncertainties, for the coefficients
of the quadratic form fory(x).
31.17 The following are the values and standard errors of a physical quantityf(θ)
measured at various values ofθ(in which there is negligible error):

θ 0 π/ 6 π/ 4 π/ 3

f(θ)3. 72 ± 0. 21. 98 ± 0. 1 − 0. 06 ± 0. 1 − 2. 05 ± 0. 1

θπ/ 22 π/ 33 π/ 4 π

f(θ) − 2. 83 ± 0. 21. 15 ± 0. 13. 99 ± 0. 29. 71 ± 0. 4

Theory suggests thatfshould be of the forma 1 +a 2 cosθ+a 3 cos 2θ. Show that

the normal equations for the coefficientsaiare

481. 3 a 1 + 158. 4 a 2 − 43. 8 a 3 = 284. 7 ,

158. 4 a 1 + 218. 8 a 2 +62. 1 a 3 =− 31. 1 ,

− 43. 8 a 1 +62. 1 a 2 + 131. 3 a 3 = 368. 4.

(a) If you have matrix inversion routines available on a computer, determine the

best values and variances for the coefficientsaiand the correlation between

the coefficientsa 1 anda 2.

(b) If you have only a calculator available, solve for the values using a Gauss–

Seidel iteration and start from the approximate solutiona 1 =2,a 2 =−2,

a 3 =4.

31.18 Prove that the expression given for the Student’st-distribution in equation (31.118)
is correctly normalised.

31.19 Verify that theF-distributionP(F) given explicitly in equation (31.126) is symme-
tric between the two data samples, i.e. that it retains the same form but withN 1
andN 2 interchanged, ifFis replaced byF′=F−^1. Symbolically, ifP′(F′)isthe
distribution ofF′andP(F)=η(F, N 1 ,N 2 ), thenP′(F′)=η(F′,N 2 ,N 1 ).
31.20 It is claimed that the two following sets of values were obtained (a) by ran-
domly drawing from a normal distribution that isN(0,1) and then (b) randomly
assigning 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. 719

0 .610 0. 482 − 1. 757 0. 058

Set B: − 0 .691 1. 515 − 1. 642 − 1. 736 1. 224 1 .423 1. 165

Make tests, includingt-andF-tests, to establish whether there is any evidence

that either claims is, or both claims are, false.