Mathematical Methods for Physics and Engineering : A Comprehensive Guide

31.6 THE METHOD OF LEAST SQUARES The other possibility is thatλis an independent parameter and not a function of the parametersa ...

STATISTICS where the quantity denoted byχ^2 is given by the quadratic form χ^2 (a)= ∑N i,j=1 [yi−f(xi;a)](N−^1 )ij[yj−f(xj;a)] = ...

31.6 THE METHOD OF LEAST SQUARES where{h 1 (x),h 2 (x),...,hM(x)}is some set of linearly independent fixed functions ofx, often ...

STATISTICS Setting the expression (31.95) equal to zero ata=aˆ, we find − 2 RTN−^1 y+2RTN−^1 Raˆ=0. Provided the matrixRTN−^1 Ri ...

31.6 THE METHOD OF LEAST SQUARES 0 0 1 1 2 2 3 3 4 4 5 5 6 7 y x Figure 31.9 A set of data points with error bars indicating the ...

STATISTICS hence does not possess an inverse. Inserting the form forRin (31.100) into the expression (31.101), we find ( ˆc mˆ ) ...

31.7 HYPOTHESIS TESTING however, such problems are best solved using one of the many commercially available software packages. O ...

STATISTICS however, one wishes to use the data to give a ‘yes’ or ‘no’ answer to a particular question. For example, one might w ...

31.7 HYPOTHESIS TESTING t P(t|H 0 ) tcrit α t tcrit P(t|H 1 ) β Figure 31.10 The sampling distributionsP(t|H 0 )andP(t|H 1 ) of ...

STATISTICS false (in which caseH 1 is true). The probabilityβ(say)thatsuchanerrorwill occur is, in general, difficult to calcula ...

31.7 HYPOTHESIS TESTING
Ten independent sample valuesxi,i=1, 2 ,..., 10 , are drawn at random from a Gaussian distribution with ...

STATISTICS properties and which reduces to the Neyman–Pearson statistic (31.108) in the special case whereH 0 andH 1 are both si ...

31.7 HYPOTHESIS TESTING
Ten independent sample valuesxi,i=1, 2 ,..., 10 , are drawn at random from a Gaussian distribution with ...

STATISTICS containingRof theM parameters.) IfH 0 is true then it follows from our discussion in subsection 31.5.6 (although we s ...

31.7 HYPOTHESIS TESTING The sum of squares in the denominator of (31.112) may be put into the form ∑ i(xi−μ^0 ) (^2) =N( ̄x−μ 0 ...

STATISTICS Using (31.113) to substitute for ̄x−μ 0 in (31.116), and noting thatdx ̄= (s/ √ N−1)dt, we find P( ̄x, s|H 0 )d ̄xds= ...

31.7 HYPOTHESIS TESTING 0 0 0. 1 0. 2 0. 3 0. 4 0. 5 − 4 − 3 − 2 − 1123 4 t P(t|H 0 ) N=2 N=3 N=5 N=10 Figure 31.11 Student’st-d ...

STATISTICS Cn(t) 0.5 0.6 0.7 0.8 0.9 0.950 0.975 0.990 0.995 0.999 n=1 0.00 0.33 0.73 1.38 3.08 6.31 12.7 31.8 63.7 318.3 2 0.00 ...

31.7 HYPOTHESIS TESTING distributions. In particular, let us consider the case where we have two independent samples of sizesN 1 ...

STATISTICS whereαis the required significance level of the test. In our case we setα=0.05, and from table 31.3 withn= 16 we find ...