The temperature dependence of resistance of a new alloy is being investigated. T

Question

The temperature dependence of resistance of a new alloy is being investigated. The table below shows values of resistance, R, of the alloy as a function of temperature, T. R(S) | 1.043 | 1.214 | 1.797 | 1.698 | 2.312 | 2.677 | 3.027 | 3.247 | 4.056 | 4.208 | 4.886 35 40 45 50 60 65 70 75 80 85 Two equations are to be fitted to the data, namely: R=a+bT a and b are constants R=A+BT+CT2 A B and C are constants Fit each equation in turn to the data in the table using unweighted least squares (Use Excel to do help you do this). Obtain from your fits best estimates for a, b, A, B and C Determine standard uncertainties in a, b, A, B and C Find the sum of squares of residuals (SSR) for each fit Calculate the Akaikes Information Criterion (AIC) for each of the equations you fitted to the data Based on your analysis, which equation better fits the data? 1. [2.5] [2.5) iii. iv.

Explanation / Answer

Here, temperature T is the independent variable and resistance R is the dependent variable.

R = a+b*T

R = A+B*T+C*T2

(i)

For First equation

LINEST(B2:B12,A2:A12,TRUE,TRUE)

a = 0.075965, b = -1.81565

For second equation

LINEST(B2:B12,C15:D25,1,1)

A =-0.0785 , B =0.013739, C = 0.000519.

(ii) Standard uncertainties

For first equation

Uncertainty in a, sa =0.227475

Uncertainty in b, sb = 0.003666

For second equation

sA =0.7252252

sB =0.02532

sC =0.000209588

(iii) SSR

For first equation

SSR = sum(y-yhat)^2= 0.3326449

For second equation

SSR = sum(y-yhat)^2=0.188448

(iv) AIC for equation

no of data, n =11

no of parameters, M = 3

K = M+1 = 4

AIC = n*Loge(SSR/n)+2*K

For first equation

AIC1 = 11*Loge(0.3326449/11)+2*4= -30.484325

For second equation

AIC2 = 11*Loge(0.188448/11)+2*4 =-36.7351129

dAIC = AIC1-AIC2=-30.484325-(-36.7351129)= 6.250788

Probability of 1 better than 2, p = exp(-dAIC/2)/{1+exp(-dAIC/2)}= 0.042071848

As the probability is small, hence model 2 is better.

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