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systems of equations with matrices; data fitting 1893. In the limit of very large numbers, we would expect a plot of ln |dN/dt| versust to be a straight line:∣ ln∆N(t)∣∣∣ ∣ ∆t ∣ ≃ ln ∆N 0∆t ∣ − 1 τ ∆t.This means that if we treat ln |∆N(t)/∆t| as the dependent variable andtime ∆t as the independent variable, we can use our linear fit results. Plotln |∆N/∆t| versus ∆t.4. Make a least-squares fit of a straight line to your data and use it to determinethe lifetime τ of the π meson. Compare your deduction to the tabulatedlifetime of 2.6 × 10 −8 s and comment on the difference.5. Plot your best fit on the same graph as the data and comment on theagreement.6. Deduce the goodness of fit of your straight line and the approximate error inyour deduced lifetime. Do these agree with what your “eye” tells you?8.7.3 Exercise: Fitting Heat FlowThe table below gives the temperature T along a metal rod whose ends are kept ata fixed constant temperature. The temperature is a function of the distance x alongthe rod.x i (cm) 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0T i (C) 14.6 18.5 36.6 30.8 59.2 60.1 62.2 79.4 99.91. Plot the data to verify the appropriateness of a linear relationT (x) ≃ a + bx. (8.62)2. Because you are not given the errors for each measurement, assume thatthe least significant figure has been rounded off and so σ ≥ 0.05. Use that tocompute a least-squares straight-line fit to these data.3. Plot your best a + bx on the curve with the data.4. After fitting the data, compute the variance and compare it to the deviationof your fit from the data. Verify that about one-third of the points miss the σerror band (that’s what is expected for a normal distribution of errors).5. Use your computed variance to determine the χ 2 of the fit. Comment on thevalue obtained.6. Determine the variances σ a and σ b and check whether it makes sense to usethem as the errors in the deduced values for a and b.−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 189