The article Polyhedral Distortions in Tourmaline (A. Ertl, J. Hughes, et al., The Canadian Mineralogist, 2002: 153162)

The article €œPolyhedral Distortions in Tourmaline€ (A. Ertl, J. Hughes, et al., The Canadian Mineralogist, 2002: 153€“162) presents a model for calculating bond length distortion in vanadium-bearing tourmaline. To check the accuracy of the model, several calculated values (x) were compared with directly observed values (y). The results (read from a graph) are presented in the following table.

a. Assume that the observed value y is an unbiased measurement of the true value. Show that if the calculated value x is accurate (i.e., equal to the true value), then y = x + ε, where ε is measurement error.

b. Compute the least-squares line y = β̂₀ + β̂₁x.

c. Show that if the calculated value is accurate, then the true coefficients are β₀ = 0 and β₁ = 1.

d. Test the null hypotheses β₀ = 0 and β₁ = 1.

e. Is it plausible that the calculated value is accurate? Or can you conclude that it is not? Explain.

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Calculated Value Observed Value Calculated Value Observed Value 0.33 0.36 0.36 0.74 0.78 0.79 0.97 0.36 0.54 0.86 0.97 0.58 0.64 0.64 0.67 1.03 1.10 1.13 1.14 0.56 1.11 0.66 1.06 0.66 0.74 1.08 0.58 1.17