The article “Polyhedral Distortions in Tourmaline” (A. Ertl, J. Hughes, et at., 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.

Observed Value |
Calculated Value |
Observed Value |
Calculated Value |

0.33 |
0.36 |
0.74 |
0.78 |

0.36 |
0.36 |
0.79 |
0.86 |

0.54 |
0.58 |
0.97 |
0.97 |

0.56 |
0.64 |
1.03 |
1.11 |

0.66 |
0.64 |
1.10 |
1.06 |

0.66 |
0.67 |
1.13 |
1.08 |

0.74 |
0.58 |
1.14 |
1.17 |

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

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.