The article “Application of Radial Basis Function Neural Networks in Optimization of Hard Turning of AISID2 Cold- Worked Tool Steel With a Ceramic Tool” (S. Basak. U. Dixit, and J. Davim, Journal of Engineering Manufacture, 2007:987-998) presents the results of an experiment in which the surface roughness (in µm) was measured for 27 D2 steel specimens and compared with the roughness predicted by a neural network model. The results are presented in the following table.

True Value (X) |
Predicted Value (Y) |
True Value (X) |
Predicted Value (Y) |
True Value (X) |
Predicted Value (Y) |

0.45 |
0.42 |
0.52 |
0.51 |
0.57 |
0.55 |

0.82 |
0.70 |
1.02 |
0.91 |
1.14 |
1.01 |

0.54 |
0.52 |
0.60 |
0.71 |
0.74 |
0.81 |

0.41 |
0.39 |
0.58 |
0.50 |
0.62 |
0.66 |

0.77 |
0.74 |
0.87 |
0.91 |
1.15 |
1.06 |

0.79 |
0.78 |
1.06 |
1.04 |
1.27 |
1.31 |

0.25 |
0.27 |
0.45 |
0.52 |
1.31 |
1.40 |

0.62 |
0.60 |
1.09 |
0.97 |
1.33 |
1.41 |

0.91 |
0.87 |
1.35 |
1.29 |
1.46 |
1.46 |

To check the accuracy of the prediction method, the linear model y = β0 + β1 x + ε is fit. If the prediction method is accurate, the value of β0 will be 0 and the value of β1 will be 1.

a. Compute the least-squares estimates and .

b. Can you reject the null hypothesis H0: β0 = 0?

c. Can you reject the null hypothesis H0: β1 = 1?

d. Do the data provide sufficient evidence to conclude that the prediction method is not accurate?

e. Compute a 95% confidence interval for the mean prediction when the true roughness is 0.8 µm.

f. Someone claims that when the true roughness is 0.8 µm, the mean prediction is only 0.75 µm. Do these data provide sufficient evidence for you to conclude that this claim is false? Explain.

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