Myers [1990] presents data on the number of fractures (y) that occur in the upper seams of coal mines in the Appalachian region of western Virginia. Four regressors were reported: \(x_{1}=\) inner burden thickness (feet), the shortest distance between seam floor and the lower seam; \(x_{2}=\) percent extraction of the lower previously mined seam; \(x_{3}=\) lower seam height (feet); and \(x_{4}=\) time (years) that the mine has been in operation. The data are shown below.

a. Fit a Poisson regression model to these data using the log link.
b. Does the model deviance indicate that the model from part \(a\) is satisfactory?
c. Perform a type 3 partial deviance analysis of the model parameters. Does this indicate that any regressors could be removed from the model?
d. Compute Wald statistics for testing the contribution of each regressor to the model. Interpret the results of these test statistics.
e. Find approximate \(95 \%\) Wald confidence intervals on the model parameters.

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Observation Number of Fractures per Subregion, y x X2 X3 X4 1234567820 9 11 12 13 14 15 16 17 18 222222222836549 19 20 21 27 30 31 33 37 40 41 43 44 2014020044L4I52555OSTI3322O15233300200323503 50 70 230 65 125 70 844 52 1.0 42 6.9 45 1.0 75 65 68 0.5 70 65 53 0.5 65 70 46 3.0 65 60 62 1.0 350 60 54 0.5 350 90 54 0.5 160 80 38 0.0 145 65 38 10.0 145 85 38 0.0 180 70 42 2.0 43 80 40 0.0 42 85 51 12.0 42 85 51 0.0 45 85 42 0.0 83 85 48 10.0 300 65 68 10.0 190 90 84 6.0 145 90 54 12.0 510 80 57 10.0 65 75 68 5.0 470 90 90 9.0 300 80 165 9.0 275 90 40 4.0 420 50 44 17.0 65 80 48 15.0 40 75 51 15.0 900 90 48 35.0 95 88 36 20.0 40 85 57 10.0 140 90 38 7.0 150 50 44 5.0 80 60 96 5.0 80 85 96 5.0 145 65 72 9.0 100 65 72 9.0 150 80 48 3.0 150 80 48 0.0 210 75 42 2.0 11 75 42 0.0 100 65 60 25.0 50 88 60 20.0