A chemical manufacturer has maintained records on the number of failures of a particular type of valve used in its processing unit and the length of time (months) since the valve was installed. The data are shown below.

a. Fit a Poisson regression model to the data.
b. Does the model deviance indicate that the Poisson regression model from part a is adequate?
c. Construct a graph of the fitted model versus months. Also plot the observed number of failures on this graph.
d. Expand the linear predictor to include a quadratic term. Is there any evidence that this term is required in the model?
e. For the model in part a, find Wald statistics for each individual model parameter.
f. Find approximate \(95 \%\) confidence intervals on the model parameters for the Poisson regression model from part a.

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Number of Number of Valve Failures Months Valve Failures Months 12345678 5 18 9 0 7 3 15 10 0 12 0 11 11 4 14 12 1 23 13 10 14 7 5 15 0 237289. 30 8