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A quadratic polynomial as a nonlinear model The data in the file swan96.txt were collected by the Minnesota Department of Natural Resources to study the abundance of black crappies, a species of fish, on Swan Lake, Minnesota in 1996. The response variable is LCPUE, the logarithm of the catch of 200 mm or longer black crappies per unit of fishing effort. It is believed that LCPUE is proportional to abundance. The single predictor is Day, the day on which the sample was taken, measured as the number of days after June 19, 1996. Some of the measurements were taken the following spring on the same population of fish before the young of the year are born in late June. No samples are taken during the winter months when the lake surface was frozen.
1. For these data, fit the quadratic polynomial
2. Using the delta method described in Section 6.1.2, obtain the estimate and variance for the value of Day that maximizes E(LCPUE|Day).
3. Another parameterization of the quadratic polynomial is