Refer to the American Scientist (Jul–Aug. 1998) study of the relationship between number of self avoiding and unrooted walks, Exercise 11.71. The data for the analysis are repeated in the accompanying table.

a. Give the equation relating the coded variable u to walk length, x, using the coding system for observational data.

b. Calculate the coded values, u.

c. Calculate the coefficient of correlation r between the variables x and x².

d. Calculate the coefficient of correlation r between the variables u and u². Compare this value to the value computed in part c.

e. Let y = number of unrooted walks. Fit the model

E(y) = β₀ + β₁u + β₂u₂²

using available statistical software. Interpret the result.

Data from Exercise 11.71

The American Scientist (July–Aug. 1998) published a study of the relationship between self-avoiding and unrooted walks. In a self-avoiding walk you never retrace or cross your own path, whereas an unrooted walk is a path in which the starting and ending points are impossible to distinguish. The possible number of walks of each type of various lengths are recorded in the accompanying table. Suppose you want to model the number of unrooted walks (y) as a function of walk length (x).

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WALK walk Length Number of Walks (number of steps) Unrooted Self-Avoiding 1 1 4 2 2 12 3 4 36 9 100 5 22 284 56 780 7 147 2172 8. 388 5916