We want to study the effect that a book monetary deposit amount at the library would have on the return rate for the book, where a book was deemed returned if it was returned within one month. A book return was scored 1 and no return was scored 0. The data below shows the number of books that were returned (Rj) out of 500 books lent (Lj), at each of 6 deposit amounts (Xj) in cents. Assume that Xj is normally distributed with estimates of mean and variance from the training data. Also assume that the probability of each of the classes are estimated from the training data. Note that is a 2-class problem with the classes Returned and Not Returned.

1. Plot the estimated proportions pj = Rj/Lj against Xj. Does the plot support the use of logistic regression? Why?
2. Find the estimates b0 and b1 for this problem.
3. Find the actual error of the training set, based on part b, where you calculate the
4. probability of a book being in the class Returned.
5. Find the misclassification error of the training set, based on part c, using 0.5 as
6. the threshold for a book to be in the class Returned. That is, book* is classified as a member of Returned if its probability of being in Returned is greater than or equal to 0.5. Also, find the answers for thresholds of 0.4 and 0.6.
7. What is the estimated probability that a book will be returned when the deposit is 15 cents?
8. Estimate the amount of the deposit so that 75% of the books are expected to be returned.

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