1. Exercise 5.5: In a class of 50 students, a logistic regression is performed of course grade (pass or fail) on midterm exam score (continuous values with mean 60 and standard deviation 15). The fitted model is Pr(pass) = logit−1(−24 + 0.4x).
(a) Graph the fitted model. Also on this graph put a scatterplot of hypothetical data consistent with the information given.
(b) Suppose the midterm scores were transformed to have a mean of 0 and standard deviation of 1. What would be the equation of the logistic regression using these transformed scores as a predictor?
(c) Create a new predictor that is pure noise (for example, in R you can create
newpred <- rnorm (n,0,1)). Add it to your model. How much does the deviance decrease?
2. Exercise 5.6: Latent-data formulation of the logistic model: take the model
Pr(y = 1) = invlogit (1 + 2x1 + 3x2) and consider a person for whom x1 = 1 and x2 = 0.5. Sketch the distribution of the latent data for this person. Figure out the probability that y=1 for the person and shade the corresponding area on your graph.
3. Exercise 5.7: Limitations of logistic regression: consider a dataset with n = 20 points, a single predictor x that takes on the values 1, . . . , 20, and binary data y. Construct data values y1, . . . , y20 that are inconsistent with any logistic regression on x. Fit a logistic regression to these data, plot the data and fitted curve, and explain why you can say that the model does not fit the data.
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