The market research department of a soft drink manufacturer is investigating the effectiveness of a price discount coupon on the purchase of a twoliter beverage product. A sample of 5500 customers was given coupons for varying price discounts between 5 and 25 cents. The response variable was the number of coupons in each price discount category redeemed after one month. The data are shown below.

a. Fit a logistic regression model to the data. Use a simple linear regression model as the structure for the linear predictor.
b. Does the model deviance indicate that the logistic regression model from part a is adequate?
c. Draw a graph of the data and the fitted logistic regression model.
d. Expand the linear predictor to include a quadratic term. Is there any evidence that this quadratic term is required in the model?
e. Draw a graph of this new model on the same plot that you prepared in part \(\mathrm{c}\). Does the expanded model visually provide a better fit to the data than the original model from part a?
f. For the quadratic model in part \(\mathrm{d}\), find Wald statistics for each individual model parameter.
g. Find approximate \(95 \%\) confidence intervals on the model parameters for the quadratic logistic regression model from part \(\mathrm{d}\).

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Discount, x Sample Size, n Number Redeemed, r 579 5 500 100 7 9 500 122 500 147 11 500 176 13 500 211 15 500 244 17 500 277 19 500 310 21 500 343 23 500 372 25 500 391