A study was performed to investigate new automobile purchases. A sample of 20 families was selected. Each family was surveyed to determine the age of their oldest vehicle and their total family income. A follow-up survey was conducted six months later to determine if they had actually purchased a new vehicle during that time period (y = 1 indicates yes and y = 0 indicates no). The data from this study are shown in the Table E11-16.

(a) Fit a logistic regression model to the data.

(b) Is the logistic regression model in part (a) adequate?

(c) Interpret the model coefficients β₁ and β₂.

(d) What is the estimated probability that a family with an income of $45,000 and a car that is five years old will purchase a new vehicle in the next six months?

(e) Expand the linear predictor to include an interaction term. Is there any evidence that this term is required in the model?

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Age, x2 Income, x, Age, x2 Income, x, 37,000 45,000 1 40,000 31,000 1 60,000 3 1 40,000 4 1 50,000 2 75,000 2 55,000 2 43,000 9. 50,000 5 1 49,000 2 35,000 1 37,500 65,000 2 71,000 53,000 34,000 5 48,000 27,000 2. Number Discount, x Sample Size, n Redeemed, r 500 100 500 122 500 147 11 500 176 13 500 211 15 500 244 17 500 277 19 500 310 21 500 343 23 500 372 25 500 391