A study was performed to investigate new automobile purchases. A sample f 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 6 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 following table.

a. Fit a logistic regression model to the data.
b. Does the model deviance indicate that the logistic regression model from part a is adequate?
c. Interpret the model coefficients \(\beta_{1}\) and \(\beta_{2}\).
d. What is the estimated probability that a family with an income of \(\$ 45,000\) and a car that is 5 years old will purchase a new vehicle in the next 6 months?
e. Expand the linear predictor to include an interaction term. Is there any evidence that this term is required in the model?
f. For the model in part a, find statistics for each individual model parameter.
g. Find approximate \(95 \%\) confidence intervals on the model parameters for the logistic regression model from part a.

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Income, x Age, X2 y Income, x Age, X2 y 45,000 40,000 60,000 50,000 55,000 50,000 35,000 65,000 53,000 48,000 1 2432257222 37,000 0 31,000 40,000 1 75,000 0 43,000 1 49,000 1 37,500 100 71,000 34,000 5742924LS 1 1 1 1 0 1000 27,000 6