Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 218 / 3 / a

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Solution

Conditional on the covariates, the binary responses are independent with Bernoulli distributions:

Y_{i} \sim Bernoulli (p_{i}), lo g \frac{p _{i}}{1 - p _{i}} = η_{i} = β_{0} + β_{I} income_{i} + β_{A} age_{i} + β_{L} amount_{i} .

Thus

p_{i} = (1 + e^{- η_{i}})^{- 1}

, the logistic regression model. The maximum-likelihood estimates are

(β_{0}, β_{I}, β_{A}, β_{L}) = (- 1.322701, - 0.005916, - 0.008580, 0.012981) .

Using the reported age standard error, an approximate 95% Wald confidence interval is

- 0.008580 \pm 1.96 (0.003833) = [- 0.01609268, - 0.00106732] \approx [- 0.01609, - 0.00107] .

This interval concerns the age coefficient in log odds per year, with income and loan amount held fixed.

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