a$)$ Use a logit model to determine the probabilities of a student commuter choosing between three modes of travel during the morning peak hour: Private car $($auto$),$ bus, and Light rail. he travel time for auto, bus, and rail are $10,13,$ and $15$ minutes, respectively. The utility functions for the three modes are estimated using the following equations.

$]=[132\mathrm{.}7313-2\mathrm{.}5946**$ tauto $]=[-25\mathrm{.}8128-26\mathrm{.}0811**$ tbus $]=[,12\mathrm{.}3078-7\mathrm{.}9316**$ trail

b$)$ To encourage more public transport use, the local transport authority has constructed a bus lane in order to alter the modal split in favour of bus usage. It is estimated that when in operation, the bus lane will reduce the bus journey time to $10$ minutes and will increase the car travel time to $12$ minutes. Based on these revised figures, determine the new modal splits.

c$)$ Students taking the EG$7031$ module were surveyed in class about their travel history $($data will be presented in class$).$ Determine the travel mode of the first $10$ students using the utility functions in $4a).$

d$)$ Discuss the appropriateness of using the utility functions given above to predict the travel patterns of EG$7031$ students.

Table $4\mathrm{.}1$ Travel times for EG$7031$ students

$\backslash$table$[[$Student$,$Travel time $($minutes$)],[,$Car,Bus,Rail$],[1,30,90,40],[2,115,95,105],[3,48,51,30],[4,15,27,35],[5,22,40,35],[6,15,20,30],[7,31,56,66],[8,25,45,30],[9,73,105,83],[10,79,243,69]]$