Problem $3(40$ points$)$

Assume we get some data from a car insurance company in Table $1,$ where there are $6$ data instances representing $6$ people, with $2$ attributes $($Age and Car$)$ and $1$ class label $($Risk$).$ Here Age is a continuous attribute. Now we will build decision trees for this data set.

Table $1$: Data for Problem $3.$ Age is numeric and Car is categorical. Risk gives the class label for each point: high $($H$)$ or low $($L$).$

$1.$ Let us consider a multi$-$way split for the Car attribute $($using its unique values for partition$).$ What is the information gain if we choose the Car attribute to split the root node? $(5$ points$)$

$2.$ Let us consider the binary splits for the Car attribute. Using information gain as the measure, which binary split of the Car attribute is the best at the root node? $(5$ points$)$

$3.$ Between $(1)$ and $(2),$ which one do you prefer for splitting the root node using the Car attribute? Hint: Consider the GainRatio measure. $(5$ points$)$

$4.$ Now, construct an entire decision tree for the given data set, using either information gain or GainRatio as the split point evaluation measure. You will need to consider the Age attribute for splitting the root node as well. Only consider binary splits of the Age attribute. You can leverage your calculations or conclusions in $(1-3).(20$ points$)$

$5.$ Classify the point $($Age$=27,$ Car$=$SUV$)$ based on the constructed decision tree in $(4).(5$ points$)$