Q$2(40$ pts$)$: Consider a dataset used by a retail company to predict whether a customer will make a purchase based on three features:

Table $1$: Customer Purchase Dataset

Questions:

$($a$)$ Calculate the entropy of the target variable $($i$.$e$.,$ the entropy of the entire dataset$).$ Show all steps in your calculation. $[5$ points$]$

$($b$)$ Calculate the entropy of the target variable for each value of Age Group, Income Level, and Online Activity. Show all steps in your calculation. $[20$ points$]$

$($c$)$ Based on the entropy calculations from parts $($a$)$ and $($b$),$ construct a decision tree by choosing the feature that provides the highest information gain. Show the steps taken to select the best feature and draw the decision tree. $[15$ points$]$

Proof: