nodo$),$ upl$1$t$,$ n$,10$ss$,$ yval, $($yprob$)$

donotes terainal nodo

root $27595$ Good $(0\mathrm{.}34545450\mathrm{.}6545455)$

ChockingAccount$3$tatus.noner $0\mathrm{.}516177$ Good $(0\mathrm{.}47$B$26090\mathrm{.}5217391)$

Aaount$)=8760\mathrm{.}5,17,0$ Bad $(1\mathrm{.}00000000\mathrm{.}0000000)*$

Aaounte $8760\mathrm{.}514460$ Good $(0\mathrm{.}41666670\mathrm{.}5833333)$

ChockingAccount$3$tatus.gt$.200<mn\; id="EditorElement\_1098021">0</mn><mn\; id="EditorElement\_1098022">.</mn><mn\; id="EditorElement\_1098023">5</mn><mn\; id="EditorElement\_1098024">1</mn><mn\; id="EditorElement\_1098025">2</mn><mn\; id="EditorElement\_1098026">0</mn><mn\; id="EditorElement\_1098027">5</mn><mn\; id="EditorElement\_1098028">7</mn>$ Good $(0\mathrm{.}47500000\mathrm{.}5250000)$

Durat$1$on $=22\mathrm{.}54818$ Bad $(0\mathrm{.}62500000\mathrm{.}3750000)$

MubberPoopleMaintenancer $1\mathrm{.}53710$ Bad $(0.$T$2972970\mathrm{.}2702703)*$

MurberPoopleMaintenance$>-1\mathrm{.}5113$ Good $(0\mathrm{.}27272730\mathrm{.}7272727)*$

Durations $22\mathrm{.}57227$ Good $(0\mathrm{.}37500000\mathrm{.}6250000)$

kaount $1282,25,11$ Bad $(0\mathrm{.}56000000\mathrm{.}4900000)$

Property. PoalEstates $0\mathrm{.}5,17,5$ Bad $(0\mathrm{.}70589240\mathrm{.}2941176)*$

Property. RoalEstate $=0\mathrm{.}5$ & $2$ Good $0\mathrm{.}7500000)*$

Raount$)=12824713$ Good $(0\mathrm{.}27659570\mathrm{.}7234043)*$

CheckingAccount$3$tatus.gt$.200=0\mathrm{.}5243$ Good $(0\mathrm{.}12500000.$g$750000)*$

ChockingAccount$3$tatus.nonop$=0\mathrm{.}511418$ Good $(0\mathrm{.}15789470\mathrm{.}8421053)*$

Is the sensitivity messure of the classification tree on these data equal to

$0\mathrm{.}8088?$ If no$,$ what is the sensitivity mensure? Justify your answer.

$($d$)$ Given a collection of competing clasifiers and some dntn$,$ does the wee of

a validation set to select the best one in the collection gunrantee that the

selected one will alwnys peovide the best predictive performance on future

unseen observations? Remernber to justify your nnswer.

$($e$)$ Two hundred labeled samples are used to trnin two binary clussifiers $M_1$ and

$M_2.$ Far classifier $M_1,$ the dataset is divided into trnining and validation

sets of $100$ samples each and the clnssifier is trained on the training set. The

performance of $M_1$ on this validntion set provides n $80\%$ nccurncy. Far dnssifier

$M_2,$ the dntnset is divided into a training set of $150$ samples nnd a validation set

of $50$ samples, and the clasifier is trained on the training set. The performnnce

of $M_2$ on the corresponding validntion set provides and accuracy of $90\%.$ Is

clnsifer $M_2$ to be preferred to classifier $M_1?$ Justify your nnswer.Provide your answer and a concise explanntion far each of the following questions.

$($a$)$ A k$-$mesans nlgorithm with $K=2$ is employed to cluster the following $52$

data matrix:

$x=\left[\begin{array}{cc}1& 3\\ 2& 4\\ 2& -1\\ 2& 3\\ 1& -2\end{array}\right]$

The algorithm is initialized from the set of centroids ${}_1=(1,1)$ and ${}_2=$

$(3,2).$ After one iteration of the algorithm, is the k$-$means objective function

value equal to $25?$ Remember to justify your answer.

$($b$)$ A marketing researcher uses the k$-$menns algorithm to cluster data concerning

a sarmple of customers of a large npparel retailer. The researcher does not

have any prior information about the number of clusters $K.$ The clustering

allocation for different values of $K$ is compared to an external classification

of the subjects into different types of purchsaing behaviors. The table below

reports the adjusted Rand index $($ARI$)$ vnluss for each value of $K$ considered:

Do you think that $K=4$ corresponds to the optimal number of clusters for

these data? Justify your nnswer.

$($c$)$ A financial institution implements a clasaification tree to classify applicants to

loans according to credit worthiness, "Good" or "Bad", with the main intent

of detecting applicennts with "Good" credit rating outbook. An excerpt of the

output of a classificntion tree implemented using the rpart function on a

sample of data is displayed below.