Data from the World Happiness Report $2019$ record composite scores measuring

the level of happiness $($happiness$),$ GDP per capita $($gdp$),$ healthy life expectancy

$(11$feexp$),$ and the perceived level of corruption $($corruption$)$ for a sample of

countries in the world. A team of data scientists wishes to investigate the potential

presence of a clustering structure in these data using k$-$means.

$($a$)$ The team uses silhouette analysis to guide the selection of the number of

clusters. The following silhouette plots are produced, for a number of clusters

$K$ ranging from $2$ to $5.$ How many clusters does the average silhouette analysis suggest? Justify your

answer. $($b$)$ To further aid the selection of the number of clusters, the team computes the

gap statistic for $K$ ranging from $1$ to $10$ using function clusGap of package

cluster. The output table from the function and the gap statistic plot are

reported below $($next page$).$ Gap statistic plot

$($i$)$ What are the quantities logW$,$ E$.$logW$,$ and SE$.$sim in the output? Explain

briefly.

$($ii$)$ How many clusters does the gap statistic method suggest? Justify your

answer.$($c$)$ The team selects $K=3$ and uses k$-$means to cluster the data. The output

from function kmeans is reported below.

$($i$)$ Compute the values of the total between cluster sum of squares and of

the total sum of squares.

$($ii$)$ The USA have the following values for the scores of interest:

To which cluster would the USA be assigned to$?$ Justify your answer

using calculations.