Question

the data represent the age of world leaders on their day of inauguration. find the five - number summary, and construct a boxplot for the data. comment on the shape of the distribution.
the five - number summary is □,□,□,□,□

Explanation:

Step1: Sort the data

First, we sort the data set. The data values are \(41,43,48,48,49,50,52,54,55,57,58,61,62,65,69\).

Step2: Find the minimum value

The minimum value (\(Q_0\)) of the sorted data set is the first value. So, \(Q_0 = 41\).

Step3: Find the first - quartile (\(Q_1\))

The first - quartile is the median of the lower half of the data. The lower half of the data is \(41,43,48,48,49,50,52\). The number of data points \(n = 7\). Using the formula \(Q_1=\frac{n + 1}{4}\)th value (for odd \(n\) in the sub - set). \(\frac{7+1}{4}=2\)nd value. So, \(Q_1 = 48\).

Step4: Find the median (\(Q_2\))

The median is the middle value of the sorted data set. Since \(n = 15\) (odd), the median is the \(\frac{15 + 1}{2}=8\)th value. So, \(Q_2=54\).

Step5: Find the third - quartile (\(Q_3\))

The third - quartile is the median of the upper half of the data. The upper half of the data is \(55,57,58,61,62,65,69\). Using the formula \(Q_3=\frac{3(n + 1)}{4}\)th value (for odd \(n\) in the sub - set). \(\frac{3\times(7 + 1)}{4}=6\)th value. So, \(Q_3 = 61\).

Step6: Find the maximum value

The maximum value (\(Q_4\)) of the sorted data set is the last value. So, \(Q_4 = 69\).

Answer:

\(41,48,54,61,69\)