Question

the following are the ratings of men by women in an experiment involving speed dating. use the given data to construct a boxplot and identify the 5 - number summary
1.0 2.0 2.5 3.0 3.5 3.5 4.5 4.5 4.5 4.5 4.5 5.5 6.5 7.0 7.0 7.0 8.0 8.0 8.5 8.5
the 5 - number summary is \\( \square, \square, \square, \square \\), and \\( \square \\)
(use ascending order. type integers or decimals. do not round )

Explanation:

Step1: Find the minimum value

The minimum value in the data set is the smallest number. Looking at the data \(1.0, 2.0, 2.5, 3.0, 3.5, 3.5, 4.5, 4.5, 4.5, 4.5, 4.5, 5.5, 6.5, 7.0, 7.0, 7.0, 8.0, 8.0, 8.5, 8.5\), the minimum \(x_{\text{min}}=1.0\).

Step2: Find the first quartile (\(Q_1\))

The data set \(n = 20\) (even number of data points). The first half of the data is \(1.0, 2.0, 2.5, 3.0, 3.5, 3.5, 4.5, 4.5, 4.5, 4.5\). The median of the first half (for \(n = 10\) (even) ) is \(\frac{3.5 + 3.5}{2}=3.5\). So \(Q_1 = 3.5\).

Step3: Find the median (\(Q_2\))

For \(n=20\) (even), the median is the average of the \(\frac{n}{2}\)th and \((\frac{n}{2}+1)\)th values. The \(\frac{20}{2}=10\)th and \(11\)th values. The \(10\)th value is \(4.5\) and the \(11\)th value is \(4.5\). So \(Q_2=\frac{4.5 + 4.5}{2}=4.5\).

Step4: Find the third quartile (\(Q_3\))

The second half of the data is \(5.5, 6.5, 7.0, 7.0, 7.0, 8.0, 8.0, 8.5, 8.5\). The median of the second half (for \(n = 10\) (even)) is \(\frac{7.0+7.0}{2}=7.0\). So \(Q_3 = 7.0\).

Step5: Find the maximum value

The maximum value in the data set is \(x_{\text{max}}=8.5\).

Answer:

\(1.0,3.5,4.5,7.0,8.5\)