Question

data analysis and probability
interpreting a box - and - whisker plot: problem type 2

use the box - and - whisker plot to answer the questions below.
(a) which of the following intervals gives the middle 50% of the data?
24 to 41 24 to 29 29 to 41 21 to 29
(b) what is the difference of the largest number and the smallest number of waves jina surfed in a day? (that is, what is
the range of the data?)
(c) which of these best describes the numbers of waves jina surfed last summer?
the number of waves she surfed was below 29 on more days than it was above 29.

Explanation:

Step1: Recall box - and - whisker plot properties

In a box - and - whisker plot, the box represents the middle 50% of the data. The left - hand side of the box is the first quartile (\(Q_1\)) and the right - hand side is the third quartile (\(Q_3\)).

Step2: Identify \(Q_1\) and \(Q_3\)

From the plot, \(Q_1 = 29\) and \(Q_3=41\). So the interval for the middle 50% of the data is \(29\) to \(41\).

Step3: Calculate the range

The range of a data set is given by \(R=\text{Max}-\text{Min}\). From the plot, \(\text{Max} = 50\) and \(\text{Min}=21\). Then \(R = 50 - 21=29\).

Step4: Analyze the position of the median

The median of a box - and - whisker plot is the line inside the box. Since the left part (below the median) and the right part (above the median) of the box are of equal length in terms of representing 50% of the data. But looking at the whiskers, the length of the whisker below \(Q_1\) (if we consider the spread) - the number of data points below \(29\) (since \(Q_1 = 29\)) is less than the number of data points above \(29\) (because the whisker above \(Q_3\) is longer and the box also contributes to the upper half).

Answer:

(a) \(29\) to \(41\)
(b) \(29\)
(c) None of the options (the number of waves she surfed was above \(29\) on more days than it was below \(29\) as the whisker above \(Q_3\) is longer, indicating a more spread out upper half. But if we assume the options are only the given one, there is an error in the problem's option set for part (c) as the correct statement should be the opposite of the given option. If we have to choose from the given options for (c) based on the box (ignoring the whisker spread in a very - basic sense, but this is not a proper statistical analysis), there is no correct option. However, if we consider only the box (middle 50%), the number of data points below \(29\) (first quartile) is 25% and above \(29\) is 75%, so the number of waves she surfed was above \(29\) on more days than it was below \(29\))