Question

use the box - and - whisker plot to answer the questions below.
(a) what is the difference of the longest distance and the shortest distance training ride? (that is, what is the range of the data?)
(b) which of these best describes the distances (in kilometers) of yolandas training rides?
there were more rides with a distance under 30 than with a distance over 30.
there were more rides with a distance over 30 than with a distance under 30.
the same number of rides had a distance under 30 as had a distance over 30.
(c) which of the following intervals gives the lowest 25% of the data?
10 to 21
21 to 23
21 to 26
23 to 26

Explanation:

Step1: Find the range for part (a)

The range of a data set in a box - and - whisker plot is calculated as \( \text{Range}=\text{Maximum}-\text{Minimum} \). From the box - and - whisker plot, the maximum value is \( 40 \) and the minimum value is \( 21 \). So, \( \text{Range}=40 - 21=19 \). But wait, no, looking at the scale, if we assume the correct maximum and minimum (by proper box - plot reading), the maximum is \( 40 \) and the minimum is \( 21 \).

Step2: Analyze part (b)

In a box - and - whisker plot, the median divides the data into two halves. The lower half (25% of the data from the minimum to the first quartile \(Q_1\)) and the upper half (25% of the data from the third quartile \(Q_3\) to the maximum). The inter - quartile range (\(IQR = Q_3 - Q_1\)) represents the middle 50% of the data. To find where 25% of the data lies:
The first quartile (\(Q_1\)) is \( 23 \) and the second quartile (median, \(Q_2\)) is \( 26 \), and the third quartile (\(Q_3\)) is \( 30 \). The lower 25% of the data is from the minimum (\(21\)) to \(Q_1 = 23\). But wait, no, the formula for the range of the lower 25% (if we consider the box - plot structure where the minimum is \(21\) and \(Q_1=23\)). The interval for the lowest 25% of the data is from \(21\) to \(23\).

Step3: Analyze part (c)

Count the number of data points on either side of \(30\) (the third quartile \(Q_3\)). The number of data points with distance under \(30\) (from minimum (\(21\)) to \(Q_3 = 30\)) is more than the number of data points with distance over \(30\) (from \(Q_3 = 30\) to maximum (\(40\))).

Answer:

(a) The range (difference between the longest and shortest distance) is \(40−21 = 19\). But if we assume the correct box - plot reading (assuming the scale is correct as per the problem's box - plot), the range is \(40 - 21=19\).
(b) The interval for the lowest 25% of the data is \(21\) to \(23\).
(c) There were more rides with a distance under \(30\) than with a distance over \(30\).