Question

(a) what is the difference of the largest number and the smallest number of people served by the tours? (that is, what is the range of the data?)
(b) which of these best describes the numbers of people served by the tours?
the number of people served was below 41 for more tours than it was above 41.
the number of people served was above 41 for more tours than it was below 41.
the number of people served was below 41 for the same number of tours that it was above 41.
(c) which of the following intervals gives the lowest 25% of the data?
20 to 30
30 to 32
30 to 35
32 to 35

Explanation:

Step1: Recall the formula for range

The range of a data - set is calculated as \(Range = Maximum - Minimum\).

Step2: Identify maximum and minimum from the box - and - whisker plot

From the box - and - whisker plot, the minimum value (the left - most end of the whisker) is \(30\) and the maximum value (the right - most end of the whisker) is \(55\).

Step3: Calculate the range

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But wait, there is a mistake. In a box - and - whisker plot, the range is indeed \(Max - Min\). Looking at the vertical axis (Number of people served):

• The minimum value (the lower extreme) is \(30\)
• The maximum value (the upper extreme) is \(55\)

$$Range=55 - 30=25$$

But if we assume that there was a mis - reading (maybe the scale is mis - interpreted). Wait, no, re - checking:
In a box - and - whisker plot:

• The left - most point (minimum) \(=30\)
• The right - most point (maximum) \(=55\)

$$Range = 55-30=25$$

But if we consider the problem (a) asks for the difference of the largest and the smallest number of people served (i.e., range).

For part (b):

• The median of the box - and - whisker plot is the line inside the box. Let's assume we mis - focused on (a). In a box - and - whisker plot, if we want to check the statements:

The median divides the data into two halves. The lower half is from the minimum (\(30\)) to the median and the upper half is from the median to the maximum (\(55\)).
If we consider the value \(41\):

• Let's assume we check the position. The inter - quartile range (\(IQR = Q3 - Q1\)). But for the statement:

The number of people served was below \(41\) for more tours than it was above \(41\). If \(41\) is less than the median (the line inside the box). If the median (let's assume from the box - and - whisker plot, if the median is \(45\) (by visual inspection of the box - position on the vertical axis). So the number of data points below the median is \(50\%\) and above is \(50\%\). But if \(41<\) median (\(45\)), the number of tours with people served below \(41\) is less than the number of tours with people served above \(41\).

For part (c):

• The lower \(25\%\) of the data (the first quartile \(Q1\)) to the minimum. In a box - and - whisker plot, the first quartile (\(Q1\)) is \(32\) (the left - end of the box). The lower \(25\%\) of the data is from \(30\) (minimum) to \(32\) ( \(Q1\)). But if we consider intervals:

The lower \(25\%\) of the data (the first quartile range) is from \(30\) to \(32\) (not in the options). Wait, no, re - checking:
In a box - and - whisker plot:

• The first quartile (\(Q1\)) is \(32\) (left end of the box)
• The second quartile (\(Q2\)) (median) is \(45\) (line inside the box)
• The third quartile (\(Q3\)) is \(50\) (right end of the box)
• The minimum is \(30\), the maximum is \(55\)

The lower \(25\%\) of the data is from \(30\) (minimum) to \(32\) (\(Q1\)). But if we consider the intervals in the options:
The inter - quartile range (\(IQR=Q3 - Q1=50 - 32 = 18\)). The lower \(25\%\) of the data (from minimum to \(Q1\)):
If we check the intervals:

• \(20 - 30\): not relevant as minimum is \(30\)
• \(30 - 32\): from minimum (\(30\)) to \(Q1\) (\(32\)) which is \(25\%\) of the data
• \(30 - 35\): larger interval
• \(32 - 35\): from \(Q1\) (\(32\)) to above \(Q1\)

Answer:

(a) The range is \(55−30 = 25\). But if we assume the problem (a) is just asking for the calculation of \(Max - Min\) from the box - and - whisker plot (where \(Max = 55\) and \(Min=30\)).
(b) None of the statements (if we assume wrong initial analysis, but if we re - check: The number of people served was below \(41\) for fewer tours than it was above \(41\) (since \(41<\) median (\(45\))).
(c) \(30\) to \(32\) (since the lower \(25\%\) of the data is from the minimum (\(30\)) to the first quartile (\(Q1 = 32\))). So for part (c) the answer is \(30\) to \(32\) (first option in part (c)'s choices).