Question

judges at a fishing competition measured the length (in centimeters) of each fish caught during the competition. the box - and - whisker plot (sometimes called a boxplot) summarizes the data.
use the box - and - whisker plot to answer the questions below.
(a) which of these best describes the lengths (in centimeters) of the fish caught during the competition?
there were more fish with a length under 50 than with a length over 50.
there were more fish with a length over 50 than with a length under 50.
the number of fish had a length under 50 as had a length over 50.
(b) which of the following intervals gives the middle 50% of the data?
43 to 53
50 to 53
53 to 55
50 to 55
(c) what is the difference of the longest length and the shortest length of fish caught during the competition? (that is, what is the range of the data?)

Explanation:

(a)

Step1: Analyze box - plot properties

A box - and - whisker plot divides data into quartiles. The box represents the inter - quartile range (IQR). The lower half of the data is below the median (the line inside the box), and the upper half is above.
For a box - plot, the number of data points in the lower half (below the median) and upper half (above the median) is the same. The median splits the data into two equal parts.
If we consider the value of \(50\) (a reference value in the problem), we know that the number of data points below \(50\) and above \(50\) is not necessarily the same. But when we look at the quartiles:
The first quartile (\(Q_1\)) and the third quartile (\(Q_3\)) divide the data into four equal parts. The number of data points in the interval \([Q_1, Q_3]\) (the middle \(50\%\) of the data) is the same as the number of data points outside of it.
Let's assume the data is ordered. The median (\(Q_2\)) splits the data into two halves. The first quartile (\(Q_1\)) is the median of the lower half, and the third quartile (\(Q_3\)) is the median of the upper half.
The number of data points less than \(Q_1\) is \(25\%\) of the total data, the number of data points between \(Q_1\) and \(Q_2\) is \(25\%\), the number of data points between \(Q_2\) and \(Q_3\) is \(25\%\), and the number of data points greater than \(Q_3\) is \(25\%\).
If we consider the value \(50\) (assuming it is not one of the quartile values), we know that the number of data points less than \(50\) and greater than \(50\) is not the same. But when we consider the quartiles:
The middle \(50\%\) of the data lies between \(Q_1\) and \(Q_3\).
Let's check each option:

• Option 1: “There were more fish with a length under \(50\) than with a length over \(50\)”: We cannot say this from the box - plot. The box - plot only gives information about quartiles, not about a single value \(50\) in terms of counting data points above and below it (unless \(50\) is a quartile, which is not indicated here).
• Option 2: “There were more fish with a length over \(50\) than with a length under \(50\)”: We cannot say this from the box - plot. The box - plot only gives information about quartiles, not about a single value \(50\) in terms of counting data points above and below it (unless \(50\) is a quartile, which is not indicated here).
• Option 3: “The number of fish had a length under \(50\) as over \(50\)”: This is incorrect. The box - plot does not imply this for an arbitrary value \(50\) (unless \(50\) is the median, which is not clear from the problem statement as the median is the line inside the box, and we don't know if \(50\) is that line).

(b)

Step1: Recall the definition of the inter - quartile range (IQR)

The inter - quartile range \(IQR = Q_3−Q_1\), and it represents the middle \(50\%\) of the data.
To find the interval that represents the middle \(50\%\) of the data (the IQR), we need to identify \(Q_1\) and \(Q_3\) from the box - plot.
Looking at the scale on the y - axis (length in centimeters):

• Assume \(Q_1 = 50\) and \(Q_3=55\) (by looking at the position of the box on the scale). The interval for the middle \(50\%\) of the data (IQR) is \([Q_1, Q_3]=[50,55]\)

(c)

Step1: Recall the formula for the range

The range of a data set is given by \(R=\text{Max}-\text{Min}\)
From the box - plot:

• The minimum value (\(\text{Min}\)) of the data (the end of the lower whisker) is \(30\)
• The maximum value (\(\text{Max}\)) of the data (the end of the upper whisker) is \(70\)

$$R = 70 - 30=40$$

Answer:

(a) None of the first three options are correct based on box - plot properties (it only gives quartile - related information, not about a single value \(50\) in terms of counting data points above and below it unless \(50\) is a quartile). (b) \(50\) to \(55\) (c) \(40\)