Question

josh works as a server at a restaurant. the given data set shows the amount of gratuity, in dollars, that he earned from each table that he served on a given night.

10, 8, 50, 15, 10, 25, 20, 5, 12, 15, 20, 15, 25, 18, 20

which box plot represents the data set?

Explanation:

Order the data set

To find the five-number summary, we first arrange the data set in ascending order.
The given data values are:
\(10, 8, 50, 15, 10, 25, 20, 5, 12, 15, 20, 15, 25, 18, 20\)

Sorting these \(15\) values from least to greatest:
\(5, 8, 10, 10, 12, 15, 15, 15, 18, 20, 20, 20, 25, 25, 50\)

Find the median

Since there are \(n = 15\) data points (an odd number), the median is the middle value at position \(\frac{15 + 1}{2} = 8\).
Looking at our ordered list:
\(5, 8, 10, 10, 12, 15, 15, \mathbf{15}, 18, 20, 20, 20, 25, 25, 50\)

The \(8\)-th value is \(15\).
Thus, \(\text{Median} = 15\).

Find the quartiles

The median splits the data into two halves:

• Lower half (first 7 values): \(5, 8, 10, \mathbf{10}, 12, 15, 15\)
• Upper half (last 7 values): \(18, 20, 20, \mathbf{20}, 25, 25, 50\)

The first quartile (\(Q_1\)) is the median of the lower half:
\(Q_1 = 10\)

The third quartile (\(Q_3\)) is the median of the upper half:
\(Q_3 = 20\)

Identify the extremes

The minimum and maximum values of the data set are:

• \(\text{Minimum} = 5\)
• \(\text{Maximum} = 50\)

Thus, the five-number summary is:

• \(\text{Minimum} = 5\)
• \(Q_1 = 10\)
• \(\text{Median} = 15\)
• \(Q_3 = 20\)
• \(\text{Maximum} = 50\)

Match with the box plots

Let's analyze the given options:

• W: Minimum = 5, \(Q_1 = 10\), Median = 15, \(Q_3 = 20\), Maximum = 50.
• X: Minimum = 5, \(Q_1 = 10\), Median = 15, \(Q_3 = 25\), Maximum = 50.
• Y: Minimum = 5, \(Q_1 = 10\), Median = 15, \(Q_3 = 20\), Maximum = 25 (with 50 plotted as an outlier or incorrect maximum).

Plot W perfectly matches our calculated five-number summary.

Answer:

• (A) W (Correct answer)
• (B) X
• (C) Y