Question

which data set could be represented by the box plot shown below?
choose 1 answer:
a 15, 16, 16, 20, 21, 21, 23, 25, 26, 26, 28, 28
b 16, 16, 16, 16, 21, 23, 23, 25, 26, 26, 28, 28
c 16, 16, 16, 20, 21, 21, 23, 25, 26, 26, 28, 28
d 16, 16, 16, 20, 21, 23, 23, 25, 26, 28, 28, 28

Explanation:

Step1: Find the minimum value

The minimum value of the box - plot is around 16 (since the left - most point is near 16). Option A has a minimum value of 15, so Option A is incorrect.

Step2: Check the data set size

A box - plot is based on 5 - number summary (minimum, first quartile \(Q_1\), median \(Q_2\), third quartile \(Q_3\), maximum). For a data set of size \(n\) (where \(n\) is even), the median is the average of the \(\frac{n}{2}\)th and \((\frac{n}{2}+ 1)\)th ordered values. For a data set of size \(n = 12\) (all options have \(n = 12\) or \(n = 11\)). Option D has \(n=11\) (count the numbers: \(1 + 3+1+1+2+3=11\)), so Option D is incorrect.

Step3: Calculate the first quartile \(Q_1\)

For a data set \(x_1,x_2,\cdots,x_{12}\) in ascending order, \(Q_1\) is the median of the first 6 values.
For Option B: The data set is \(16,16,16,16,21,23\). The median of these 6 values (since \(n = 6\) (even)) is \(\frac{16 + 16}{2}=16\).
For Option C: The data set is \(16,16,16,20,21,21\). The median of these 6 values (since \(n = 6\) (even)) is \(\frac{16+20}{2}=18\).
Looking at the box - plot, the left - hand side of the box (which represents \(Q_1\)) is at 18.

Answer:

C. \(16,16,16,20,21,21,23,25,26,26,28,28\)