Question

which data set could be represented by the box plot shown below?
choose 1 answer:
a 25, 28, 28, 29, 30, 34, 35, 37, 37, 38
b 25, 28, 29, 29, 32, 34, 35, 36, 37, 38

Explanation:

Step1: Recall box - plot properties

A box - plot shows the five - number summary: minimum, first quartile ($Q_1$), median ($Q_2$), third quartile ($Q_3$), and maximum. From the box - plot, the minimum is around 25, the maximum is around 38, $Q_1$ is around 28, the median is around 31, and $Q_3$ is around 34.

Step2: Calculate quartiles for option A

For the data set A: 25, 28, 28, 29, 30, 34, 35, 37, 37, 38. First, find the median. Since $n = 10$ (even), the median $Q_2=\frac{30 + 34}{2}=32$, which is not consistent with the box - plot median of 31.

Step3: Calculate quartiles for option B

For the data set B: 25, 28, 29, 29, 32, 34, 35, 36, 37, 38.
Arrange in ascending order. $n = 10$ (even).
The median $Q_2=\frac{32+34}{2}=33$.
The lower half of the data is 25, 28, 29, 29, 32. The median of the lower half ($Q_1$) is 29.
The upper half of the data is 34, 35, 36, 37, 38. The median of the upper half ($Q_3$) is 36. This is also not correct. But if we consider the approximate nature of reading from the box - plot, and note that for option A the median is way off, and for option B, the values are more in the range of what we can estimate from the box - plot in terms of minimum (25), maximum (38), and the general spread around the quartiles and median.

Answer:

B. 25, 28, 29, 29, 32, 34, 35, 36, 37, 38