Question

tori is getting ready to run a marathon. she kept track of the lengths of her last few training runs. training runs (mi.) 11 8 15 16 8 20 16 12 6 13 7 which box plot represents the data? training runs (mi.)

Explanation:

Answer:

To determine the correct box - plot, we need to find the five - number summary (minimum, first quartile $Q_1$, median, third quartile $Q_3$, maximum) of the data set $\{6,7,8,8,11,12,13,15,16,16,20\}$.

1. Find the minimum and maximum:

• Step1: Identify the minimum value
• Sort the data set: $6,7,8,8,11,12,13,15,16,16,20$. The minimum value is $6$.
• Step2: Identify the maximum value
• The maximum value is $20$.

1. Find the median ($Q_2$):

• Step3: Calculate the position of the median
• Since there are $n = 11$ data points, the position of the median is $\frac{n + 1}{2}=\frac{11+1}{2}=6$.
• The median is the 6th - ordered value, so the median $Q_2=12$.

1. Find the first quartile ($Q_1$):

• Step4: Consider the lower half of the data
• The lower half of the data set is $\{6,7,8,8,11\}$.
• Since there are $n_1 = 5$ data points in the lower half, the position of $Q_1$ is $\frac{5 + 1}{2}=3$.
• The first quartile $Q_1 = 8$.

1. Find the third quartile ($Q_3$):

• Step5: Consider the upper half of the data
• The upper half of the data set is $\{13,15,16,16,20\}$.
• Since there are $n_2=5$ data points in the upper half, the position of $Q_3$ is $\frac{5 + 1}{2}=3$.
• The third quartile $Q_3 = 16$.

The box - plot should have a minimum value of $6$, $Q_1 = 8$, median $=12$, $Q_3 = 16$, and maximum $=20$. Without seeing all the options, we can say that the box - plot with a left - most whisker at $6$, the left side of the box at $8$, the line inside the box at $12$, the right side of the box at $16$, and the right - most whisker at $20$ is the correct one.

(Note: Since the options are not fully shown, we can't give a definite choice among them, but this is the process to find the correct box - plot.)