Question

the players on matthew’s soccer team each bring a water bottle with them to practice. for a statistics project, matthew asked each player how much water his bottle held. this box plot shows the results. what fraction of the water bottles held at least 24 ounces?

Explanation:

Step1: Understand Box - Plot Quartiles

In a box - plot, the data is divided into four equal parts (quartiles). The total number of data points can be thought of as being split into 4 groups, each representing $\frac{1}{4}$ of the data. The upper quartile (Q3) is the value such that 75% of the data is less than or equal to it, and 25% of the data is greater than or equal to it. In a box - plot, the part of the data that is at least the value of the third quartile (the end of the box or beyond the box towards the maximum) represents the upper 25% of the data. Looking at the box - plot for the water bottle volumes, the value 24 oz is at the third quartile (Q3) or the end of the box. So the fraction of data (water bottles) with volume at least 24 ounces is the fraction of data in the upper quartile. Since a box - plot divides data into 4 equal parts, the fraction of data with value at least Q3 is $\frac{1}{4}$ (or 25% of the data). Wait, actually, let's recall: the box - plot has the minimum, Q1, median (Q2), Q3, and maximum. The data between Q1 and Q3 is the middle 50%, below Q1 is 25%, and above Q3 is 25%. So if we want the fraction of water bottles that held at least 24 ounces (and 24 is Q3), then the fraction is the proportion of data above Q3. Since the data is divided into 4 quartiles, each quartile is $\frac{1}{4}$ of the data. So the fraction of data with volume $\geq24$ (Q3) is $\frac{1}{4}$? Wait, no, wait. Wait, the box - plot: the lower whisker is from min to Q1, the box is from Q1 to Q3, and the upper whisker is from Q3 to max. So the data points are: 25% below Q1, 25% between Q1 and median, 25% between median and Q3, and 25% above Q3. Wait, no, actually, the quartiles divide the data into four equal parts. So Q1 is the 25th percentile, median (Q2) is the 50th percentile, Q3 is the 75th percentile. So the percentage of data $\geq$ Q3 is $100 - 75=25\%$, which is $\frac{1}{4}$. But wait, maybe the number of data points: let's assume the total number of data points is 4 parts. So the fraction of data with volume at least 24 (Q3) is $\frac{1}{4}$? Wait, no, maybe I made a mistake. Wait, the box - plot: the length from Q1 to Q3 is the inter - quartile range (IQR), and the data above Q3 is 25% of the total data. So if we consider the total number of data points as $n$, the number of data points with volume $\geq24$ (Q3) is $\frac{n}{4}$, so the fraction is $\frac{1}{4}$. Wait, but let's think again. Suppose we have 4 data points (for simplicity). Q1 is the first, median the second, Q3 the third, and max the fourth. Then the data above Q3 (the third point) is the fourth point, which is $\frac{1}{4}$ of the data. So in general, the fraction of data with value $\geq$ Q3 is $\frac{1}{4}$. So in this case, since 24 is Q3 (from the box - plot, the end of the box is at 24), the fraction of water bottles that held at least 24 ounces is $\frac{1}{4}$. Wait, but maybe the total number of quartiles: the box - plot has 4 parts, so the upper 25% is above Q3. So the fraction is $\frac{1}{4}$.

Step2: Confirm the Fraction

Since the box - plot divides the data into four equal - sized groups (quartiles), the group of data with values at least the third quartile (Q3, which is 24 oz here) makes up $\frac{1}{4}$ of the total data. So the fraction of water bottles that held at least 24 ounces is $\frac{1}{4}$.

Answer:

$\frac{1}{4}$