Question

students at javi’s school perform community service hours by volunteering at nursing homes, hospitals, and other places in their community. the box plot shows the number of hours each student volunteered. at least what percent of the students volunteered between 15 and 30 hours? box plot image options: 50%, 25%, 75%, 100%

Explanation:

Step1: Recall Box Plot Percentiles

A box plot has quartiles: the first quartile (\(Q_1\)) is the 25th percentile, the median (\(Q_2\)) is the 50th percentile, and the third quartile (\(Q_3\)) is the 75th percentile. The interquartile range (IQR) is \(Q_3 - Q_1\), and data within \(Q_1\) to \(Q_3\) represents about 50%? Wait, no—wait, the box in a box plot spans from \(Q_1\) (25th percentile) to \(Q_3\) (75th percentile), so the middle 50%? Wait, no, actually, the box contains the middle 50%? Wait, no: \(Q_1\) is the 25th percentile (25% of data below), \(Q_3\) is 75th percentile (25% above). So the data between \(Q_1\) and \(Q_3\) is \(75\% - 25\% = 50\%\)? Wait, no, wait. Wait, the box plot: the left end of the box is \(Q_1\) (25th percentile), right end is \(Q_3\) (75th percentile). So the data between \(Q_1\) and \(Q_3\) is \(75 - 25 = 50\%\)? Wait, no, that's the IQR, which is the middle 50%? Wait, no, actually, the percentage of data between \(Q_1\) and \(Q_3\) is 50%? Wait, no, let's think again. The first quartile (\(Q_1\)) is the value where 25% of the data is below it, and 75% is above. The third quartile (\(Q_3\)) is where 75% is below it, and 25% is above. So the data between \(Q_1\) and \(Q_3\) is \(75\% - 25\% = 50\%\)? Wait, no, that can't be. Wait, no: if \(Q_1\) is 25th percentile, then 25% of data is ≤ \(Q_1\), and 75% is ≥ \(Q_1\). \(Q_3\) is 75th percentile, so 75% of data is ≤ \(Q_3\), 25% is ≥ \(Q_3\). So the data between \(Q_1\) and \(Q_3\) is \(75\% - 25\% = 50\%\)? Wait, no, that's the interquartile range, which is the middle 50%? Wait, no, actually, the percentage of data between \(Q_1\) and \(Q_3\) is 50%? Wait, no, let's take an example. Suppose we have 100 data points. \(Q_1\) is the 25th data point (value), \(Q_3\) is the 75th data point (value). So the number of data points between \(Q_1\) and \(Q_3\) is \(75 - 25 = 50\) (since from 26th to 75th, which is 50 points), so 50% of the data is between \(Q_1\) and \(Q_3\). Wait, but in the box plot here, looking at the axis: the first box starts at 15 (maybe \(Q_1\)) and ends at 25? Wait, no, the box plot has two boxes? Wait, no, the image shows a box plot with two adjacent boxes? Wait, maybe it's a modified box plot, but typically a box plot has one box. Wait, maybe the left box is from 15 to 25 (Q1 to Q2, median), and the right box is from 25 to 30 (Q2 to Q3). Wait, so Q1 is 15, Q3 is 30? Wait, no, let's check the axis: the numbers are 5,10,15,20,25,30,35,40,45. The first box (left) starts at 15, ends at 25? Then the second box starts at 25, ends at 30? So Q1 is 15 (25th percentile), Q2 (median) is 25, Q3 is 30 (75th percentile). Wait, so then the data between Q1 (15) and Q3 (30) is from 25th percentile to 75th percentile, so the percentage is \(75\% - 25\% = 50\%\)? Wait, but the options include 50%? Wait, the options are 50%, 25%, 75%, 100%. Wait, maybe I misread the box plot. Wait, maybe the left end of the first box is Q1=15, and the right end of the second box is Q3=30. So the data between 15 and 30 is from Q1 to Q3, which is 75% - 25% = 50%? Wait, no, wait: Q1 is 25th percentile (25% below), Q3 is 75th percentile (25% above). So the data between Q1 and Q3 is 75 - 25 = 50%? Wait, but maybe the box plot here has Q1 at 15 and Q3 at 30, so the percentage of data between 15 and 30 is 50%? Wait, but let's confirm. The key is that in a box plot, the box represents the middle 50% of the data (from Q1 to Q3), which is 50% of the data. Wait, no, Q1 is 25th percentile, Q3 is 75th percentile, so the data between Q1 and Q3 is 75 - 25 = 50% of the data. So if 15 is Q1 and 30 is Q3, then the percentage of students who volunteered between 15 and 30 hours is at least 50%? Wait, but the options have 50% as an option. Wait, maybe the box plot is split into two boxes, but the first quartile is 15 and third quartile is 30. So the answer is 50%? Wait, but let's check again. Wait, the problem says "at least what percent", so using the interquartile range (IQR), which is Q3 - Q1, and the percentage of data within IQR is at least 50%? Wait, no, actually, the IQR contains 50% of the data (from 25th to 75th percentile). So if 15 is Q1 and 30 is Q3, then the data between 15 and 30 is 50% of the data. So the answer should be 50%.

Step2: Confirm with Box Plot Percentiles

In a box plot, the first quartile (\(Q_1\)) marks the 25th percentile (25% of data ≤ \(Q_1\)), and the third quartile (\(Q_3\)) marks the 75th percentile (75% of data ≤ \(Q_3\)). The data between \(Q_1\) and \(Q_3\) is \(75\% - 25\% = 50\%\). From the box plot, \(Q_1 = 15\) (left end of the box) and \(Q_3 = 30\) (right end of the box). Thus, the percentage of students volunteering between 15 and 30 hours is at least \(50\%\).

Answer:

50%