two students compared the scores on their math tests. their results on 9 tests are shown in the box plots below. compare the box plots. which statements are true? check all that apply. nadia ben 76 80 84 88 92 96 100 □ ben has the smaller range of test scores. □ the lower quartile and upper quartile of nadia’s data are both higher than the lower quartile and upper quartile of ben’s data. □ the median of nadia’s data is equal to the median of ben’s data. □ nadia had the highest score on a test. □ ben’s lowest score was equal to nadia’s lowest score.

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# Brief Explanations:
1. **Range**: Range = Maximum - Minimum. Ben's scores range from 80 to 98 (approx), Nadia's from 82 to 100 (approx). Ben's range: $98 - 80 = 18$, Nadia's: $100 - 82 = 18$? Wait, no—wait, looking at the box plot: Nadia's whiskers: left at ~82, right at 100. Ben's left at 80, right at ~98. Wait, no, maybe I misread. Wait, the x - axis: 76, 80, 84, 88, 92, 96, 100. Nadia's left whisker starts at, say, 82 (between 80 and 84), right at 100. Ben's left at 80, right at 98 (between 96 and 100). So Ben's range: $98 - 80 = 18$, Nadia's: $100 - 82 = 18$? No, maybe the right whisker for Nadia is at 100, Ben's at 98. Wait, the first statement: "Ben has the smaller range"—no, maybe my initial thought is wrong. Wait, let's check each statement:

- **Statement 1: Ben has the smaller range of test scores.**
  Range = Max - Min. From the box plot:
  - Ben’s min: 80, max: ~98 (since his right whisker ends before 100).
  - Nadia’s min: ~82 (her left whisker starts after 80), max: 100.
  Ben’s range: $98 - 80 = 18$; Nadia’s range: $100 - 82 = 18$? Wait, no, maybe the right whisker for Ben is at 98, Nadia at 100. Left whisker: Ben at 80, Nadia at 82. So Ben’s range: $98 - 80 = 18$, Nadia’s: $100 - 82 = 18$? No, maybe I’m misinterpreting the plot. Wait, maybe the right whisker for Nadia is at 100, Ben at 98. So Ben’s range is $98 - 80 = 18$, Nadia’s is $100 - 82 = 18$? No, that can’t be. Wait, maybe the left whisker for Nadia is at 80? No, the plot shows Nadia’s box is to the right of Ben’s. Wait, maybe I made a mistake. Let's move to other statements.

- **Statement 2: The lower quartile (Q1) and upper quartile (Q3) of Nadia’s data are both higher than Q1 and Q3 of Ben’s data.**
  The box in a box - plot represents Q1 (lower quartile, start of box) to Q3 (upper quartile, end of box). Nadia’s box is to the right of Ben’s box. So Nadia’s Q1 and Q3 are higher than Ben’s. So this statement is true.

- **Statement 3: The median of Nadia’s data is equal to the median of Ben’s data.**
  The median is the line inside the box. Nadia’s median (line in her box) and Ben’s median (line in his box) seem to be at the same position (around 90? Wait, no, looking at the plot: Nadia’s box has a median line, Ben’s box has a median line. Wait, the plot shows Nadia’s box and Ben’s box: the median lines (the vertical lines inside the boxes) are at the same x - value? Wait, no, maybe. Wait, the boxes: Nadia’s box is from, say, 86 to 94, median at 90. Ben’s box from 88 to 94, median at 90? Wait, maybe. But let's check the next statement.

- **Statement 4: Nadia had the highest score on a test.**
  Nadia’s right whisker ends at 100, Ben’s at ~98. So Nadia’s max is 100, which is higher than Ben’s max (~98). So this is true.

- **Statement 5: Ben’s lowest score was equal to Nadia’s lowest score.**
  Ben’s min: 80, Nadia’s min: ~82 (her left whisker starts after 80). So they are not equal. So this is false.

Wait, let's re - evaluate:

1. **Ben has the smaller range**:
   Ben’s range: Max - Min = 98 - 80 = 18. Nadia’s range: 100 - 82 = 18. Wait, no, maybe the right whisker for Ben is at 98, Nadia at 100. Left whisker: Ben at 80, Nadia at 82. So ranges are equal? But the statement says "Ben has the smaller range"—so this is false.

2. **Lower quartile (Q1) and upper quartile (Q3) of Nadia’s data are both higher than Ben’s.**
   The box in a box - plot: Q1 is the start of the box, Q3 is the end. Nadia’s box is to the right of Ben’s box. So Nadia’s Q1 (start of her box) is higher than Ben’s Q1 (start of his box), and Nadia’s Q3 (end of her box) is higher than Ben’s Q3 (end of his box). So this is true.

3. **Median of Nadia’s data = median of Ben’s data.**
   The median is the line inside the box. From the plot, the median lines (the vertical lines in the boxes) seem to be at the same position (e.g., both at 90). So this is true? Wait, maybe. But let's check the boxes: Nadia’s box and Ben’s box—if the median lines are aligned, then yes.

4. **Nadia had the highest score**:
   Nadia’s right whisker is at 100, Ben’s at ~98. So Nadia’s max is 100, highest. True.

5. **Ben’s lowest = Nadia’s lowest**:
   Ben’s min: 80, Nadia’s min: 82 (or higher). False.

Wait, maybe my initial analysis of the range was wrong. Let's check the box - plot again. The x - axis: 76, 80, 84, 88, 92, 96, 100. Nadia’s left whisker: starts at 80? No, the plot shows Nadia’s left whisker is to the right of Ben’s. Ben’s left whisker is at 80, Nadia’s at 82 (between 80 and 84). So Ben’s min is 80, Nadia’s min is 82. Ben’s max: 98 (between 96 and 100), Nadia’s max: 100. So Ben’s range: 98 - 80 = 18. Nadia’s range: 100 - 82 = 18. So range is same. So statement 1 is false.

Statement 2: Q1 and Q3 of Nadia are higher than Ben’s. Nadia’s box is to the right, so Q1 (start of box) and Q3 (end of box) are higher. True.

Statement 3: Median (line in box) of Nadia and Ben—if the lines are at the same x - value, then yes. Let's say the median for both is 90. So true.

Statement 4: Nadia’s max is 100, Ben’s is 98. So Nadia has the highest score. True.

Statement 5: Ben’s min (80) vs Nadia’s min (82). Not equal. False.

Wait, but maybe the correct statements are:

- The lower quartile and upper quartile of Nadia’s data are both higher than the lower quartile and upper quartile of Ben’s data. (True, since her box is to the right)
- The median of Nadia’s data is equal to the median of Ben’s data. (If the median lines are at the same position)
- Nadia had the highest score on a test. (True, since her max is 100)

Wait, let's check with the box - plot structure:

- **Box - plot components**:
  - Min (left whisker start), Q1 (box start), Median (box line), Q3 (box end), Max (right whisker end).

From the plot:

- **Nadia’s box**: Starts (Q1) at a higher x - value than Ben’s box start (Q1). Ends (Q3) at a higher x - value than Ben’s box end (Q3). So Q1 and Q3 of Nadia > Ben’s Q1 and Q3. So statement 2 is true.

- **Median**: The vertical line inside Nadia’s box and Ben’s box—if they are at the same x - coordinate, then median is equal. So statement 3 is true.

- **Highest score**: Nadia’s right whisker ends at 100, Ben’s at <100. So Nadia has the highest score. Statement 4 is true.

Wait, maybe I made a mistake with statement 1. Let's confirm:

- **Statement 1: Ben has the smaller range.**
  Ben’s range: Max - Min = 98 - 80 = 18. Nadia’s range: 100 - 82 = 18. So range is equal. So statement 1 is false.

So the true statements are:

- The lower quartile and upper quartile of Nadia’s data are both higher than the lower quartile and upper quartile of Ben’s data.
- The median of Nadia’s data is equal to the median of Ben’s data.
- Nadia had the highest score on a test.

But let's check again:

- **Nadia’s Q1 and Q3 higher than Ben’s**: Yes, because her box is shifted right.
- **Median equal**: If the median lines (inside the boxes) are at the same x - value, then yes.
- **Nadia’s highest score**: Her max is 100, Ben’s is less than 100. Yes.

So the correct options are:

- The lower quartile and upper quartile of Nadia’s data are both higher than the lower quartile and upper quartile of Ben’s data.
- The median of Nadia’s data is equal to the median of Ben’s data.
- Nadia had the highest score on a test.

(Note: Depending on the exact plot, but based on the x - axis and box positions, these are the true statements.)
# Answer:
- The lower quartile and upper quartile of Nadia’s data are both higher than the lower quartile and upper quartile of Ben’s data.
- The median of Nadia’s data is equal to the median of Ben’s data.
- Nadia had the highest score on a test.

(If we list the options with checkboxes as per the question, the correct ones are:

- The lower quartile and upper quartile of Nadia’s data are both higher than the lower quartile and upper quartile of Ben’s data.
- The median of Nadia’s data is equal to the median of Ben’s data.
- Nadia had the highest score on a test.  )

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