Question

in the box plots shown. plot a (box plot with axis 4–16) plot b (box plot with axis 4–14) which statements comparing the two box plots are correct? choose all the correct statements. a. the mean of plot b is greater than the mean of plot a. b. the range of plot b is greater than the range of plot a. c. the median of plot b is the same as the median of plot a. d. the interquartile range of plot b is the same as the interquartile range of plot a.

Explanation:

Step1: Analyze Range

For Plot A: Min = 4, Max = 10. Range = \( 10 - 4 = 6 \).
For Plot B: Min = 8, Max = 14. Range = \( 14 - 8 = 6 \). So B is incorrect.

Step2: Analyze Median

Box plot’s median is the middle line of the box. Both plots have boxes with median at the same relative position (since boxes look symmetric in structure, median value: Plot A’s median is around 7 - 8? Wait, no—wait, the box is split into two equal parts. Wait, actually, in box plots, the median is the line inside the box. Looking at the scales: Plot A’s box spans from 5 to 9? Wait, no, the plot A: the whiskers start at 4, box from 5 to 9? Wait, no, the x - axis for Plot A: 4,5,6,7,8,9,10,... The box has two parts, so median is the middle. Wait, maybe better: Plot A: min = 4, Q1, median, Q3, max = 10? Wait, no, the right whisker ends at 10? Wait, the plot A’s right whisker is at 10? Wait, the diagram: Plot A has left whisker at 4, box from 5 to 9 (split into two parts, so median at 7? Wait, no, the box is two rectangles? Wait, maybe the box is split into two, so median is the middle. Wait, Plot B: left whisker at 8, box from 9 to 13? Wait, no, the x - axis for Plot B: 4,5,6,7,8,9,10,11,12,13,14. The box starts at 9, ends at 13, split into two, so median at 11? Wait, no, maybe I misread. Wait, the key is: in box plots, the median is the line inside the box. If both boxes are split into two equal - length parts (visually), then the median’s position relative to the data. Wait, actually, the range of Plot A: max - min = 10 - 4 = 6. Plot B: 14 - 8 = 6. So range is same. So B is wrong.

Now, median: For Plot A, the data is from 4 to 10. The box is in the middle, so median is around 7? Wait, no, maybe the box is from 5 to 9, so median at 7. For Plot B, box from 9 to 13, median at 11? Wait, that can't be. Wait, no, maybe the x - axis labels: Plot A: 4,5,6,7,8,9,10,11,12,13,14,15,16. Plot B: 4,5,6,7,8,9,10,11,12,13,14. Wait, Plot A’s max is 10? No, the right whisker is at 10? Wait, the diagram shows Plot A’s right whisker at 10? And Plot B’s right whisker at 14. Wait, Plot A: min = 4, max = 10. Plot B: min = 8, max = 14. So range for A: 10 - 4 = 6. Range for B: 14 - 8 = 6. So range is same. So B is incorrect.

Median: The box in Plot A: let's see, the box is divided into two parts, so median is the middle. Similarly for Plot B. Wait, maybe the median of Plot A: let's assume the box is from 5 to 9, so median at 7. Plot B’s box from 9 to 13, median at 11? No, that can't be. Wait, maybe I made a mistake. Wait, the problem says "the median of plot B is the same as the median of plot A". Wait, no, maybe the boxes are of the same width (interquartile range) and the median position relative to the box is same, but the actual median values? Wait, no, let's check the interquartile range (IQR). IQR is Q3 - Q1. For Plot A: Q1 and Q3: let's say Q1 = 5, Q3 = 9, so IQR = 4. For Plot B: Q1 = 9, Q3 = 13, so IQR = 4. So IQR is same (D is correct).

Mean: Box plots don't show mean, but since the distributions are symmetric (boxes split into two), the mean should be equal to the median. But Plot B’s data is shifted right? Wait, Plot A’s data is from 4 - 10, Plot B’s from 8 - 14. So Plot B’s median (and mean, if symmetric) is greater than Plot A’s? Wait, no, wait Plot A’s median: if the box is from 5 to 9, median at 7. Plot B’s box from 9 to 13, median at 11. So mean of B is greater than A (A is correct)? Wait, but earlier range: Plot A: 10 - 4 = 6, Plot B:14 - 8 = 6, so range same (B is wrong). Median: Plot B’s median is 11, Plot A’s is 7, so C is wrong. IQR: Plot A: Q3 - Q1 = 9 - 5 = 4, Plot B:13 - 9 = 4, so D is correct. Wait, now I'm confused. Let's re - examine:

Wait, the box plot structure: each box has two parts (maybe representing lower and upper quartiles, with median in the middle). So for Plot A:

- Min: 4 (left whisker start)
- Q1: 5 (start of box)
- Median: 7 (middle of box)
- Q3: 9 (end of box)
- Max: 10 (right whisker end)

For Plot B:

- Min: 8 (left whisker start)
- Q1: 9 (start of box)
- Median: 11 (middle of box)
- Q3: 13 (end of box)
- Max: 14 (right whisker end)

Now:

- Range (Max - Min): Plot A: 10 - 4 = 6; Plot B:14 - 8 = 6. So B is incorrect.
- Mean: Since the distribution is symmetric (box split into two equal parts, whiskers same length? Wait, left whisker of A: 5 - 4 = 1, right whisker:10 - 9 = 1. Left whisker of B:9 - 8 = 1, right whisker:14 - 13 = 1. So symmetric, so mean = median. Plot A’s median:7, Plot B’s median:11. So mean of B > mean of A (A is correct).
- Median: 7 vs 11, so C is incorrect.
- IQR (Q3 - Q1): Plot A:9 - 5 = 4; Plot B:13 - 9 = 4. So D is correct.

Wait, but the options: A, D? Wait, let's check again.

Wait, the problem's box plots: maybe the boxes are of the same width (IQR) and the median is in the same position within the box, but the overall data is shifted. So:

- A: Mean of B > Mean of A: Correct, because B’s data is shifted right (min, max, Q1, Q3, median all higher by 4).
- B: Range of B > Range of A: Range is 6 for both, so incorrect.
- C: Median same: No, B’s median is higher, incorrect.
- D: IQR same: 4 for both, correct.

So correct options are A and D? Wait, but let's check the original diagram again. Wait, maybe I misread the max of Plot A. If Plot A’s max is 10? And Plot B’s max is 14. Plot A’s min is 4, Plot B’s min is 8. So range is 6 for both. IQR: if the box in A is from 5 to 9 (length 4), box in B from 9 to 13 (length 4), so IQR same (D correct). Median: A’s median is 7, B’s is 11, so not same (C incorrect). Mean: since symmetric, mean = median, so B’s mean (11) > A’s mean (7) (A correct).

Answer:

A. The mean of plot B is greater than the mean of plot A, D. The interquartile range of plot B is the same as the interquartile range of plot A