Question

jen created a box plot based on this data set: 4 12 13 17 19 19 20 24 25 28 31 35 35 39 then, jen learned that the 4 was supposed to be a 6, so she updated her box plot. what did she have to change? select all that apply. the maximum the upper quartile the minimum the median the lower quartile

Explanation:

Step1: Analyze the original data set

The original data set is: \(4, 12, 13, 17, 19, 19, 20, 24, 25, 28, 31, 35, 35, 39, 42\) (15 data points). The minimum value is \(4\).

Step2: Analyze the updated data set

After changing \(4\) to \(6\), the new data set is: \(6, 12, 13, 17, 19, 19, 20, 24, 25, 28, 31, 35, 35, 39, 42\).

Step3: Check each measure

- **Maximum**: The maximum value was \(42\) and remains \(42\), so no change.
- **Upper Quartile**: The upper half of the data (after median) is for original: \(24, 25, 28, 31, 35, 35, 39, 42\) (wait, no, 15 data points, median is at 8th term (\(24\))? Wait, 15 data points: positions 1 - 15. Median is position \(\frac{15 + 1}{2}=8\), so 8th term is \(24\). Lower half: positions 1 - 7: \(4, 12, 13, 17, 19, 19, 20\) (7 data points), lower quartile (Q1) is the median of lower half, position \(\frac{7 + 1}{2}=4\), so 4th term \(17\). Upper half: positions 9 - 15: \(25, 28, 31, 35, 35, 39, 42\) (7 data points), upper quartile (Q3) is median of upper half, position \(\frac{7 + 1}{2}=4\), so 4th term \(35\).

After changing \(4\) to \(6\), lower half: \(6, 12, 13, 17, 19, 19, 20\) (7 data points), Q1 is still 4th term \(17\). Upper half remains same, so Q3 same. Median: 8th term is \(24\) in both, so median same. Minimum: original minimum \(4\), new minimum \(6\), so minimum changes. Wait, but also, the lower quartile: wait, no, the lower half is positions 1 - 7 (for 15 data points). Original lower half: \(4,12,13,17,19,19,20\) (Q1 is 4th term \(17\)). New lower half: \(6,12,13,17,19,19,20\) (Q1 is 4th term \(17\)), so lower quartile same? Wait, no, maybe I made a mistake. Wait, 15 data points: median at 8th term. Lower half: first 7 terms (positions 1 - 7), upper half: last 7 terms (positions 9 - 15). So original lower half: \(4,12,13,17,19,19,20\) (Q1 is median of these 7, which is 4th term: \(17\)). New lower half: \(6,12,13,17,19,19,20\) (Q1 is 4th term: \(17\)), so lower quartile same. Wait, but the minimum changes from \(4\) to \(6\). What about median? Median is 8th term, which is \(24\) in both, so median same. Upper quartile: upper half is positions 9 - 15: \(25,28,31,35,35,39,42\) (median is 4th term \(35\)) in both, so upper quartile same. Maximum same. Wait, but the minimum changes. Also, wait, the original minimum was \(4\), new minimum is \(6\), so the minimum changes. But also, does the lower quartile change? Wait, no, because the lower half's median (Q1) is based on the 4th term of the lower half. In original lower half, the data is \(4,12,13,17,19,19,20\). The 4th term is \(17\). In new lower half, data is \(6,12,13,17,19,19,20\). The 4th term is still \(17\), so Q1 doesn't change. Wait, but the minimum is the smallest value, so changing the smallest value from \(4\) to \(6\) will change the minimum. What about the median? The median is the middle value (8th term), which is \(24\) in both, so median doesn't change. Wait, maybe I messed up the data count. Wait, original data: let's count again. The original data: \(4,12,13,17,19,19,20,24,25,28,31,35,35,39,42\) – that's 15 numbers? Wait, \(4,12,13,17,19,19,20\) (7), \(24\) (median), \(25,28,31,35,35,39,42\) (7). Yes. So after changing \(4\) to \(6\), the lower half is \(6,12,13,17,19,19,20\) (7), median (Q1) is 4th term \(17\), same as before. So the measures that change: minimum (since the smallest value changes from \(4\) to \(6\)). Wait, but what about the options? The options are the maximum, upper quartile, minimum, median, lower quartile. Wait, maybe I made a mistake. Wait, let's list the data points again. Original data: 15 points. Positions:

1: 4

2:12

3:13

4:17

5:19

6:19

7:20

8:24 (median)

9:25

10:28

11:31

12:35

13:35

14:39

15:42

Lower half (positions 1 - 7): [4,12,13,17,19,19,20]

Q1 is the median of lower half: position (7+1)/2 = 4th term: 17

Upper half (positions 9 - 15): [25,28,31,35,35,39,42]

Q3 is median of upper half: position (7+1)/2 = 4th term: 35

After changing 4 to 6:

Lower half (positions 1 - 7): [6,12,13,17,19,19,20]

Q1: 4th term: 17 (same as before)

Median: position 8: 24 (same)

Upper half: same as before, Q3: 35 (same)

Maximum: 42 (same)

Minimum: was 4, now 6 (changes)

Wait, but the options are "the maximum", "the upper quartile", "the minimum", "the median", "the lower quartile". So the only one that changes is the minimum? Wait, no, wait, maybe I miscounted the data points. Wait, the original data: let's count the numbers. The user wrote: 4,12,13,17,19,19,20,24,25,28,31,35,35,39,42 – that's 15 numbers? Wait, 4 (1),12(2),13(3),17(4),19(5),19(6),20(7),24(8),25(9),28(10),31(11),35(12),35(13),39(14),42(15). Yes. So after changing 4 to 6, the minimum changes. What about the lower quartile? The lower quartile is the median of the lower half. The lower half is the first 7 numbers. In original, first 7: 4,12,13,17,19,19,20. Median of these 7 is the 4th term, which is 17. In new, first 7: 6,12,13,17,19,19,20. Median is 4th term, 17. So lower quartile doesn't change. Median is 8th term, 24, same. Upper quartile: upper half is last 7 numbers: 25,28,31,35,35,39,42. Median is 4th term, 35, same. Maximum is 42, same. So the only measure that changes is the minimum. Wait, but the options are to select all that apply. Wait, maybe I made a mistake. Wait, let's check again.

Wait, the original minimum is 4, new minimum is 6, so minimum changes. What about the lower quartile? Wait, the lower quartile is calculated as the median of the data below the median. The median is at position 8 (24). The data below the median is positions 1 - 7: [4,12,13,17,19,19,20]. The median of this subset (Q1) is the 4th term, 17. After changing 4 to 6, the subset is [6,12,13,17,19,19,20], median is 4th term, 17. So Q1 doesn't change. Median is 24, same. Upper quartile: data above median (positions 9 - 15): [25,28,31,35,35,39,42], median is 4th term, 35, same. Maximum: 42, same. So the only one that changes is the minimum. Wait, but the options include "the minimum" as a choice. So the answer should be the minimum. Wait, but maybe I messed up. Let's confirm:

- Maximum: 42 (original) vs 42 (new) → no change.
- Upper Quartile: 35 (original) vs 35 (new) → no change.
- Minimum: 4 (original) vs 6 (new) → change.
- Median: 24 (original) vs 24 (new) → no change.
- Lower Quartile: 17 (original) vs 17 (new) → no change.

So the only one that changes is the minimum.

Answer:

the minimum