Question

question 6 of 10
on a piece of paper, draw a box plot to represent the data
below. then determine which answer choice matches the box
plot you drew.

11, 12, 13, 15, 17, 19, 22, 24, 29, 33, 38

a.

b.

c.

Explanation:

Step1: Find the median (Q2)

The data set is \(11, 12, 13, 15, 17, 19, 22, 24, 29, 33, 38\) (11 values). The median is the 6th value, so \(Q2 = 19\)? Wait, no, wait: 11 values, the middle is the 6th? Wait, 11 numbers: positions 1 - 11. The median (Q2) is at position \(\frac{11 + 1}{2}=6\), so the 6th value is \(19\)? Wait, no, let's list them: 1 (11), 2 (12), 3 (13), 4 (15), 5 (17), 6 (19), 7 (22), 8 (24), 9 (29), 10 (33), 11 (38). So median (Q2) is 19? Wait, no, wait, 11 numbers: the median is the 6th term? Wait, no, for odd number of observations, median is the \((n + 1)/2\)th term. So \(n = 11\), \((11 + 1)/2 = 6\), so 6th term is 19. Then, the lower half is the first 5 numbers: \(11, 12, 13, 15, 17\). The median of the lower half (Q1) is the 3rd term: 13. The upper half is the last 5 numbers: \(22, 24, 29, 33, 38\). The median of the upper half (Q3) is the 3rd term: 29. Wait, no, wait: lower half is first 5 (positions 1 - 5), so median of lower half is position 3: 13. Upper half is positions 7 - 11 (5 numbers), median is position 9: 29? Wait, no, positions 7 - 11: 7 (22), 8 (24), 9 (29), 10 (33), 11 (38). So median of upper half is 29. Wait, but let's check again. Wait, the data set is 11 numbers:

1: 11

2: 12

3: 13

4: 15

5: 17

6: 19 (median, Q2)

7: 22

8: 24

9: 29

10: 33

11: 38

So lower quartile (Q1) is the median of the first 5 numbers (11,12,13,15,17). The median of these 5 is the 3rd number: 13. Upper quartile (Q3) is the median of the last 5 numbers (22,24,29,33,38). The median of these 5 is the 3rd number: 29. The minimum is 11, maximum is 38.

Now, let's check the box plots:

- The box should have Q1 (13), Q2 (19), Q3 (29). Wait, no, wait, maybe I made a mistake. Wait, the data set: 11,12,13,15,17,19,22,24,29,33,38. Let's count the number of elements: 11. So median is the 6th element: 19. Then lower half is elements 1 - 5: 11,12,13,15,17. Median of lower half (Q1) is 13 (3rd element). Upper half is elements 7 - 11: 22,24,29,33,38. Median of upper half (Q3) is 29 (3rd element of upper half). So the box plot should have:

- Minimum: 11

- Q1: 13

- Q2: 19

- Q3: 29

- Maximum: 38

Now let's check the options:

Option A: The left whisker goes to 11 (min), Q1 is around 13? Wait, the first box starts at 12? No, wait the x-axis is 10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40. Wait, maybe the ticks are at 10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40. So let's see the box positions.

Wait maybe I miscalculated Q1, Q2, Q3. Let's use the formula for quartiles:

For a data set with n elements, the position of Q1 is \(\frac{n + 1}{4}\), Q2 is \(\frac{n + 1}{2}\), Q3 is \(\frac{3(n + 1)}{4}\).

n = 11, so:

Q1 position: \(\frac{11 + 1}{4}=3\), so 3rd element: 13 (correct, as before)

Q2 position: \(\frac{11 + 1}{2}=6\), 6th element: 19 (correct)

Q3 position: \(\frac{3(11 + 1)}{4}=9\), 9th element: 29 (correct)

So min = 11, Q1 = 13, Q2 = 19, Q3 = 29, max = 38.

Now let's check the box plots:

Option A: The left whisker (min) is at 11 (since the first line is at 11). The box starts at Q1 (13? Wait the first box's left edge is at 12? No, the x-axis ticks: 10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40. So between 10 and 12 is 11? Wait maybe the ticks are labeled at 10,12,14,... so the interval between ticks is 2. So 10,12 (11 is between 10 and 12), 14 (13 between 12 and 14), etc.

Wait Option C: The left whisker is at 11 (min), Q1 is at 13 (between 12 and 14), Q2 at 19 (between 18 and 20), Q3 at 29 (between 28 and 30), max at 38. Wait but let's check the box lengths.

Wait maybe I made a mistake in quartile calculation. Let's list the data in order:

11, 12, 13, 15, 17, 19, 22, 24, 29, 33, 38

n = 11.

Another method: split into lower half (first 5: 11,12,13,15,17) and upper half (last 5: 22,24,29,33,38), median is the middle (6th) term: 19.

So Q1 is the median of lower half: (13 + 13)/2? No, lower half has 5 elements, median is 3rd: 13.

Q3 is median of upper half: 3rd element: 29.

So min = 11, Q1 = 13, Q2 = 19, Q3 = 29, max = 38.

Now look at the options:

Option C: The left whisker (min) is at 11 (the first vertical line), Q1 is at 13 (the left edge of the first box), Q2 at 19 (middle line of the box), Q3 at 29 (right edge of the box), and the right whisker at 38 (max). Wait the x-axis in option C has ticks at 10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40. So 11 is between 10 and 12, 13 between 12 and 14, 19 between 18 and 20, 29 between 28 and 30, 38 at the end.

Option A: The left whisker is at 11, but Q1 is at 12? No, the first box starts at 12, which would be Q1 = 12, but our Q1 is 13.

Option B: Q1 is at 14? No, the box starts at 14? No, the box starts at 14? No, the box in B starts at 14? Wait no, the box in B is between 14 and 22? No, maybe I'm misreading.

Wait let's check the number of data points: 11. So the five-number summary is min=11, Q1=13, Q2=19, Q3=29, max=38.

Now, looking at the options, Option C has the left whisker at 11 (min), the box from Q1=13 (between 12 and 14) to Q3=29 (between 28 and 30), with the median line at 19 (between 18 and 20), and the right whisker at 38 (max). So Option C matches.

Wait but let's confirm:

In Option C, the leftmost line (min) is at 11 (between 10 and 12), the first box (Q1 to Q2? No, the box is Q1 to Q3, with median inside). Wait no, the box plot has a box from Q1 to Q3, with a line at Q2 (median), and whiskers from min to Q1 and Q3 to max.

So the box is between Q1 and Q3, median inside. So Q1=13, Q3=29, median=19.

Looking at the x-axis of Option C: the box starts at 13 (between 12 and 14) and ends at 29 (between 28 and 30), with the median line at 19 (between 18 and 20). The left whisker is from 11 (min) to 13 (Q1), right whisker from 29 (Q3) to 38 (max). That matches our five-number summary.

Option A: The box starts at 12 (Q1=12) and ends at 24 (Q3=24), median at 19? No, that doesn't match.

Option B: The box starts at 14 (Q1=14) and ends at 24 (Q3=24), median at 19? No.

So the correct answer is Option C.

Answer:

C