Question

what is true of the data in the dot plot? check all that apply.
number of minutes shelly spent waiting for the bus each morning
dot plot with 10: 2 dots, 11: 4 dots, 12: 0 dots, 13: 3 dots, 14: 6 dots, 15: 1 dot
□ the center is 13.
□ the center is 14.
□ the peak is at 14.
□ it has two clusters.

Explanation:

Step1: Analyze the dot plot

First, count the number of dots at each value:
- 10: 2 dots
- 11: 4 dots
- 12: 0 dots
- 13: 3 dots
- 14: 6 dots
- 15: 1 dot

Step2: Determine the center

To find the center (median), we first find the total number of data points. The total number of dots is \(2 + 4+0 + 3+6 + 1=16\). Since there are 16 data points (even number), the median is the average of the 8th and 9th values. Let's list the values in order:
- Values at 10: 2 (positions 1 - 2)
- Values at 11: 4 (positions 3 - 6)
- Values at 13: 3 (positions 7 - 9)
- So the 8th value is 13 and the 9th value is 13? Wait, no, wait. Wait, after 10 (2), 11 (4) – so positions 1 - 2: 10, positions 3 - 6: 11, then position 7: 13, position 8: 13, position 9: 13, position 10: 13, then position 11: 14, etc. Wait, maybe I made a mistake. Let's list all data points:

10, 10, 11, 11, 11, 11, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15.

Now, the 8th value is 13, the 9th value is 13? Wait, no, the 8th value: let's count:

1:10, 2:10, 3:11, 4:11, 5:11, 6:11, 7:13, 8:13, 9:13, 10:14, 11:14, 12:14, 13:14, 14:14, 15:14, 16:15.

Wait, so the 8th value is 13, 9th is 13? No, 7th is 13, 8th is 13, 9th is 13? Wait, no, after 11 (4 values: positions 3 - 6), then the next is 13 (3 values: positions 7 - 9), then 14 (6 values: positions 10 - 15), then 15 (position 16). So the 8th value is 13, 9th value is 13? Wait, no, 7th:13, 8th:13, 9th:13, 10th:14? Wait, no, 13 has 3 values: 13,13,13 (positions 7,8,9), then 14 starts at position 10. So the 8th value is 13, 9th value is 13. So the median is \(\frac{13 + 13}{2}=13\)? Wait, but the options are center is 13 or 14. Wait, maybe I miscounted. Let's check the dot plot again.

Wait the dot plot:

10: 2 dots

11: 4 dots

12: 0

13: 3 dots

14: 6 dots

15: 1 dot

Total data points: 2 + 4 + 0 + 3 + 6 + 1 = 16.

So the ordered data:

10,10,

11,11,11,11,

13,13,13,

14,14,14,14,14,14,

15.

Wait, that's 2 + 4 + 3 + 6 + 1 = 16. So positions:

1:10, 2:10,

3:11, 4:11, 5:11, 6:11,

7:13, 8:13, 9:13,

10:14, 11:14, 12:14, 13:14, 14:14, 15:14,

16:15.

Ah, so the 8th value is 13, 9th value is 13? Wait, no, 7th is 13, 8th is 13, 9th is 13, 10th is 14. Wait, no, 13 has 3 values (positions 7,8,9), then 14 has 6 values (positions 10 - 15), then 15 (position 16). So the 8th value is 13, 9th value is 13? Wait, no, 7th:13, 8th:13, 9th:13, 10th:14. Wait, that can't be. Wait, 2 (10) + 4 (11) = 6, so positions 1 - 6: 10,10,11,11,11,11. Then position 7:13, position 8:13, position 9:13, position 10:14, position 11:14, position 12:14, position 13:14, position 14:14, position 15:14, position 16:15. Ah, there we go. So the 8th value is 13, 9th value is 13? Wait, no, position 7:13, 8:13, 9:13, 10:14. So the 8th and 9th values are 13 and 13? Wait, no, 8th is 13, 9th is 13? Wait, 7th:13, 8th:13, 9th:13, 10th:14. So the median is (13 + 13)/2 = 13? Wait, but the options are center is 13 or 14. Wait, maybe the center is the median, but maybe the question is about the mean? Let's calculate the mean.

Mean = (2*10 + 4*11 + 3*13 + 6*14 + 1*15)/16

= (20 + 44 + 39 + 84 + 15)/16

= (20 + 44 = 64; 64 + 39 = 103; 103 + 84 = 187; 187 + 15 = 202)/16

= 202/16 ≈ 12.625? No, that can't be. Wait, 2*10=20, 4*11=44, 3*13=39, 6*14=84, 1*15=15. Sum: 20+44=64, 64+39=103, 103+84=187, 187+15=202. 202/16=12.625. But the options are 13 or 14. Maybe the center is the median, but I must have messed up the data points. Wait, the dot plot: at 10: 2 dots, 11: 4 dots, 13: 3 dots, 14: 6 dots, 15:1 dot. Wait, 2+4+3+6+1=16. So the data points are:

10,10,

11,11,11,11,

13,13,13,

14,14,14,14,14,14,

15.

So when ordered, the 8th and 9th terms: let's list them in order:

1. 10

2. 10

3. 11

4. 11

5. 11

6. 11

7. 13

8. 13

9. 13

10. 14

11. 14

12. 14

13. 14

14. 14

15. 14

16. 15

Ah, so the 8th term is 13, 9th term is 13. So median is 13. So the center is 13? Wait, but the options include "The center is 14" – maybe I made a mistake. Wait, maybe the dot plot is different. Wait, the dot plot: at 14, there are 6 dots, which is the most. The peak is at 14 (since it has the most dots). Now, clusters: are there two clusters? Let's see: the dots at 10,11 (a cluster), then a gap at 12, then dots at 13,14,15 (another cluster). So two clusters.

So let's re-express:

- Peak: the value with the most dots is 14 (6 dots), so peak is at 14.

- Center: median is 13 (as calculated), but wait, maybe the question considers the center as the median or the mean? Wait, the mean was ~12.6, median 13. So "The center is 13" – is that correct? Wait, the 8th and 9th terms are 13 and 13, so median is 13. So "The center is 13" is correct? Wait, but let's check the options.

Options:

- The center is 13: yes (median 13)

- The center is 14: no (median 13, mean ~12.6)

- The peak is at 14: yes (most dots at 14)

- It has two clusters: yes (cluster at 10,11; cluster at 13,14,15 with a gap at 12)

Wait, but maybe I made a mistake in the median. Wait, let's list all data points:

10,10,11,11,11,11,13,13,13,14,14,14,14,14,14,15.

Now, the 8th term: index 8 (starting at 1) is 13, 9th term is 13. So median is (13 + 13)/2 = 13. So "The center is 13" is correct.

Peak at 14: correct, since 14 has 6 dots, more than others.

Two clusters: correct, because there's a gap at 12 between the first cluster (10,11) and the second cluster (13,14,15).

So the correct options are:

- The center is 13.

- The peak is at 14.

- It has two clusters.

Wait, but let's confirm the clusters: dots at 10,11 (cluster 1), then no dots at 12, then dots at 13,14,15 (cluster 2). So two clusters.

So the correct options are:

- The center is 13.

- The peak is at 14.

- It has two clusters.

Wait, but let's check again.

Number of dots:

10: 2

11: 4

13: 3

14: 6

15: 1

So peak at 14 (most dots).

Clusters: yes, two clusters (10-11 and 13-15, with a gap at 12).

Center: median is 13 (as 8th and 9th terms are 13 and 13).

So the correct options are:

- The center is 13.

- The peak is at 14.

- It has two clusters.

Wait, but the option "The center is 14" is incorrect. So the correct options are:

- The center is 13.

- The peak is at 14.

- It has two clusters.

Answer:

The center is 13, The peak is at 14, It has two clusters (i.e., the options: "The center is 13", "The peak is at 14", "It has two clusters" are correct).