Question

eighteen cyclists were each asked the number of miles they biked last week.
their responses are given below:
number of miles
23, 4, 9, 6, 23, 23
28, 4, 10, 28, 28, 25
31, 12, 5, 15, 4, 7
(a) complete the grouped frequency distribution for
the data. (note that the class width is 6.)
number of miles | frequency
2 to 7 |
8 to 13 |
14 to 19 |
20 to 25 |
(b) construct a histogram for the data.

Explanation:

Step1: List all data points

The data points are: 13, 4, 9, 6, 23, 23, 28, 4, 10, 28, 18, 25, 31, 12, 5, 15, 4, 7.

Step2: Count for 2 to 7

Numbers in [2,7]: 4,6,4,5,4,7. Count: 6.

Step3: Count for 8 to 13

Numbers in [8,13]: 9,10,12,13. Wait, 9,10,12,13? Wait, 9,10,12,13? Wait, original data: 9,10,12,13? Wait, no, let's recheck. Data: 13,4,9,6,23,23,28,4,10,28,18,25,31,12,5,15,4,7. So 8 - 13: 9,10,12,13. Wait, 9,10,12,13: that's 4? Wait, 9,10,12,13: 9,10,12,13. Wait, 13 is included? The class is 8 to 13, so inclusive? So 9,10,12,13: 4 numbers? Wait, 9,10,12,13: yes. Wait, also 12? 12 is between 8-13. 13 is between 8-13. So 9,10,12,13: 4? Wait, no, 9,10,12,13: that's 4? Wait, maybe I missed. Wait, 9,10,12,13: 4 numbers. Then 14-19: 18,15. Wait, 18 and 15: that's 2? Wait, no, 15,18: two numbers. Then 20-25: 23,23,25. Wait, 23,23,25: three numbers? Wait, 23 is between 20-25, 25 is between 20-25. Wait, 23,23,25: three? Wait, but the total data points are 18. Let's check: 6 (2-7) + 4 (8-13) + 2 (14-19) + 3 (20-25) = 15? No, that's not right. Wait, maybe I made a mistake. Let's list all data points with their classes:

2-7: 4,6,4,5,4,7. Let's count: 4 (first), 6, 4 (second), 5, 4 (third), 7. That's 6 numbers. Correct.

8-13: 9,10,12,13. Wait, 9,10,12,13: 4 numbers. Wait, but also 12? 12 is 12, which is 8-13. 13 is 13, 8-13. So 9,10,12,13: 4.

14-19: 15,18. 15 and 18: two numbers.

20-25: 23,23,25. 23,23,25: three numbers.

Wait, but then 28,28,31: these are above 25. Wait, the classes are 2-7, 8-13, 14-19, 20-25. But the data has 28,28,31. Oh, maybe the class for 20-25 is actually 20-25, and the next class is 26-31? But the problem says "20 to 25". Wait, maybe the original data has some errors, or I misread. Wait, the problem says "Eighteen cyclists" so 18 data points. Let's count all data points:

1. 13

2. 4

3. 9

4. 6

5. 23

6. 23

7. 28

8. 4

9. 10

10. 28

11. 18

12. 25

13. 31

14. 12

15. 5

16. 15

17. 4

18. 7

Yes, 18 points. Now, let's reclassify:

2 to 7: numbers ≥2 and ≤7. So 4,6,4,5,4,7. Let's count: 4 (2nd), 6 (4th), 4 (8th), 5 (15th), 4 (17th), 7 (18th). That's 6 numbers. Correct.

8 to 13: ≥8 and ≤13. So 9 (3rd), 10 (9th), 12 (14th), 13 (1st). Wait, 9,10,12,13: that's 4 numbers? Wait, 9,10,12,13: 4. But 13 is 13, which is ≤13. Correct.

14 to 19: ≥14 and ≤19. So 15 (16th), 18 (11th). Wait, 15 and 18: two numbers? Wait, 15 is 15, 18 is 18. What about 14? There's no 14. So 15 and 18: two numbers.

20 to 25: ≥20 and ≤25. So 23 (5th), 23 (6th), 25 (12th). Wait, 23,23,25: three numbers.

Now, 6 + 4 + 2 + 3 = 15. But we have 18 data points. The remaining are 28 (7th), 28 (10th), 31 (13th). So these are in 26-31, but the problem's classes are 2-7, 8-13, 14-19, 20-25. So maybe the problem has a typo, or the classes are different. Wait, maybe the class width is 6? Wait, 2-7: width 5 (7-2=5). 8-13: 13-8=5. 14-19: 19-14=5. 20-25: 25-20=5. So class width 5. Then 26-31 would be next, but the problem only has four classes. So maybe the data includes 28,28,31 as part of 20-25? No, 28>25. So maybe the problem's data is different. Wait, maybe I misread the data. Let's check again:

The first row: 13, 4, 9, 6, 23, 23

Second row: 28, 4, 10, 28, 18, 25

Third row: 31, 12, 5, 15, 4, 7

Ah! Third row: 31,12,5,15,4,7. So 31 is there, 12,5,15,4,7. So total data points: 6 (first row) + 6 (second row) + 6 (third row) = 18. Correct.

So 2-7: 4 (first row, 2nd), 6 (first row, 4th), 4 (second row, 2nd), 5 (third row, 3rd), 4 (third row, 5th), 7 (third row, 6th). 6 numbers.

8-13: 9 (first row, 3rd), 10 (second row, 3rd), 12 (third row, 2nd), 13 (first row, 1st). 4 numbers.

14-19: 15 (third row, 4th), 18 (second row, 5th). 2 numbers.

20-25: 23 (first row, 5th), 23 (first row, 6th), 25 (second row, 6th). 3 numbers.

Now, 6+4+2+3=15. But we have 18 data points. The remaining are 28 (second row, 1st), 28 (second row, 4th), 31 (third row, 1st). So these three are in 26-31, but the problem's classes don't include that. Maybe the problem's data is different, or the classes are 2-7, 8-13, 14-19, 20-31? No, the problem says "20 to 25". Maybe it's a mistake, but assuming the problem expects us to use the given classes, let's proceed with the counts we have: 6, 4, 2, 3.

Now, for part (a), the grouped frequency table:

Number of miles | Frequency

2 to 7 | 6

8 to 13 | 4

14 to 19 | 2

20 to 25 | 3

For part (b), the histogram would have bars with heights 6, 4, 2, 3 for the classes 2-7, 8-13, 14-19, 20-25 respectively.

Answer:

(a) The grouped frequency table is:

Number of miles	Frequency
8 to 13	4
14 to 19	2
20 to 25	3

(b) The histogram will have bars with heights 6, 4, 2, 3 for the respective classes.