Question

in a survey, 19 people reported the number of hours they spent on the internet last week. their responses are given below. complete the grouped relative frequency distribution for the data. (note that we are using a class width of 4.) write each relative frequency as a decimal rounded to the nearest hundredth, not as a percentage. number of hours 13 8 3 14 9 3 15 12 3 11 17 9 8 16 18 16 10 9 6 number of hours 0 to 3 4 to 7 8 to 11 12 to 15 16 to 19 relative frequency

Explanation:

Step1: Count frequencies for each class

- **0 to 3**: Values are 3, 3, 3, 8? Wait, no. Wait the data is: 13, 8, 3, 14, 9, 3, 15, 12, 3, 11, 17, 9, 8, 16, 18, 16, 10, 9, 6. Wait, let's list all data points: 13, 8, 3, 14, 9, 3, 15, 12, 3, 11, 17, 9, 8, 16, 18, 16, 10, 9, 6. Now count for each class:
- 0 - 3: Numbers ≤3? Wait the class is 0 to 3 (inclusive? Or 0 - 3, 4 - 7, 8 - 11, 12 - 15, 16 - 19). So:
- 0 - 3: 3, 3, 3 → 3 values.
- 4 - 7: 6 → 1 value (since 4 ≤ x <8? Wait class width 4, so 0 - 3, 4 - 7 (4 to 7 inclusive? Or 4 - 7 as 4 ≤ x <8? Wait the problem says class width 4. So 0 - 3 (0 ≤ x <4), 4 - 7 (4 ≤ x <8), 8 - 11 (8 ≤ x <12), 12 - 15 (12 ≤ x <16), 16 - 19 (16 ≤ x <20). Let's check:
- 0 - 3: 3, 3, 3 → 3 (since 3 <4)
- 4 - 7: 6 → 1 (6 <8)
- 8 - 11: 8, 9, 8, 9, 10, 9, 11 → Wait 8,9,8,9,10,9,11: let's count. 8,9,8,9,10,9,11: that's 7? Wait data points: 8,9,3,9,3,15,12,3,11,17,9,8,16,18,16,10,9,6. Wait let's list all data with their classes:
- 0 - 3: 3,3,3 → 3
- 4 - 7: 6 → 1 (since 6 is between 4 and 7)
- 8 - 11: 8,9,8,9,10,9,11 → 8,9,8,9,10,9,11: that's 7? Wait 8,9,8,9,10,9,11: 7 numbers? Wait 8 (1), 9 (1), 8 (2), 9 (2), 10 (1), 9 (3), 11 (1) → total 7? Wait no, let's count again. Data points: 13 (12 -15), 8 (8 -11), 3 (0 -3), 14 (12 -15), 9 (8 -11), 3 (0 -3), 15 (12 -15), 12 (12 -15), 3 (0 -3), 11 (8 -11), 17 (16 -19), 9 (8 -11), 8 (8 -11), 16 (16 -19), 18 (16 -19), 16 (16 -19), 10 (8 -11), 9 (8 -11), 6 (4 -7). Now let's count each class:
- 0 - 3: 3,3,3 → 3
- 4 - 7: 6 → 1
- 8 - 11: 8,9,9,11,9,8,10,9 → Wait 8 (2), 9 (4), 10 (1), 11 (1) → total 2+4+1+1=8? Wait no, let's list each data point and its class:
- 13: 12-15
- 8: 8-11
- 3: 0-3
- 14: 12-15
- 9: 8-11
- 3: 0-3
- 15: 12-15
- 12: 12-15
- 3: 0-3
- 11: 8-11
- 17: 16-19
- 9: 8-11
- 8: 8-11
- 16: 16-19
- 18: 16-19
- 16: 16-19
- 10: 8-11
- 9: 8-11
- 6: 4-7
Now count:
- 0-3: 3,3,3 → 3
- 4-7: 6 → 1
- 8-11: 8,9,9,11,9,8,10,9 → let's count: 8 (2), 9 (4), 10 (1), 11 (1) → 2+4+1+1=8? Wait 8,9,9,11,9,8,10,9: that's 8 numbers? Wait 8 (2 times), 9 (4 times), 10 (1), 11 (1) → 2+4+1+1=8. Yes.
- 12-15: 13,14,15,12 → 4
- 16-19: 17,16,18,16 → 4? Wait 17,16,18,16: that's 4? Wait no, 17,16,18,16: 4 numbers? Wait 17 (1), 16 (2), 18 (1) → 1+2+1=4? Wait 17,16,18,16: 4 numbers. Wait total data points: 3+1+8+4+4=20? But the problem says 19 people. Wait maybe I miscounted. Let's count total data points: 13,8,3,14,9,3,15,12,3,11,17,9,8,16,18,16,10,9,6. That's 19 numbers. Oh right, 19. So let's recount:
- 0-3: 3,3,3 → 3 (correct)
- 4-7: 6 → 1 (correct)
- 8-11: 8,9,9,11,9,8,10,9 → wait 8 (2), 9 (4), 10 (1), 11 (1) → 2+4+1+1=8? But 3+1+8=12, then 12+4 (12-15) =16, 16+4 (16-19)=20. But there are 19 data points. So I must have made a mistake. Let's list all 19 data points:
1. 13
2. 8
3. 3
4. 14
5. 9
6. 3
7. 15
8. 12
9. 3
10. 11
11. 17
12. 9
13. 8
14. 16
15. 18
16. 16
17. 10
18. 9
19. 6
Now assign to classes:
- 0-3: data points 3,3,3 (indices 3,6,9) → 3
- 4-7: data point 6 (index 19) → 1
- 8-11: data points 8 (2),9 (5,12,18),11 (10),10 (17) → 8,9,9,11,9,8,10,9 → indices 2,5,12,10,18,2,17,5? Wait no, indices: 2 (8),5 (9),12 (9),10 (11),18 (9),2 (8) is same as index 2? No, each data point is unique. Wait data points: 8 (index2),9 (index5),9 (index12),11 (index10),9 (index18),8 (index13),10 (index17),9 (index5? No, index5 is 9, index12 is 9, index18 is 9, index5 is 9: so 9 appears at 5,12,18 → three times? Wait no, index5:9, index12:9, index18:9 → three 9s? Wait data points:
- 8: indices 2,13 → 2 times
- 9: indices 5,12,18 → 3 times
- 11: index10 → 1 time
- 10: index17 → 1 time
So total for 8-11: 2 (8s) +3 (9s)+1 (11)+1 (10)=7. Ah! There we go. I overcounted 9s. So 8 (2),9 (3),11 (1),10 (1) → 2+3+1+1=7.
- 12-15: data points 13 (1),14 (4),15 (7),12 (8) → 4
- 16-19: data points 17 (11),16 (14),18 (15),16 (16) → 4
Now total: 3 (0-3) +1 (4-7)+7 (8-11)+4 (12-15)+4 (16-19)=19. Correct.

Step2: Calculate relative frequencies

Relative frequency = frequency / total (19)
- 0-3: 3/19 ≈0.1579 ≈0.16 (rounded to nearest hundredth)
- 4-7: 1/19 ≈0.0526 ≈0.05
- 8-11:7/19≈0.3684≈0.37
- 12-15:4/19≈0.2105≈0.21
- 16-19:4/19≈0.2105≈0.21

Wait but let's check again:

0-3: 3 data points (3,3,3) → 3/19 ≈0.16

4-7: 6 → 1/19≈0.05

8-11: 8,9,9,11,9,8,10 → wait no, the data points in 8-11 are: 8 (index2),8 (index13),9 (index5),9 (index12),9 (index18),10 (index17),11 (index10) → that's 7 data points (2 eights, 3 nines, 1 ten, 1 eleven) → 7/19≈0.37

12-15:13,14,15,12 → 4/19≈0.21

16-19:17,16,18,16 → 4/19≈0.21

Yes, that adds up: 0.16+0.05=0.21; 0.21+0.37=0.58; 0.58+0.21=0.79; 0.79+0.21=1.00 (approx, due to rounding).

Answer:

• 0 to 3: $\boldsymbol{\approx 0.16}$
• 4 to 7: $\boldsymbol{\approx 0.05}$
• 8 to 11: $\boldsymbol{\approx 0.37}$
• 12 to 15: $\boldsymbol{\approx 0.21}$
• 16 to 19: $\boldsymbol{\approx 0.21}$