Question

there are 18 students in a certain painting class. their ages are shown below. complete the grouped relative frequency distribution for the data. (note that we are using a class width of 5.) write each relative frequency as a decimal rounded to the nearest hundredth, not as a percentage. age: 24, 33, 30, 30, 31, 29, 28, 21, 26, 21, 18, 22, 29, 29, 19, 23, 37, 29. age | relative frequency: 18 to 22, 23 to 27, 28 to 32, 33 to 37.

Explanation:

Step1: Count data in each class

- 18 to 22: Ages are 18, 21, 21, 22, 19. Wait, no: 18, 21, 21, 22, 19? Wait, let's list all ages: 24,33,30,30,31,29,28,21,26,21,18,22,29,29,19,23,37,29. Wait, total 18? Wait original data: first row 5, second 5, third 5, fourth 3: 5+5+5+3=18. Let's list all 18 ages: 24,33,30,30,31,29,28,21,26,21,18,22,29,29,19,23,37,29. Now categorize:

18 - 22: 18,21,21,22,19. Wait 18,21,21,22,19: that's 5? Wait 18,19,21,21,22: yes, 5.

23 - 27: 24,26,23. Wait 24,26,23: and is there more? 24,26,23: any others? 24 (23-27: 23<=x<=27), 26 (yes), 23 (yes), and 24,26,23, and what about 24,26,23, and is there 25? No. Wait 24,26,23: that's 3? Wait wait, 24 is 23-27, 26 is 23-27, 23 is 23-27, and also 24,26,23: any other? Wait 24,26,23, and 24: yes, 24 is in 23-27. Wait maybe I missed: 24,26,23, and also 24? Wait no, 24 is one, 26 is one, 23 is one, and is there 25? No. Wait 24,26,23: that's 3? Wait no, wait the data: 24,33,30,30,31,29,28,21,26,21,18,22,29,29,19,23,37,29. Let's list each age and its class:

18: 18-22

21: 18-22

21: 18-22

22: 18-22

19: 18-22

24: 23-27

26: 23-27

23: 23-27

28: 28-32

29: 28-32 (multiple times)

30: 28-32

30: 28-32

31: 28-32

29: 28-32

29: 28-32

29: 28-32

33: 33-37

37: 33-37

Wait let's count each class:

18-22: 18,21,21,22,19 → 5 ages? Wait no, 18,21,21,22,19: that's 5? Wait 18 (1),19(1),21(2),22(1): total 5.

23-27: 24(1),26(1),23(1): and is there 25? No. Wait 24,26,23: that's 3? Wait wait, 24 is 23-27, 26 is 23-27, 23 is 23-27, and also 24: yes, 24 is in 23-27. Wait any other? 24,26,23: that's 3? Wait no, wait 24,26,23: 3? Wait original data: 24,26,23, and also 24? No, 24 is once. Wait 24,26,23: 3.

28-32: 28(1),29(4 times: 29,29,29,29),30(2),31(1). Let's count: 28 (1), 29 (4), 30 (2), 31 (1): total 1+4+2+1=8? Wait 28,29,29,29,29,30,30,31: that's 8.

33-37: 33(1),37(1): total 2.

Wait let's verify total: 5 (18-22) + 3 (23-27) + 8 (28-32) + 2 (33-37) = 5+3+8+2=18. Correct.

Now relative frequency is frequency / total (18).

Step2: Calculate each relative frequency

18-22: frequency 5 → 5/18 ≈ 0.28 (rounded to nearest hundredth: 5÷18≈0.277...→0.28)

23-27: frequency 3 → 3/18 = 0.17 (3÷18=0.166...→0.17)

28-32: frequency 8 → 8/18 ≈ 0.44 (8÷18≈0.444...→0.44)

33-37: frequency 2 → 2/18 ≈ 0.11 (2÷18≈0.111...→0.11)

Wait wait, let's recheck the frequency counts:

18-22: ages 18,19,21,21,22. Let's list all in 18-22:

18 (yes), 19 (yes), 21 (yes), 21 (yes), 22 (yes). Any others? 21,21,22,18,19: that's 5. Correct.

23-27: ages 23,24,26. 23 (yes), 24 (yes), 26 (yes). Any others? 23,24,26: 3. Correct.

28-32: ages 28,29,29,29,29,30,30,31. Let's list:

28 (yes), 29 (four times), 30 (two times), 31 (one time). Total: 1+4+2+1=8. Correct.

33-37: ages 33,37. Two. Correct.

Now calculate each relative frequency:

18-22: 5/18 ≈ 0.28 (rounded to nearest hundredth)

23-27: 3/18 = 0.17 (3÷18=0.166..., rounded to 0.17)

28-32: 8/18 ≈ 0.44 (8÷18≈0.444..., rounded to 0.44)

33-37: 2/18 ≈ 0.11 (2÷18≈0.111..., rounded to 0.11)

Wait but let's check the original data again. Wait the fourth row: 23,37,29. So 23 is 23-27, 37 is 33-37, 29 is 28-32. So that's correct.

So:

18 to 22: 5/18 ≈ 0.28

23 to 27: 3/18 ≈ 0.17

28 to 32: 8/18 ≈ 0.44

33 to 37: 2/18 ≈ 0.11

Wait but let's confirm the frequency counts again:

Total data points: 18.

18-22: 18,19,21,21,22 → 5. Correct.

23-27: 23,24,26 → 3. Correct.

28-32: 28,29,29,29,29,30,30,31 → 8. Correct.

33-37: 33,37 → 2. Correct. 5+3+8+2=18. Perfect.

Now calculate each relative frequency:

18-22: 5/18 ≈ 0.28 (5 divided by 18 is approximately 0.277..., rounded to nearest hundredth is 0.28)

23-27: 3/18 = 0.166..., rounded to 0.17

28-32: 8/18 ≈ 0.444..., rounded to 0.44

33-37: 2/18 ≈ 0.111..., rounded to 0.11

Answer:

18 to 22: 0.28
23 to 27: 0.17
28 to 32: 0.44
33 to 37: 0.11