Question

maya is taking a self - paced math class. her scores for each of the 19 quizzes during the academic year are shown 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.
quiz score
72 to 75
76 to 79
80 to 83
84 to 87
88 to 91
relative frequency

Explanation:

Step1: Count total number of scores

First, we count the total number of quiz scores. Let's list out the scores: 79, 89, 86, 86, 77, 80, 81, 76, 84, 82, 80, 78, 74, 77, 73, 78, 74, 83, 79. Wait, let's count them one by one. Wait, the problem says there are 19 quizzes. Let's count the number of scores: Let's list all the scores from the table (assuming the table has 19 scores). Wait, maybe I miscounted. Wait, let's check the rows. Let's see, the first column: 79, 89, 86, 86, 77 (5 scores), second column: 80, 81, 76, 84, 82 (5 scores), third column: 80, 78, 74, 77, 73 (5 scores), fourth column: 78, 74, 83, 79 (wait, no, maybe the table is 4 rows? Wait, the original problem says 19 quizzes. Wait, maybe I made a mistake. Wait, let's count the number of scores: Let's list all the scores:

From the given table (the lower table with quiz scores):

First column (let's say first row): 79, 89, 86, 86, 77 (5)

Second column: 80, 81, 76, 84, 82 (5)

Third column: 80, 78, 74, 77, 73 (5)

Fourth column: 78, 74, 83, 79 (wait, that's 4? Wait, maybe there's a typo, but the problem says 19. Wait, maybe I missed one. Wait, 5 + 5 + 5 + 4 = 19. Yes, 5+5=10, 10+5=15, 15+4=19. So total number of scores \( n = 19 \).

Step2: Count scores in each class

Now, let's determine which scores fall into each class interval (class width 4, so each class is 4 units wide):

1. **72 to 75**: Scores between 72 (inclusive) and 75 (inclusive). Let's check the scores: 73, 74, 74. Wait, let's list all scores and check:

Scores: 79, 89, 86, 86, 77, 80, 81, 76, 84, 82, 80, 78, 74, 77, 73, 78, 74, 83, 79.

Now, 72 to 75: scores ≥72 and ≤75. So 73, 74, 74. Wait, 73, 74, 74. Let's count: 73 (1), 74 (2), 74 (3). Wait, is that right? Wait, 73 is in 72-75, 74 is in 72-75, another 74. So that's 3 scores? Wait, no, let's check again. Wait, 72 to 75: the class is 72 ≤ score < 76? Wait, because class width is 4, so 72-75 (72 to 75, inclusive? Or 72 to 76? Wait, the problem says "72 to 75", "76 to 79", etc. So 72-75: scores from 72 to 75 (inclusive), 76-79: 76 to 79 (inclusive), 80-83: 80 to 83 (inclusive), 84-87: 84 to 87 (inclusive), 88-91: 88 to 91 (inclusive).

So:

- 72-75: scores 72,73,74,75. So in our data, 73, 74, 74. Wait, 73 (1), 74 (2), 74 (3). Wait, is there a 72 or 75? No. So frequency for 72-75: 3?

Wait, no, let's list all scores:

79, 89, 86, 86, 77, 80, 81, 76, 84, 82, 80, 78, 74, 77, 73, 78, 74, 83, 79.

Now, 72-75:

Scores: 73 (1), 74 (2), 74 (3). So 3 scores.

76-79:

Scores: 76, 77, 77, 78, 78, 79, 79. Wait, 76 (1), 77 (2), 77 (3), 78 (4), 78 (5), 79 (6), 79 (7). Wait, let's check: 76 (from second column: 76), 77 (first column: 77), 77 (third column: 77), 78 (third column: 78), 78 (fourth column: 78), 79 (first column: 79), 79 (fourth column: 79). So that's 7 scores? Wait, no, let's count:

76: 1

77: 2 (two 77s)

78: 2 (two 78s)

79: 2 (two 79s)

Wait, 1+2+2+2=7? Wait, 76 (1), 77 (2), 77 (3), 78 (4), 78 (5), 79 (6), 79 (7). Yes, 7 scores.

80-83:

Scores: 80, 80, 81, 82, 83. Let's check: 80 (first column? No, 80 is in second column: 80, third column: 80), 81 (second column: 81), 82 (second column: 82), 83 (fourth column: 83). So 80 (2), 81 (1), 82 (1), 83 (1). Total: 2+1+1+1=5? Wait, 80, 80, 81, 82, 83. So 5 scores.

84-87:

Scores: 84, 86, 86, 87? Wait, 84 (second column: 84), 86 (first column: 86, 86), 87? No, 84, 86, 86. Wait, 84 (1), 86 (2), 86 (3). Wait, is there an 85 or 87? No. So 3 scores? Wait, 84, 86, 86. So 3 scores?

Wait, no, 84-87: scores 84,85,86,87. So 84 (1), 86 (2), 86 (3). So 3 scores.

88-91:

Scores: 89. So 89 is in 88-91. So 1 score.

Wait, but let's check the total: 3 (72-75) + 7 (76-79) + 5 (80-83) + 3 (84-87) + 1 (88-91) = 3+7=10, 10+5=15, 15+3=18, 18+1=19. Perfect, that matches the total number of scores (19).

Now, relative frequency is frequency divided by total number of scores (19).

So:

- 72-75: frequency = 3, relative frequency = \( \frac{3}{19} \approx 0.1579 \)
- 76-79: frequency = 7, relative frequency = \( \frac{7}{19} \approx 0.3684 \)
- 80-83: frequency = 5, relative frequency = \( \frac{5}{19} \approx 0.2632 \)
- 84-87: frequency = 3, relative frequency = \( \frac{3}{19} \approx 0.1579 \)
- 88-91: frequency = 1, relative frequency = \( \frac{1}{19} \approx 0.0526 \)

Wait, but let's confirm the counts again:

72-75: 73,74,74 → 3 ✔️

76-79: 76,77,77,78,78,79,79 → 7 ✔️ (76, two 77s, two 78s, two 79s)

80-83: 80,80,81,82,83 → 5 ✔️ (two 80s, 81, 82, 83)

84-87: 84,86,86 → 3 ✔️ (84, two 86s)

88-91: 89 → 1 ✔️

Now, let's calculate each relative frequency:

1. 72 to 75: \( \frac{3}{19} \approx 0.1579 \) (rounded to nearest hundredth: 0.16? Wait, 3 divided by 19: 3 ÷ 19 ≈ 0.15789... ≈ 0.16? Wait, no, 0.15789 is approximately 0.16 when rounded to nearest hundredth? Wait, 0.15789: the third decimal is 7, which is ≥5, so round up the second decimal: 0.16? Wait, no, 0.15789: first decimal 1, second 5, third 7. So 0.15789 ≈ 0.16? Wait, no, 0.15789 is 0.1579, which is 0.16 when rounded to two decimal places? Wait, 0.1579: the second decimal is 5, third is 7, so we round the second decimal up: 0.16.

Wait, but let's check:

3 ÷ 19 ≈ 0.1578947... ≈ 0.16 (rounded to nearest hundredth).

7 ÷ 19 ≈ 0.368421... ≈ 0.37 (rounded to nearest hundredth).

5 ÷ 19 ≈ 0.263157... ≈ 0.26 (rounded to nearest hundredth).

3 ÷ 19 ≈ 0.1578947... ≈ 0.16 (rounded to nearest hundredth).

1 ÷ 19 ≈ 0.052631... ≈ 0.05 (rounded to nearest hundredth).

Wait, but let's confirm the total: 0.16 + 0.37 + 0.26 + 0.16 + 0.05 = 1.00? 0.16+0.37=0.53, +0.26=0.79, +0.16=0.95, +0.05=1.00. Perfect.

But wait, maybe I made a mistake in the frequency counts. Let's recheck the scores:

List of all 19 scores:

1. 79

2. 89

3. 86

4. 86

5. 77

6. 80

7. 81

8. 76

9. 84

10. 82

11. 80

12. 78

13. 74

14. 77

15. 73

16. 78

17. 74

18. 83

19. 79

Now, let's categorize each:

1. 79 → 76-79

2. 89 → 88-91

3. 86 → 84-87

4. 86 → 84-87

5. 77 → 76-79

6. 80 → 80-83

7. 81 → 80-83

8. 76 → 76-79

9. 84 → 84-87

10. 82 → 80-83

11. 80 → 80-83

12. 78 → 76-79

13. 74 → 72-75

14. 77 → 76-79

15. 73 → 72-75

16. 78 → 76-79

17. 74 → 72-75

18. 83 → 80-83

19. 79 → 76-79

Now, count each category:

- 72-75: scores 13 (74), 15 (73), 17 (74) → 3 scores (correct)

- 76-79: scores 1 (79), 5 (77), 8 (76), 12 (78), 14 (77), 16 (78), 19 (79) → 7 scores (correct: 79,77,76,78,77,78,79)

- 80-83: scores 6 (80), 7 (81), 10 (82), 11 (80), 18 (83) → 5 scores (correct: 80,81,82,80,83)

- 84-87: scores 3 (86), 4 (86), 9 (84) → 3 scores (correct: 86,86,84)

- 88-91: score 2 (89) → 1 score (correct)

So frequencies are correct. Now, relative frequencies:

- 72-75: 3/19 ≈ 0.1579 ≈ 0.16

- 76-79: 7/19 ≈ 0.3684 ≈ 0.37

- 80-83: 5/19 ≈ 0.2632 ≈ 0.26

- 84-87: 3/19 ≈ 0.1579 ≈ 0.16

- 88-91: 1/19 ≈ 0.0526 ≈ 0.05

So the relative frequency distribution (rounded to nearest hundredth) is:

- 72 to 75: 0.16

- 76 to 79: 0.37

- 80 to 83: 0.26

- 84 to 87: 0.16

- 88 to 91: 0.05

Answer:

Quiz score	Relative frequency
76 to 79	0.37
80 to 83	0.26
84 to 87	0.16
88 to 91	0.05