Question

1. for the following set of scores:

6 3 7 6 1 2 7 3 6 4
1 2 4 7 4 4 5 5 6 6
(a) construct a frequency distribution table to organize the scores. include cumulative frequency and cumulative percent.

Explanation:

Step1: Identify unique scores

First, we list the unique scores from the data set. The scores are: 1, 2, 3, 4, 5, 6, 7.

Step2: Count frequency of each score

• For score 1: Appears 2 times.
• For score 2: Appears 2 times.
• For score 3: Appears 2 times.
• For score 4: Appears 3 times.
• For score 5: Appears 2 times.
• For score 6: Appears 5 times.
• For score 7: Appears 3 times.

Step3: Calculate cumulative frequency

Cumulative frequency is the sum of frequencies up to that score.

• Cumulative frequency for 1: \( 2 \)
• Cumulative frequency for 2: \( 2 + 2 = 4 \)
• Cumulative frequency for 3: \( 4 + 2 = 6 \)
• Cumulative frequency for 4: \( 6 + 3 = 9 \)
• Cumulative frequency for 5: \( 9 + 2 = 11 \)
• Cumulative frequency for 6: \( 11 + 5 = 16 \)
• Cumulative frequency for 7: \( 16 + 3 = 19 \) (Wait, total number of scores: Let's count. First row: 10 scores, second row: 10 scores? Wait, first row: 6,3,7,6,1,2,7,3,6,4 (10), second row:1,2,4,7,4,4,5,5,6,6 (10). Total 20 scores. Wait, my mistake earlier. Let's recount frequencies correctly.

Let's list all scores:

First row: 6, 3, 7, 6, 1, 2, 7, 3, 6, 4 (10 scores)

Second row: 1, 2, 4, 7, 4, 4, 5, 5, 6, 6 (10 scores)

Now count each score:

• 1: appears in first row (1) and second row (1) → 2 times. Correct.
• 2: first row (1), second row (1) → 2 times. Correct.
• 3: first row (2) → 2 times. Correct.
• 4: first row (1), second row (3) → 1 + 3 = 4 times? Wait second row: 4,7,4,4 → three 4s? Wait second row: 1,2,4,7,4,4,5,5,6,6. So 4 appears at positions 3,5,6 → three times. First row: 4 appears once (position 10). So total 1 + 3 = 4 times. Wait earlier I thought 3, that was wrong.

• 5: second row (2 times) → 2 times. Correct.
• 6: first row (3 times: positions 1,4,9), second row (2 times: positions 9,10) → 3 + 2 = 5 times. Correct.
• 7: first row (2 times: positions 3,7), second row (1 time: position 4) → 2 + 1 = 3 times. Correct.

Total scores: 2 (1s) + 2 (2s) + 2 (3s) + 4 (4s) + 2 (5s) + 5 (6s) + 3 (7s) = 2+2=4, +2=6, +4=10, +2=12, +5=17, +3=20. Correct, 20 scores.

So cumulative frequency:

• 1: 2
• 2: 2 + 2 = 4
• 3: 4 + 2 = 6
• 4: 6 + 4 = 10
• 5: 10 + 2 = 12
• 6: 12 + 5 = 17
• 7: 17 + 3 = 20

Step4: Calculate cumulative percent

Cumulative percent is (cumulative frequency / total number of scores) * 100. Total number of scores \( N = 20 \).

• For 1: \( \frac{2}{20} \times 100 = 10\% \)
• For 2: \( \frac{4}{20} \times 100 = 20\% \)
• For 3: \( \frac{6}{20} \times 100 = 30\% \)
• For 4: \( \frac{10}{20} \times 100 = 50\% \)
• For 5: \( \frac{12}{20} \times 100 = 60\% \)
• For 6: \( \frac{17}{20} \times 100 = 85\% \)
• For 7: \( \frac{20}{20} \times 100 = 100\% \)

Step5: Construct the table

Now we construct the frequency distribution table with columns: Score (X), Frequency (f), Cumulative Frequency (cf), Cumulative Percent (c%).

Score (X)	Frequency (f)	Cumulative Frequency (cf)	Cumulative Percent (c%)
2	2	4	20%
3	2	6	30%
4	4	10	50%
5	2	12	60%
6	5	17	85%
7	3	20	100%

Answer:

The frequency distribution table with cumulative frequency and cumulative percent is as follows:

Score (X)	Frequency (f)	Cumulative Frequency (cf)	Cumulative Percent (c%)
2	2	4	20%
3	2	6	30%
4	4	10	50%
5	2	12	60%
6	5	17	85%
7	3	20	100%