Bot, please answer this % score 11 - 20 21 - 30 31 - 40 41 - 50 51 - 60 61 - 70 71 - 80 81- 90

Question

No of pupils 60 190 300 400 480 320 170 80
a) Prepare a cumulative frequency table and draw a cumulative frequency curve for the
distribution
b) Use your curve to estimate the
i. number of students that failed if the cut off mark is 55%
ii. number of students that passed if the cut off mark is 75%
iii. cut off mark if 300 students are to be offered admission
iv. 35th percentile mark.

Answer 1

a)

| Score Range | Frequency | Cumulative Frequency |
|-------------|-----------|---------------------|
| 11-20 | 60 | 60 |
| 21-30 | 190 | 250 |
| 31-40 | 300 | 550 |
| 41-50 | 400 | 950 |
| 51-60 | 480 | 1430 |
| 61-70 | 320 | 1750 |
| 71-80 | 170 | 1920 |
| 81-90 | 80 | 2000 |

b)

i. To find the number of students that failed if the cut off mark is 55%, we need to look for the cumulative frequency closest to 55%. From the table, we see that the cumulative frequency at the score range of 50-60 is 1430. We also see that the frequency at the score range of 51-60 is 480. This means that the number of students who scored less than 55% is 950 (cumulative frequency up to score range of 50-60) + (480/2) = 1195. Therefore, the number of students that failed is 2000-1195 = 805.

ii. To find the number of students that passed if the cut off mark is 75%, we need to look for the cumulative frequency closest to 75%. From the table, we see that the cumulative frequency at the score range of 71-80 is 1920. Therefore, the number of students that passed is 1920.

iii. To find the cut off mark if 300 students are to be offered admission, we need to look for the score range that corresponds to a cumulative frequency closest to 300. From the table, we see that the cumulative frequency at the score range of 31-40 is 550, and the cumulative frequency at the score range of 41-50 is 950. Therefore, we know that the cut off mark must be in the range of 41-50. To find the exact cut off, we can use proportional reasoning:

(total number of students in range of 41-50) / (total frequency in range of 41-50) = (number of students needed to reach 300) / (total frequency in range of 41-50)

or

(400) / (400+480) = (x) / (950-400)

where x is the number of students needed to reach 300. Solving for x, we get:

x = 236

This means that we need 236 more students in the range of 41-50 to reach a total of 300 students. To find the cut off, we add this number to the frequency at the score range of 41-50 (which is 400), and get:

cut off = 50 + 236/400 = 50.59

Therefore, the cut off is approximately 50.59.

iv. To find the 35th percentile mark, we need to find the score range that corresponds to a cumulative frequency closest to 35% of the total frequency (which is 2000). From the table, we see that the cumulative frequency at the score range of 21-30 is 250, and the cumulative frequency at the score range of 31-40 is 550. Therefore, we know that the 35th percentile mark must be in the range of 31-40. To find the exact percentile mark, we can use proportional reasoning:

(total frequency in range of 31-40) / (total frequency) = (35/100)

or

(300) / (2000) = (x) / (250+300)

where x is the frequency in the range of 31-40 that corresponds to the 35th percentile mark. Solving for x, we get:

x = 102.9

This means that the 35th percentile mark is approximately at the score range of:

30 + (102.9/300) * 10 = 33.43

Therefore, the 35th percentile mark is approximately 33.43.