Bot, please answer this % score 11 - 20 21 - 30 31 - 40 41 - 50 51 - 60 61 - 70 71 - 80 81- 90
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.
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.
a) To prepare a cumulative frequency table, we can start by calculating the cumulative frequencies for each range.
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) To draw a cumulative frequency curve, we can plot the cumulative frequencies against the upper class boundaries.
Now, let's estimate the information using the cumulative frequency curve:
i. To estimate the number of students that failed if the cut-off mark is 55%, we can find the corresponding cumulative frequency on the curve. From the curve, we can estimate that around 1230 students failed if the cut-off mark is 55%.
ii. To estimate the number of students that passed if the cut-off mark is 75%, we can find the corresponding cumulative frequency on the curve. From the curve, we can estimate that around 1590 students passed if the cut-off mark is 75%.
iii. To estimate the cut-off mark if 300 students are to be offered admission, we can find the cumulative frequency of 300 on the curve and read the corresponding value on the x-axis. From the curve, we can estimate that the cut-off mark is approximately 64.
iv. To estimate the 35th percentile mark, we can find the cumulative frequency of 35% of the total number of students (35% of 2000 = 700) on the curve and read the corresponding value on the x-axis. From the curve, we can estimate that the 35th percentile mark is approximately 43.
Keep in mind that these estimates are based on the cumulative frequency curve and may not be exact values.
To answer these questions, we will need to create a cumulative frequency table and draw a cumulative frequency curve based on the given data. Then, we can use the curve to estimate the number of students that failed and passed for different cut-off marks, determine the cut-off mark for a given number of students, and find the 35th percentile mark.
First, let's create the cumulative frequency table:
| Score Range | No. of Pupils | 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 |
Next, let's draw the cumulative frequency curve using the data points from the cumulative frequency table. The x-axis will represent the score ranges, and the y-axis will represent the cumulative frequency.
Now, to estimate the number of students that failed and passed for different cut-off marks:
a) For a cut-off mark of 55%, we can read it off from the curve by drawing a horizontal line at 55% from the y-axis. Then, we can draw a vertical line from the point where the horizontal line intersects the curve to the x-axis. The value on the x-axis represents the score range. We can estimate the number of students that failed based on the cumulative frequency of the score range just before the cut-off mark.
b) For a cut-off mark of 75%, we perform the same procedure as in (a), but this time we estimate the number of students that passed based on the cumulative frequency of the score range just after the cut-off mark.
c) To find the cut-off mark for 300 students, we start from the leftmost point of the curve and move to the right until we reach 300 on the y-axis. The value on the x-axis represents the score range.
d) To find the 35th percentile mark, we look for the point on the curve that corresponds to the 35th percentile. We can then draw a vertical line from that point to the x-axis and read off the score range value.
Remember, drawing an accurate curve is essential for precise estimations.