The table below shows the frequency distribution of the marks of 800 candidates in an examination. Marks 0-9 10-19 20-29 30-39 40-49 50-59 60-69 70-79 80-89...
Question 1 Report
The table below shows the frequency distribution of the marks of 800 candidates in an examination.
| Marks | 0-9 | 10-19 | 20-29 | 30-39 | 40-49 | 50-59 | 60-69 | 70-79 | 80-89 | 90-99 |
| Freq | 10 | 40 | 80 | 140 | 170 | 130 | 100 | 70 | 40 | 20 |
(a) (i) Construct a cumulative frequency table ; (ii) Draw the Ogive ; (iii) Use your ogive to determine the 50th percentile.
(b) The candidates that scored less than 25% are to be withdrawn from the institution, while those that scored than 75% are to be awarded scholarship. Estimate the number of students that will be retained, but will not enjoy the award.
Answer Details
(a)(i) Cumulative frequency table. \(N=800\).
| Marks | Frequency | Upper boundary | Cumulative frequency |
|---|---|---|---|
| 0 - 9 | 10 | 9.5 | 10 |
| 10 - 19 | 40 | 19.5 | 50 |
| 20 - 29 | 80 | 29.5 | 130 |
| 30 - 39 | 140 | 39.5 | 270 |
| 40 - 49 | 170 | 49.5 | 440 |
| 50 - 59 | 130 | 59.5 | 570 |
| 60 - 69 | 100 | 69.5 | 670 |
| 70 - 79 | 70 | 79.5 | 740 |
| 80 - 89 | 40 | 89.5 | 780 |
| 90 - 99 | 20 | 99.5 | 800 |
(a)(ii) Ogive. Plot cumulative frequency against upper class boundaries and join with a smooth curve.
(a)(iii) 50th percentile (median). At \(\frac{N}{2}=400\), interpolating in the 40 - 49 class:
\[P_{50}=39.5+\frac{400-270}{170}\times10 \approx 47.\]
(b) Students retained but not awarded. Withdraw those scoring under 25 marks; award scholarships to those scoring over 75 marks. From the ogive:
\[\text{below }25:\ 50+\frac{25-19.5}{10}\times80 \approx 94,\]
\[\text{below }75:\ 670+\frac{75-69.5}{10}\times70 \approx 708.5.\]
Retained but not awarded \(=708.5-94 \approx 615\) students. So about 615 candidates are kept without a scholarship.
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