The table shows the scores of 2000 candidates in an entrance examination into a private secondary school. % Mark 11-20 21-30 31-40 41-50 51-60 61-70 71-80 8...
Question 1 Report
The table shows the scores of 2000 candidates in an entrance examination into a private secondary school.
| % Mark | 11-20 | 21-30 | 31-40 | 41-50 | 51-60 | 61-70 | 71-80 | 81-90 |
| No of pupils | 68 | 184 | 294 | 402 | 480 | 310 | 164 | 98 |
(a) Prepare a cumulative frequency table and draw the cumulative frequency curve for the distribution.
(b) Use your curve to estimate the : (i) cut off mark, if 300 candidates are to be offered admission ; (ii) probability that a candidate picked at random, scored at least 45%.
Answer Details
(a) Cumulative frequency table
| Marks (%) | Number of pupils, \(f\) | Cumulative frequency |
|---|---|---|
| 11 - 20 | 68 | 68 |
| 21 - 30 | 184 | 252 |
| 31 - 40 | 294 | 546 |
| 41 - 50 | 402 | 948 |
| 51 - 60 | 480 | 1428 |
| 61 - 70 | 310 | 1738 |
| 71 - 80 | 164 | 1902 |
| 81 - 90 | 98 | 2000 |
Using upper class boundaries, plot the cumulative frequencies and join the points with a smooth increasing curve.
(b)(i) Cut-off mark
The 300 candidates admitted are the highest scorers. Hence the number below the cut-off mark is
\[2000-300=1700.\]
From the ogive, at cumulative frequency \(1700\), the corresponding mark is approximately \(68\).
Therefore, the cut-off mark is 68 marks (68%).
(b)(ii) Probability of scoring at least \(45\%\)
From the ogive, the cumulative frequency at \(45\%\) is approximately \(720\).
\[\text{Number scoring at least }45\%=2000-720=1280.\]
\[P(\text{score at least }45\%)=\frac{1280}{2000}=0.64.\]
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