Theory question 112
StatisticsThe table shows the distribution of marks scored by some students in a test.
Marks: 1 − 10, 11 − 20, 21 − 30, 31 − 40, 41 − 50, 51 − 60, 61 − 70, 71 − 80, 81 − 90, 91 − 100
Number of students: 3, 17, 41, 85, 97, 115, 101, 64, 21, 6
(a)(i) Construct a cumulative frequency table for the distrbution. (ii) Draw a cumulative frequency curve for the distribution. (b) Use the curve to estimate the: (i) number of students who scored marks between 32 and 74; (ii) pass mark, if 18% of the students failed; (iii) lowest mark for distinction, if 8% of the students passed with distinction.
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(a)(i) Upper boundaries 10.5, 20.5, …, 100.5; cumulative frequencies 3, 20, 61, 146, 243, 358, 459, 523, 544, 550.
(ii) Plot these (start at (0.5, 0)) and join with a smooth curve.
(b)(i) About 74 scored up to 32 and about 481 up to 74: about 407 students.
(ii) 18% of 550 = 99: pass mark ≈ 35.
(iii) Top 8% = 44, so cumulative frequency 506: lowest distinction mark ≈ 78.
(Graph readings may differ slightly.)
Read values off the curve at the required cumulative frequency.