A-LevelMathematicsProbability and statistics9709/51 · Oct/Nov 2020 · Q610 Marks

The times, t minutes, taken by 150 students to complete a particular challenge are summarised in the following cumulative frequency table. Time taken (t minutes) t ≤ 20 | t ≤ 30 | t ≤ 40 | t ≤ 60 | t ≤ 100 Cumulative frequency 12 | 48 | 106 | 134 | 150

(a)Draw a cumulative frequency graph to illustrate the data.[2]
(b)24% of the students take k minutes or longer to complete the challenge. Use your graph to estimate the value of k.[2]
(c)Calculate estimates of the mean and the standard deviation of the time taken to complete the challenge.[6]

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Mark scheme

Marking points from the official mark scheme (Oct/Nov 2020). The mark or mark type is shown in brackets.

[M1] At least 4 points plotted at upper end points, with both scales linear with at least 3 values indicated
[A1] All plotted correctly with curve drawn joined to (0, 0), axes labelled cumulative frequency, time, minutes
[M1] 114 SOI, may be on graph
[A1FT] Clear indication that their graph has been used, tolerance ±1mm. k = 45 (mins)
[B1] Correct frequencies seen: 12, 36, 58, 28, 16
[B1] At least 4 correct midpoints seen and used: 10, 25, 35, 50, 80
[M1] Correct formula for mean with their midpoints (not upper boundary, lower boundary, class width or frequency density). (e.g., (10×12 + 25×36 + 35×58 + 50×28 + 80×16) / 150)
[A1] Mean = 38.2
[M1] Substitute their midpoints and frequencies (condone use of cumulative frequency) in correct variance formula, must have '– their mean²'. (e.g., (12×10² + 36×25² + 58×35² + 28×50² + 16×80²) / 150 - (mean)²)
[A1] Standard deviation = 17.9 (sqrt(321.76) or similar value based on mean from candidate)

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