A-LevelMathematicsProbability and statistics9709/53 · Oct/Nov 2022 · Q37 Marks

The times, t minutes, taken to complete a walking challenge by 250 members of a club are summarised in the table. Time taken (t minutes) t≤20 t≤30 t≤35 t≤40 t≤50 t ≤ 60 Cumulative frequency 32 66 112 178 228 250

(a)Draw a cumulative frequency graph to illustrate the data.[2]
(b)Use your graph to estimate the 60th percentile of the data.[1]
(c)Calculate an estimate for the standard deviation of the times taken to complete the challenge by these 250 members.[4]

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Mark scheme

Marking points from the official mark scheme (Oct/Nov 2022). The mark or mark type is shown in brackets.

[M1] At least 3 points plotted accurately at class upper end points: (20,32), (30, 66), (35, 112), (40, 178), (50, 228), (60, 250). Linear cf scale 0 < cf < 250 and linear time scale 0 < time < 60 with at least 3 values identified on each.
[A1] All points plotted correct, curve drawn (within tolerance) and joined to (0,0). Axes labelled cumulative frequency (cf), time (t) and minutes (min or m) – or a suitable title. Axes can be the other way round.
[B1 FT] Must be an increasing cf graph with correct upper bounds. Use of graph must be seen. Expect an answer in range 37 ≤ t ≤ 39 for a correct graph
[B1] [Frequencies] 32, 34, 46, 66, 50, 22 May be unsimplified and/or in variance calculation.
[M1] At least 5 correct midpoints seen, may be unsimplified.
[M1] Correct unsimplified Variance formula with their midpoints and their frequencies for var or sd. (- mean² included)
[A1] Awrt WWW SC B1 for 12.3 if second M1 not awarded.

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