A-LevelMathematicsProbability and statistics9709/52 · May/June 2025 · Q510 Marks

The times taken, t minutes, by 300 students to travel to Hollowton College are recorded. The results are summarised in the table below. Time (t minutes) | t ≤ 10 | t ≤ 20 | t ≤ 30 | t ≤ 40 | t ≤ 60 | t ≤ 90 Cumulative frequency | 34 | 86 | 142 | 208 | 265 | 300

(a)On the grid, draw a cumulative frequency graph to illustrate this information.[2]
(b)120 students take more than k minutes to travel to college. Use your graph to estimate the value of k.[2]
(c)Calculate estimates of the mean and standard deviation of the times taken to travel to college by the 300 students.[6]

This question uses a diagram from the original paper, which isn't shown here. Open the full paper from the past papers archive to see it.

Mark scheme

Marking points from the official mark scheme (May/June 2025). The mark or mark type is shown in brackets.

[B1] At least 5 points plotted accurately at class upper end points: (10, 34), (20, 86), (30, 142), (40, 208), (60, 265), (90, 300). Linear cf scale 0 < cf < 300 and linear time scale 0 < time < 90 with at least 3 values identified on each. Bar/histograms score B0. Required axes must be over 50% of grid size. Axes can be the other way round.
[B1] All points plotted correct, curve drawn, no line segments and joined to (0,0) passing within ½ square of points. Axes labelled cumulative frequency (cf), time (t) and minutes (min or m) or a suitable title. Curve must be <300 for 0 < t < 90.
[M1] Use of graph must be seen. Must be an increasing graph. Annotate this mark on 5(a) grid.
[A1] FT Strict FT, reading at their curve condone use of t. [k=] 35 minutes
[B1] Frequencies: [34,] 52, 56, 66, 57, 35. May be un-simplified and/or in variance calculation.
[B1] Midpoints: 5, 15, 25, 35, 50, 75. At least 5 correct midpoints seen, may be un-simplified, may be in calculation, may be by data table.
[M1] Correct un-simplified mean formula with their 6 midpoints (not upper bound, lower bound, upper limits, lower limits, class width, frequency density, frequency or cumulative frequency and must be within class) and their frequencies (not cw, cf, or fd), accept un-simplified. Condone 1 value error on numerator. The 'table' approach may be used with multiplications and additions implied by appropriate values.
[A1] WWW. 33.78 ≤ mean < 33.8, 2027/60 , 47/60. If M not scored, SCB1 for 2027/60 , 47/60, 33.78 WWW. (Note: 10135/300 scores A0).
[M1] Correct un-simplified Variance formula with their midpoints and their frequencies – their mean² for variance or standard deviation. Condone one value error on numerator. Condone their Σf from denominator of mean calculation if not 300.
[A1] 20.4 ≤ σ < 20.45 WWW. At least one of the mean or the standard deviation must be linked to the value, and no incorrect identifications, for this mark to be scored.