A-LevelMathematicsProbability and statistics9709/52 · Feb/Mar 2020 · Q79 Marks

Helen measures the lengths of 150 fish of a certain species in a large pond. These lengths, correct to the nearest centimetre, are summarised in the following table. Length (cm) | 0–9 | 10–14 | 15–19 | 20–30 Frequency | 15 | 48 | 66 | 21

(a)Draw a cumulative frequency graph to illustrate the data.[4]
(b)40% of these fish have a length of d cm or more. Use your graph to estimate the value of d.[2]
(c)The mean length of these 150 fish is 15.295 cm. Calculate an estimate for the variance of the lengths of the fish.[3]

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Mark scheme

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

B1 Correct cumulative frequencies seen (may be on graph)
B1 0 ≤ Horizontal axis < 30, 0 ≤ vertical axis < 150 Labels correct: length cm, cf
M1 At least 3 points plotted at upper end points (e.g. allow 9, 9.5, 10) with a linear horizontal scale.
A1 Linear vertical scale, all points at correct upper end points (9.5 etc.), curve drawn accurately, joined to (0,0) (condone (-0.5, 0))
M1 90 seen or implied by use on graph
A1FT FT their increasing cumulative frequency graph, Use of graph must be seen. If no clear evidence of use of graph SCB1FT correct value from their graph
M1 At least 3 correct midpoints used (39449.4375 implies M1)
M1 Using midpoints ±0.5 in correct var formula, including subtraction of their μ².
A1

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