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Learn/ A Level Mathematics/ Past Papers/ 2020/ May/June 2020 /Paper 6 (Variant 1)

Probability & Statistics 2 — Variant 1

A Level Mathematics · 9709/61 · May/June 2020

Question Paper

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Questions

4 of 6 questions shown here, checked against the paper. The others are in the PDF above — they contain diagrams or maths we could not yet verify, and we would rather show you the original than a transcription we are not sure of.

2
In the past the yield of a certain crop, in tonnes per hectare, had mean 0.560.56 and standard deviation 0.080.08. Following the introduction of a new fertilizer, the farmer intends to test at the 2.5%2.5\% significance level whether the mean yield has increased. He finds that the mean yield over 1010 years is 0.610.61 tonnes per hectare.
2(a) [2]
State two assumptions that are necessary for the test.
2(b) [5]
Carry out the test.
3
The masses, in kilograms, of large sacks of flour and small sacks of flour have the independent distributions N(40,1.52)\mathrm{N}(40, 1.5^{2}) and N(12,0.72)\mathrm{N}(12, 0.7^{2}) respectively.
3(a) [4]
Find the probability that the total mass of 66 randomly chosen large sacks of flour is more than 245245 kg.
3(b) [6]
Find the probability that the mass of a randomly chosen large sack of flour is less than 44 times the mass of a randomly chosen small sack of flour.
5
Each week a sports team plays one home match and one away match. In their home matches they score goals at a constant average rate of 2.12.1 goals per match. In their away matches they score goals at a constant average rate of 0.80.8 goals per match. You may assume that goals are scored at random times and independently of one another.
5(a)
A week is chosen at random.
5(a)(i) [2]
Find the probability that the team scores a total of 44 goals in their two matches.
5(a)(ii) [3]
Find the probability that the team scores a total of 44 goals, with more goals scored in the home match than in the away match.
5(b) [4]
Use a suitable approximating distribution to find the probability that the team scores fewer than 2525 goals in 1010 randomly chosen weeks.
5(c) [1]
Justify the use of the approximating distribution used in part (b).
6
The length of time, TT minutes, that a passenger has to wait for a bus at a certain bus stop is modelled by the probability density function given by f(t)={34000(20t−t2)0⩽t⩽20,0otherwise.f(t) = \begin{cases} \frac{3}{4000}(20t - t^{2}) & 0 \leqslant t \leqslant 20, \\ 0 & \text{otherwise.} \end{cases}
6(a) [1]
Sketch the graph of y=f(t)y = f(t).
6(b) [1]
Hence explain, without calculation, why E(T)=10\mathrm{E}(T) = 10.
6(c) [3]
Find Var(T)\mathrm{Var}(T).
6(d) [2]
It is given that P(T<10+a)=p\mathrm{P}(T < 10 + a) = p, where 0<a<100 < a < 10. Find P(10−a<T<10+a)\mathrm{P}(10 - a < T < 10 + a) in terms of pp.
6(e) [3]
Find P(8<T<12)\mathrm{P}(8 < T < 12).
6(f) [1]
Give one reason why this model may be unrealistic.

Published by Cambridge Assessment International Education. Reproduced for study use.