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Learn/ A Level Mathematics/ Past Papers/ 2020/ Feb/March 2020 /Paper 6 (Variant 2)

Probability & Statistics 2 — Variant 2

A Level Mathematics · 9709/62 · Feb/March 2020

Question Paper

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Questions

4 of 7 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.

3
In the past, the mean time taken by Freda for a particular daily journey was 39.239.2 minutes. Following the introduction of a one-way system, Freda wishes to test whether the mean time for the journey has decreased. She notes the times, tt minutes, for 4040 randomly chosen journeys and summarises the results as follows.\n\nn=40n = 40 Σt=1504\Sigma t = 1504 Σt2=57760\Sigma t^{2} = 57760
3(a) [3]
Calculate unbiased estimates of the population mean and variance of the new journey time.
3(b) [5]
Test, at the 5%5\% significance level, whether the population mean time has decreased.
5
Bottles of Lanta contain approximately 300300 ml of juice. The volume of juice, in millilitres, in a bottle is 300+X300 + X, where XX is a random variable with probability density function given by\n\nf(x)={34000(100−x2)−10⩽x⩽10,0otherwise.\mathrm{f}(x) = \begin{cases} \frac{3}{4000}(100 - x^{2}) & -10 \leqslant x \leqslant 10, \\ 0 & \text{otherwise}. \end{cases}
5(a) [3]
Find the probability that a randomly chosen bottle of Lanta contains more than 305305 ml of juice.
5(b) [4]
Given that 25%25\% of bottles of Lanta contain more than (300+p)(300 + p) ml of juice, show that\n\np3−300p+1000=0p^{3} - 300p + 1000 = 0.
5(c) [2]
Given that p=3.47p = 3.47, and that 50%50\% of bottles of Lanta contain between (300−q)(300 - q) and (300+q)(300 + q) ml of juice, find qq. Justify your answer.
6
The volumes, in millilitres, of large and small cups of tea are modelled by the distributions N(200,30)\mathrm{N}(200, 30) and N(110,20)\mathrm{N}(110, 20) respectively.
6(a) [4]
Find the probability that the total volume of a randomly chosen large cup of tea and a randomly chosen small cup of tea is less than 300300 ml.
6(b) [6]
Find the probability that the volume of a randomly chosen large cup of tea is more than twice the volume of a randomly chosen small cup of tea.
7
A national survey shows that 95%95\% of year 1212 students use social media. Arvin suspects that the percentage of year 1212 students at his college who use social media is less than the national percentage. He chooses a random sample of 2020 students at his college and notes the number who use social media. He then carries out a test at the 2%2\% significance level.
7(a) [4]
Find the rejection region for the test.
7(b) [1]
Find the probability of a Type I error.
7(c) [3]
Jimmy believes that the true percentage at Arvin's college is 70%70\%. Assuming that Jimmy is correct, find the probability of a Type II error.

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