StudentAid
HomeLearn
9th Class10th Class11th Class12th ClassCambridge past papersBrowse everything
Attendance
Class attendanceVisitor register
For schoolsFor driversFor parentsBook a demo
Learn/ A Level Mathematics/ Past Papers/ 2022/ Feb/March 2022 /Paper 6 (Variant 2)

Probability & Statistics 2 — Variant 2

A Level Mathematics · 9709/62 · Feb/March 2022

Question Paper

PDF · 106 KB · 12 pages

Mark Scheme

PDF · 306 KB · 11 pages

This browser cannot show the PDF inline.

Open the question paper · Open the mark scheme

Questions

6 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.

1
The lengths, in millimetres, of a random sample of 1212 rods made by a certain machine are as follows. 200,201,198,202,200,199,199,201,197,202,200,199200, 201, 198, 202, 200, 199, 199, 201, 197, 202, 200, 199
1(a) [3]
Find unbiased estimates of the population mean and variance.
1(b) [1]
Give a statistical reason why these estimates may not be reliable.
2 [5]
Harry has a five-sided spinner with sectors coloured blue, green, red, yellow and black. Harry thinks the spinner may be biased. He plans to carry out a hypothesis test with the following hypotheses. H0:P(the spinner lands on blue)=15H_{0}: P(\text{the spinner lands on blue}) = \frac{1}{5} H1:P(the spinner lands on blue)≠15H_{1}: P(\text{the spinner lands on blue}) \neq \frac{1}{5} Harry spins the spinner 300300 times. It lands on blue on 4545 spins. Use a suitable approximation to carry out Harry's test at the 5%5\% significance level.
3 [5]
A random sample of 500500 households in a certain town was chosen. Using this sample, a confidence interval for the proportion, pp, of all households in that town that owned two or more cars was found to be 0.355<p<0.4450.355 < p < 0.445. Find the confidence level of this confidence interval. Give your answer correct to the nearest integer.
4
In the past the time, in minutes, taken by students to complete a certain challenge had mean 25.525.5 and standard deviation 5.25.2. A new challenge is devised and it is expected that students will take, on average, less than 25.525.5 minutes to complete this challenge. A random sample of 4040 students is chosen and their mean time for the new challenge is found to be 23.723.7 minutes.
4(a) [5]
Assuming that the standard deviation of the time for the new challenge is 5.25.2 minutes, test at the 1%1\% significance level whether the population mean time for the new challenge is less than 25.525.5 minutes.
4(b) [1]
State, with a reason, whether it is possible that a Type I error was made in the test in part (a).
5
The heights of buildings in a large city are normally distributed with mean 18.318.3 m and standard deviation 2.52.5 m.
5(a) [4]
Find the probability that the total height of 55 randomly chosen buildings in the city is more than 9595 m.
5(b) [5]
Find the probability that the difference between the heights of two randomly chosen buildings in the city is less than 11 m.
7
7(a)
Two ponds, AA and BB, each contain a large number of fish. It is known that 2.4%2.4\% of fish in pond AA are carp and 1.8%1.8\% of fish in pond BB are carp. Random samples of 5050 fish from pond AA and 6060 fish from pond BB are selected. Use appropriate Poisson approximations to find the following probabilities.
7(a)(i) [3]
The samples contain at least 22 carp from pond AA and at least 22 carp from pond BB.
7(a)(ii) [3]
The samples contain at least 44 carp altogether.
7(b) [4]
The random variables XX and YY have the distributions Po(λ)\mathrm{Po}(\lambda) and Po(μ)\mathrm{Po}(\mu) respectively. It is given that P(X=0)=[P(Y=0)]2P(X = 0) = [P(Y = 0)]^{2}, P(X=2)=k[P(Y=1)]2P(X = 2) = k[P(Y = 1)]^{2}, where kk is a non-zero constant. Find the value of kk.

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