Questions — CAIE S2 (717 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
CAIE S2 2015 June Q4
4 In the past, the time taken by vehicles to drive along a particular stretch of road has had mean 12.4 minutes and standard deviation 2.1 minutes. Some new signs are installed and it is expected that the mean time will increase. In order to test whether this is the case, the mean time for a random sample of 50 vehicles is found. You may assume that the standard deviation is unchanged.
  1. The mean time for the sample of 50 vehicles is found to be 12.9 minutes. Test at the \(2.5 \%\) significance level whether the population mean time has increased.
  2. State what is meant by a Type II error in this context.
  3. State what extra piece of information would be needed in order to find the probability of a Type II error.
CAIE S2 2015 June Q5
5 The masses, \(m\) grams, of a random sample of 80 strawberries of a certain type were measured and summarised as follows. $$n = 80 \quad \Sigma m = 4200 \quad \Sigma m ^ { 2 } = 229000$$
  1. Find unbiased estimates of the population mean and variance.
  2. Calculate a 98\% confidence interval for the population mean. 50 random samples of size 80 were taken and a \(98 \%\) confidence interval for the population mean, \(\mu\), was found from each sample.
  3. Find the number of these 50 confidence intervals that would be expected to include the true value of \(\mu\).
CAIE S2 2015 June Q6
6 A publishing firm has found that errors in the first draft of a new book occur at random and that, on average, there is 1 error in every 3 pages of a first draft. Find the probability that in a particular first draft there are
  1. exactly 2 errors in 10 pages,
  2. at least 3 errors in 6 pages,
  3. fewer than 50 errors in 200 pages.
CAIE S2 2015 June Q7
7 The independent variables \(X\) and \(Y\) are such that \(X \sim \mathrm {~B} ( 10,0.8 )\) and \(Y \sim \mathrm { Po } ( 3 )\). Find
  1. \(\mathrm { E } ( 7 X + 5 Y - 2 )\),
  2. \(\operatorname { Var } ( 4 X - 3 Y + 3 )\),
  3. \(\mathrm { P } ( 2 X - Y = 18 )\).
CAIE S2 2015 June Q1
1 The independent random variables \(X\) and \(Y\) have standard deviations 3 and 6 respectively. Calculate the standard deviation of \(4 X - 5 Y\).
CAIE S2 2015 June Q2
2 Cloth made at a certain factory has been found to have an average of 0.1 faults per square metre. Suki claims that the cloth made by her machine contains, on average, more than 0.1 faults per square metre. In a random sample of \(5 \mathrm {~m} ^ { 2 }\) of cloth from Suki's machine, it was found that there were 2 faults. Assuming that the number of faults per square metre has a Poisson distribution,
  1. state null and alternative hypotheses for a test of Suki's claim,
  2. test at the \(10 \%\) significance level whether Suki's claim is justified.
CAIE S2 2015 June Q3
3 In a golf tournament, the number of times in a day that a 'hole-in-one' is scored is denoted by the variable \(X\), which has a Poisson distribution with mean 0.15 . Mr Crump offers to pay \(
) 200$ each time that a hole-in-one is scored during 5 days of play. Find the expectation and variance of the amount that Mr Crump pays.
CAIE S2 2015 June Q4
4 In the past, the flight time, in hours, for a particular flight has had mean 6.20 and standard deviation 0.80 . Some new regulations are introduced. In order to test whether these new regulations have had any effect upon flight times, the mean flight time for a random sample of 40 of these flights is found.
  1. State what is meant by a Type I error in this context.
  2. The mean time for the sample of 40 flights is found to be 5.98 hours. Assuming that the standard deviation of flight times is still 0.80 hours, test at the \(5 \%\) significance level whether the population mean flight time has changed.
  3. State, with a reason, which of the errors, Type I or Type II, might have been made in your answer to part (ii).
CAIE S2 2015 June Q5
5 The volumes, \(v\) millilitres, of juice in a random sample of 50 bottles of Cooljoos are measured and summarised as follows. $$n = 50 \quad \Sigma v = 14800 \quad \Sigma v ^ { 2 } = 4390000$$
  1. Find unbiased estimates of the population mean and variance.
  2. An \(\alpha \%\) confidence interval for the population mean, based on this sample, is found to have a width of 5.45 millilitres. Find \(\alpha\). Four random samples of size 10 are taken and a \(96 \%\) confidence interval for the population mean is found from each sample.
  3. Find the probability that these 4 confidence intervals all include the true value of the population mean.
CAIE S2 2015 June Q6
6 The waiting time, \(T\) minutes, for patients at a doctor's surgery has probability density function given by $$\mathrm { f } ( t ) = \begin{cases} k \left( 225 - t ^ { 2 } \right) & 0 \leqslant t \leqslant 15
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 1 } { 2250 }\).
  2. Find the probability that a patient has to wait for more than 10 minutes.
  3. Find the mean waiting time.
CAIE S2 2015 June Q7
7 In a certain lottery, 10500 tickets have been sold altogether and each ticket has a probability of 0.0002 of winning a prize. The random variable \(X\) denotes the number of prize-winning tickets that have been sold.
  1. State, with a justification, an approximating distribution for \(X\).
  2. Use your approximating distribution to find \(\mathrm { P } ( X < 4 )\).
  3. Use your approximating distribution to find the conditional probability that \(X < 4\), given that \(X \geqslant 1\).
CAIE S2 2016 June Q1
1 The time taken for a particular type of paint to dry was measured for a sample of 150 randomly chosen points on a wall. The sample mean was 192.4 minutes and an unbiased estimate of the population variance was 43.6 minutes \({ } ^ { 2 }\). Find a \(98 \%\) confidence interval for the mean drying time.
CAIE S2 2016 June Q2
2 In the past, the mean annual crop yield from a particular field has been 8.2 tonnes. During the last 16 years, a new fertiliser has been used on the field. The mean yield for these 16 years is 8.7 tonnes. Assume that yields are normally distributed with standard deviation 1.2 tonnes. Carry out a test at the \(5 \%\) significance level of whether the mean yield has increased.
\(31 \%\) of adults in a certain country own a yellow car.
  1. Use a suitable approximating distribution to find the probability that a random sample of 240 adults includes more than 2 who own a yellow car.
  2. Justify your approximation.
CAIE S2 2016 June Q4
4 The number of sightings of a golden eagle at a certain location has a Poisson distribution with mean 2.5 per week. Drilling for oil is started nearby. A naturalist wishes to test at the \(5 \%\) significance level whether there are fewer sightings since the drilling began. He notes that during the following 3 weeks there are 2 sightings.
  1. Find the critical region for the test and carry out the test.
  2. State the probability of a Type I error.
  3. State why the naturalist could not have made a Type II error.
CAIE S2 2016 June Q5
3 marks
5 The time, \(T\) minutes, taken by people to complete a test has probability density function given by $$\mathrm { f } ( t ) = \begin{cases} k \left( 10 t - t ^ { 2 } \right) & 5 \leqslant t \leqslant 10
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 3 } { 250 }\).
  2. Find \(\mathrm { E } ( T )\).
  3. Find the probability that a randomly chosen value of \(T\) lies between \(\mathrm { E } ( T )\) and the median of \(T\). [3]
  4. State the greatest possible length of time taken to complete the test.
    \(6 X\) and \(Y\) are independent random variables with distributions \(\operatorname { Po } ( 1.6 )\) and \(\operatorname { Po } ( 2.3 )\) respectively.
CAIE S2 2016 June Q7
7 Bags of sugar are packed in boxes, each box containing 20 bags. The masses of the boxes, when empty, are normally distributed with mean 0.4 kg and standard deviation 0.01 kg . The masses of the bags are normally distributed with mean 1.02 kg and standard deviation 0.03 kg .
  1. Find the probability that the total mass of a full box of 20 bags is less than 20.6 kg .
  2. Two full boxes are chosen at random. Find the probability that they differ in mass by less than 0.02 kg .
CAIE S2 2018 June Q1
1 A random sample of 75 values of a variable \(X\) gave the following results. $$n = 75 \quad \Sigma x = 153.2 \quad \Sigma x ^ { 2 } = 340.24$$ Find unbiased estimates for the population mean and variance of \(X\).
CAIE S2 2018 June Q2
2 A six-sided die is suspected of bias. The die is thrown 100 times and it is found that the score is 2 on 20 throws. It is given that the probability of obtaining a score of 2 on any throw is \(p\).
  1. Find an approximate \(94 \%\) confidence interval for \(p\).
  2. Use your answer to part (i) to comment on whether the die may be biased.
CAIE S2 2018 June Q3
3 The number of e-readers sold in a 10-day period in a shop is modelled by the distribution \(\operatorname { Po } ( 5.1 )\). Use an approximating distribution to find the probability that fewer than 140 e-readers are sold in a 300-day period.
CAIE S2 2018 June Q4
4 The volume, in millilitres, of a small cup of coffee has the distribution \(\mathrm { N } ( 103.4,10.2 )\). The volume of a large cup of coffee is 1.5 times the volume of a small cup of coffee.
  1. Find the mean and standard deviation of the volume of a large cup of coffee.
  2. Find the probability that the total volume of a randomly chosen small cup of coffee and a randomly chosen large cup of coffee is greater than 250 ml .
CAIE S2 2018 June Q5
5 The mass, in kilograms, of rocks in a certain area has mean 14.2 and standard deviation 3.1.
  1. Find the probability that the mean mass of a random sample of 50 of these rocks is less than 14.0 kg .
  2. Explain whether it was necessary to assume that the population of the masses of these rocks is normally distributed.
  3. A geologist suspects that rocks in another area have a mean mass which is less than 14.2 kg . A random sample of 100 rocks in this area has sample mean 13.5 kg . Assuming that the standard deviation for rocks in this area is also 3.1 kg , test at the \(2 \%\) significance level whether the geologist is correct.
CAIE S2 2018 June Q6
6 The time, in minutes, taken by people to complete a test is modelled by the continuous random variable \(X\) with probability density function given by $$f ( x ) = \begin{cases} \frac { k } { x ^ { 2 } } & 5 \leqslant x \leqslant 10
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = 10\).
  2. Show that \(\mathrm { E } ( X ) = 10 \ln 2\).
  3. Find \(\mathrm { P } ( X > 9 )\).
  4. Given that \(\mathrm { P } ( X < a ) = 0.6\), find \(a\).
CAIE S2 2018 June Q7
7 The number of absences by girls from a certain class on any day is modelled by a random variable with distribution \(\operatorname { Po } ( 0.2 )\). The number of absences by boys from the same class on any day is modelled by an independent random variable with distribution \(\operatorname { Po } ( 0.3 )\).
  1. Find the probability that, during a randomly chosen 2-day period, the total number of absences is less than 3 .
  2. Find the probability that, during a randomly chosen 5-day period, the number of absences by boys is more than 3.
  3. The teacher claims that, during the football season, there are more absences by boys than usual. In order to test this claim at the 5\% significance level, he notes the number of absences by boys during a randomly chosen 5-day period during the football season.
    (a) State what is meant by a Type I error in this context.
    (b) State appropriate null and alternative hypotheses and find the probability of a Type I error.
    (c) In fact there were 4 absences by boys during this period. Test the teacher's claim at the 5\% significance level.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE S2 2018 June Q1
1 The numbers of alpha, beta and gamma particles emitted per minute by a certain piece of rock have independent distributions \(\operatorname { Po } ( 0.2 ) , \operatorname { Po } ( 0.3 )\) and \(\operatorname { Po } ( 0.6 )\) respectively. Find the probability that the total number of particles emitted during a 4 -minute period is less than 4.
CAIE S2 2018 June Q2
2 The random variable \(X\) has the distribution \(\mathrm { N } ( 3,1.2 )\). The random variable \(A\) is defined by \(A = 2 X\). The random variable \(B\) is defined by \(B = X _ { 1 } + X _ { 2 }\), where \(X _ { 1 }\) and \(X _ { 2 }\) are independent random values of \(X\). Describe fully the distribution of \(A\) and the distribution of \(B\). Distribution of \(A\) : \(\_\_\_\_\)
Distribution of \(B\) : \(\_\_\_\_\)