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 2020 March Q4
4 The number of accidents on a certain road has a Poisson distribution with mean 0.4 per 50-day period.
  1. Find the probability that there will be fewer than 3 accidents during a year (365 days).
  2. The probability that there will be no accidents during a period of \(n\) days is greater than 0.95 . Find the largest possible value of \(n\).
CAIE S2 2020 March Q5
5 Bottles of Lanta contain approximately 300 ml of juice. The volume of juice, in millilitres, in a bottle is \(300 + X\), where \(X\) is a random variable with probability density function given by $$f ( x ) = \begin{cases} \frac { 3 } { 4000 } \left( 100 - x ^ { 2 } \right) & - 10 \leqslant x \leqslant 10
0 & \text { otherwise } \end{cases}$$
  1. Find the probability that a randomly chosen bottle of Lanta contains more than 305 ml of juice.
  2. Given that \(25 \%\) of bottles of Lanta contain more than \(( 300 + p ) \mathrm { ml }\) of juice, show that $$p ^ { 3 } - 300 p + 1000 = 0 .$$
  3. Given that \(p = 3.47\), and that \(50 \%\) of bottles of Lanta contain between ( \(300 - q\) ) and ( \(300 + q\) ) ml of juice, find \(q\). Justify your answer.
CAIE S2 2020 March Q6
6 The volumes, in millilitres, of large and small cups of tea are modelled by the distributions \(\mathrm { N } ( 200,30 )\) and \(\mathrm { N } ( 110,20 )\) respectively.
  1. 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 300 ml .
  2. 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.
CAIE S2 2020 March Q7
7 A national survey shows that \(95 \%\) of year 12 students use social media. Arvin suspects that the percentage of year 12 students at his college who use social media is less than the national percentage. He chooses a random sample of 20 students at his college and notes the number who use social media. He then carries out a test at the \(2 \%\) significance level.
  1. Find the rejection region for the test.
  2. Find the probability of a Type I error.
  3. Jimmy believes that the true percentage at Arvin's college is \(70 \%\). Assuming that Jimmy is correct, find the probability of a Type II error.
    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 2021 March Q1
1 A construction company notes the time, \(t\) days, that it takes to build each house of a certain design. The results for a random sample of 60 such houses are summarised as follows. $$\Sigma t = 4820 \quad \Sigma t ^ { 2 } = 392050$$
  1. Calculate a 98\% confidence interval for the population mean time.
  2. Explain why it was necessary to use the Central Limit theorem in part (a).
CAIE S2 2021 March Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{image-not-found} The diagram shows the graph of the probability density function, f , of a random variable \(X\).
  1. Find the value of the constant \(k\).
  2. Using this value of \(k\), find \(\mathrm { f } ( x )\) for \(0 \leqslant x \leqslant k\) and hence find \(\mathrm { E } ( X )\).
  3. Find the value of \(p\) such that \(\mathrm { P } ( p < X < 1 ) = 0.25\).
CAIE S2 2021 March Q3
3 An architect wishes to investigate whether the buildings in a certain city are higher, on average, than buildings in other cities. He takes a large random sample of buildings from the city and finds the mean height of the buildings in the sample. He calculates the value of the test statistic, \(z\), and finds that \(z = 2.41\).
  1. Explain briefly whether he should use a one-tail test or a two-tail test.
  2. Carry out the test at the \(1 \%\) significance level.
CAIE S2 2021 March Q4
4 On average, 1 in 400 microchips made at a certain factory are faulty. The number of faulty microchips in a random sample of 1000 is denoted by \(X\).
  1. State the distribution of \(X\), giving the values of any parameters.
  2. State an approximating distribution for \(X\), giving the values of any parameters.
  3. Use this approximating distribution to find each of the following.
    1. \(\mathrm { P } ( X = 4 )\).
    2. \(\mathrm { P } ( 2 \leqslant X \leqslant 4 )\).
  4. Use a suitable approximating distribution to find the probability that, in a random sample of 700 microchips, there will be at least 1 faulty one.
CAIE S2 2021 March Q5
5 The volumes, in litres, of juice in large and small bottles have the distributions \(\mathrm { N } ( 5.10,0.0102 )\) and \(\mathrm { N } ( 2.51,0.0036 )\) respectively.
  1. Find the probability that the total volume of juice in 3 randomly chosen large bottles and 4 randomly chosen small bottles is less than 25.5 litres.
  2. Find the probability that the volume of juice in a randomly chosen large bottle is at least twice the volume of juice in a randomly chosen small bottle.
CAIE S2 2021 March Q6
6 It is known that \(8 \%\) of adults in a certain town own a Chantor car. After an advertising campaign, a car dealer wishes to investigate whether this proportion has increased. He chooses a random sample of 25 adults from the town and notes how many of them own a Chantor car.
  1. He finds that 4 of the 25 adults own a Chantor car. Carry out a hypothesis test at the 5\% significance level.
  2. Explain which of the errors, Type I or Type II, might have been made in carrying out the test in part (a).
    Later, the car dealer takes another random sample of 25 adults from the town and carries out a similar hypothesis test at the 5\% significance level.
  3. Find the probability of a Type I error.
    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 2022 March Q1
1 The lengths, in millimetres, of a random sample of 12 rods made by a certain machine are as follows.
200
201
198
202
200
199
199
201
197
202
200
199
  1. Find unbiased estimates of the population mean and variance.
  2. Give a statistical reason why these estimates may not be reliable.
CAIE S2 2022 March Q2
2 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. $$\begin{aligned} & \mathrm { H } _ { 0 } : \mathrm { P } ( \text { the spinner lands on blue } ) = \frac { 1 } { 5 }
& \mathrm { H } _ { 1 } : \mathrm { P } ( \text { the spinner lands on blue } ) \neq \frac { 1 } { 5 } \end{aligned}$$ Harry spins the spinner 300 times. It lands on blue on 45 spins.
Use a suitable approximation to carry out Harry's test at the \(5 \%\) significance level.
CAIE S2 2022 March Q3
3 A random sample of 500 households in a certain town was chosen. Using this sample, a confidence interval for the proportion, \(p\), of all households in that town that owned two or more cars was found to be \(0.355 < p < 0.445\). Find the confidence level of this confidence interval. Give your answer correct to the nearest integer.
CAIE S2 2022 March Q4
4 In the past the time, in minutes, taken by students to complete a certain challenge had mean 25.5 and standard deviation 5.2. A new challenge is devised and it is expected that students will take, on average, less than 25.5 minutes to complete this challenge. A random sample of 40 students is chosen and their mean time for the new challenge is found to be 23.7 minutes.
  1. Assuming that the standard deviation of the time for the new challenge is 5.2 minutes, test at the \(1 \%\) significance level whether the population mean time for the new challenge is less than 25.5 minutes.
  2. State, with a reason, whether it is possible that a Type I error was made in the test in part (a).
CAIE S2 2022 March Q5
5 The heights of buildings in a large city are normally distributed with mean 18.3 m and standard deviation 2.5 m .
  1. Find the probability that the total height of 5 randomly chosen buildings in the city is more than 95 m .
  2. Find the probability that the difference between the heights of two randomly chosen buildings in the city is less than 1 m .
CAIE S2 2022 March Q6
6 In a game a ball is rolled down a slope and along a track until it stops. The distance, in metres, travelled by the ball is modelled by the random variable \(X\) with probability density function $$f ( x ) = \begin{cases} - k ( x - 1 ) ( x - 3 ) & 1 \leqslant x \leqslant 3
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Without calculation, explain why \(\mathrm { E } ( X ) = 2\).
  2. Show that \(k = \frac { 3 } { 4 }\).
  3. Find \(\operatorname { Var } ( X )\).
    One turn consists of rolling the ball 3 times and noting the largest value of \(X\) obtained. If this largest value is greater than 2.5, the player scores a point.
  4. Find the probability that on a particular turn the player scores a point.
CAIE S2 2022 March Q7
7
  1. Two ponds, \(A\) and \(B\), each contain a large number of fish. It is known that \(2.4 \%\) of fish in pond \(A\) are carp and \(1.8 \%\) of fish in pond \(B\) are carp. Random samples of 50 fish from pond \(A\) and 60 fish from pond \(B\) are selected. Use appropriate Poisson approximations to find the following probabilities.
    1. The samples contain at least 2 carp from pond \(A\) and at least 2 carp from pond \(B\).
    2. The samples contain at least 4 carp altogether.
  2. The random variables \(X\) and \(Y\) have the distributions \(\operatorname { Po } ( \lambda )\) and \(\operatorname { Po } ( \mu )\) respectively. It is given that
    • \(\mathrm { P } ( X = 0 ) = [ \mathrm { P } ( Y = 0 ) ] ^ { 2 }\),
    • \(\mathrm { P } ( X = 2 ) = k [ \mathrm { P } ( Y = 1 ) ] ^ { 2 }\), where \(k\) is a non-zero constant.
    Find the value of \(k\).
    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 2014 June Q1
1 The weights, in grams, of a random sample of 8 packets of cereal are as follows. $$\begin{array} { l l l l l l l l } 250 & 248 & 255 & 244 & 259 & 250 & 242 & 258 \end{array}$$ Calculate unbiased estimates of the population mean and variance.
CAIE S2 2014 June Q2
5 marks
2 Each day Samuel travels from \(A\) to \(B\) and from \(B\) to \(C\). He then returns directly from \(C\) to \(A\). The times, in minutes, for these three journeys have the independent distributions \(\mathrm { N } \left( 20,2 ^ { 2 } \right) , \mathrm { N } \left( 18,1.5 ^ { 2 } \right)\) and \(\mathrm { N } \left( 30,1.8 ^ { 2 } \right)\), respectively. Find the probability that, on a randomly chosen day, the total time for his two journeys from \(A\) to \(B\) and \(B\) to \(C\) is less than the time for his return journey from \(C\) to \(A\). [5]
CAIE S2 2014 June Q3
3 The number of calls per day to an enquiry desk has a Poisson distribution. In the past the mean has been 5 . In order to test whether the mean has changed, the number of calls on a random sample of 10 days was recorded. The total number of calls was found to be 61 . Use an approximate distribution to test at the 10\% significance level whether the mean has changed.
CAIE S2 2014 June Q4
4
  1. The random variable \(W\) has the distribution \(\operatorname { Po } ( 1.5 )\). Find the probability that the sum of 3 independent values of \(W\) is greater than 2 .
  2. The random variable \(X\) has the distribution \(\operatorname { Po } ( \lambda )\). Given that \(\mathrm { P } ( X = 0 ) = 0.523\), find the value of \(\lambda\) correct to 3 significant figures.
  3. The random variable \(Y\) has the distribution \(\operatorname { Po } ( \mu )\), where \(\mu \neq 0\). Given that $$\mathrm { P } ( Y = 3 ) = 24 \times \mathrm { P } ( Y = 1 )$$ find \(\mu\).
CAIE S2 2014 June Q5
5 Mahmoud throws a coin 400 times and finds that it shows heads 184 times. The probability that the coin shows heads on any throw is denoted by \(p\).
  1. Calculate an approximate \(95 \%\) confidence interval for \(p\).
  2. Mahmoud claims that the coin is not fair. Use your answer to part (i) to comment on this claim.
  3. Mahmoud's result of 184 heads in 400 throws gives an \(\alpha \%\) confidence interval for \(p\) with width 0.1 . Calculate the value of \(\alpha\).
CAIE S2 2014 June Q6
6 The time, \(T\) hours, spent by people on a visit to a museum has probability density function $$\mathrm { f } ( t ) = \begin{cases} k t \left( 16 - t ^ { 2 } \right) & 0 \leqslant t \leqslant 4
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 1 } { 64 }\).
  2. Calculate the probability that two randomly chosen people each spend less than 1 hour on a visit to the museum.
  3. Find the mean time spent on a visit to the museum.
CAIE S2 2014 June Q7
7 A researcher is investigating the actual lengths of time that patients spend with the doctor at their appointments. He plans to choose a sample of 12 appointments on a particular day.
  1. Which of the following methods is preferable, and why?
    • Choose the first 12 appointments of the day.
    • Choose 12 appointments evenly spaced throughout the day.
    Appointments are scheduled to last 10 minutes. The actual lengths of time, in minutes, that patients spend with the doctor may be assumed to have a normal distribution with mean \(\mu\) and standard deviation 3.4. The researcher suspects that the actual time spent is more than 10 minutes on average. To test this suspicion, he recorded the actual times spent for a random sample of 12 appointments and carried out a hypothesis test at the 1\% significance level.
  2. State the probability of making a Type I error and explain what is meant by a Type I error in this context.
  3. Given that the total length of time spent for the 12 appointments was 147 minutes, carry out the test.
  4. Give a reason why the Central Limit theorem was not needed in part (iii).
CAIE S2 2015 June Q1
1 Jyothi wishes to choose a representative sample of 5 students from the 82 members of her school year.
  1. She considers going into the canteen and choosing a table with five students from her year sitting at it, and using these five people as her sample. Give two reasons why this method is unsatisfactory.
  2. Jyothi decides to use another method. She numbers all the students in her year from 1 to 82 . Then she uses her calculator and generates the following random numbers. $$231492 \quad 762305 \quad 346280$$ From these numbers, she obtains the student numbers \(23,14,76,5,34\) and 62 . Explain how Jyothi obtained these student numbers from the list of random numbers.