Questions — CAIE S2 (737 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure 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 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
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 PURE 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 PURE S1 S2 S3 S4 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 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 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
Sort by: Default | Easiest first | Hardest first
CAIE S2 2012 November Q2
5 marks Moderate -0.3
2 The heights of a certain type of plant have a normal distribution. When the plants are grown without fertilizer, the population mean and standard deviation are 24.0 cm and 4.8 cm respectively. A gardener wishes to test, at the \(2 \%\) significance level, whether Hiergro fertilizer will increase the mean height. He treats 150 randomly chosen plants with Hiergro and finds that their mean height is 25.0 cm . Assuming that the standard deviation of the heights of plants treated with Hiergro is still 4.8 cm , carry out the test.
CAIE S2 2012 November Q3
6 marks Moderate -0.8
3 The cost of hiring a bicycle consists of a fixed charge of 500 cents together with a charge of 3 cents per minute. The number of minutes for which people hire a bicycle has mean 142 and standard deviation 35.
  1. Find the mean and standard deviation of the amount people pay when hiring a bicycle.
  2. 6 people hire bicycles independently. Find the mean and standard deviation of the total amount paid by all 6 people.
CAIE S2 2012 November Q4
8 marks Moderate -0.3
4 A cereal manufacturer claims that \(25 \%\) of cereal packets contain a free gift. Lola suspects that the true proportion is less than \(25 \%\). In order to test the manufacturer's claim at the \(5 \%\) significance level, she checks a random sample of 20 packets.
  1. Find the critical region for the test.
  2. Hence find the probability of a Type I error. Lola finds that 2 packets in her sample contain a free gift.
  3. State, with a reason, the conclusion she should draw.
CAIE S2 2012 November Q5
8 marks Standard +0.3
5 A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { k } { x - 1 } & 3 \leqslant x \leqslant 5 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 1 } { \ln 2 }\).
  2. Find \(a\) such that \(\mathrm { P } ( X < a ) = 0.75\).
CAIE S2 2012 November Q6
9 marks Moderate -0.3
6 In order to obtain a random sample of people who live in her town, Jane chooses people at random from the telephone directory for her town.
  1. Give a reason why Jane's method will not give a random sample of people who live in the town. Jane now uses a valid method to choose a random sample of 200 people from her town and finds that 38 live in apartments.
  2. Calculate an approximate \(99 \%\) confidence interval for the proportion of all people in Jane's town who live in apartments.
  3. Jane uses the same sample to give a confidence interval of width 0.1 for this proportion. This interval is an \(x \%\) confidence interval. Find the value of \(x\).
CAIE S2 2012 November Q7
11 marks Standard +0.3
7 A random variable \(X\) has the distribution \(\operatorname { Po } ( 1.6 )\).
  1. The random variable \(R\) is the sum of three independent values of \(X\). Find \(\mathrm { P } ( R < 4 )\).
  2. The random variable \(S\) is the sum of \(n\) independent values of \(X\). It is given that $$\mathrm { P } ( S = 4 ) = \frac { 16 } { 3 } \times \mathrm { P } ( S = 2 )$$ Find \(n\).
  3. The random variable \(T\) is the sum of 40 independent values of \(X\). Find \(\mathrm { P } ( T > 75 )\).
CAIE S2 2012 November Q1
3 marks Easy -1.2
1 \includegraphics[max width=\textwidth, alt={}, center]{0cd5fc36-486d-4c24-b809-907b3e87cfd7-2_371_531_255_806} The diagram shows the graph of the probability density function, f , of a random variable \(X\). Find the median of \(X\).
CAIE S2 2012 November Q1
3 marks Moderate -0.5
1 The lengths of logs are normally distributed with mean 3.5 m and standard deviation 0.12 m . Describe fully the distribution of the total length of 8 randomly chosen logs.
CAIE S2 2012 November Q2
6 marks Moderate -0.8
2
  1. A random variable \(X\) has mean \(\mu\) and variance \(\sigma ^ { 2 }\). The mean of a random sample of \(n\) values of \(X\) is denoted by \(\bar { X }\). Give expressions for \(\mathrm { E } ( \bar { X } )\) and \(\operatorname { Var } ( \bar { X } )\).
  2. The heights, in centimetres, of adult males in Brancot are normally distributed with mean 177.8 and standard deviation 6.1. Find the probability that the mean height of a random sample of 12 adult males from Brancot is less than 176 cm .
  3. State, with a reason, whether it was necessary to use the Central Limit Theorem in the calculation in part (ii).
CAIE S2 2012 November Q3
7 marks Moderate -0.3
3 Joshi suspects that a certain die is biased so that the probability of showing a six is less than \(\frac { 1 } { 6 }\). He plans to throw the die 25 times and if it shows a six on fewer than 2 throws, he will conclude that the die is biased in this way.
  1. Find the probability of a Type I error and state the significance level of the test. Joshi now decides to throw the die 100 times. It shows a six on 9 of these throws.
  2. Calculate an approximate \(95 \%\) confidence interval for the probability of showing a six on one throw of this die.
CAIE S2 2012 November Q4
7 marks Challenging +1.8
4 The masses of a certain variety of potato are normally distributed with mean 180 g and variance \(1550 \mathrm {~g} ^ { 2 }\). Two potatoes of this variety are chosen at random. Find the probability that the mass of one of these potatoes is at least twice the mass of the other.
CAIE S2 2012 November Q5
8 marks Standard +0.3
5 It is claimed that, on average, people following the Losefast diet will lose more than 2 kg per month. The weight losses, \(x\) kilograms per month, of a random sample of 200 people following the Losefast diet were recorded and summarised as follows. $$n = 200 \quad \Sigma x = 460 \quad \Sigma x ^ { 2 } = 1636$$
  1. Calculate unbiased estimates of the population mean and variance.
  2. Test the claim at the \(1 \%\) significance level.
CAIE S2 2012 November Q6
9 marks Moderate -0.3
6 Darts are thrown at random at a circular board. The darts hit the board at distances \(X\) centimetres from the centre, where \(X\) is a random variable with probability density function given by $$f ( x ) = \begin{cases} \frac { 2 } { a ^ { 2 } } x & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) is a positive constant.
  1. Verify that f is a probability density function whatever the value of \(a\). It is now given that \(\mathrm { E } ( X ) = 8\).
  2. Find the value of \(a\).
  3. Find the probability that a dart lands more than 6 cm from the centre of the board.
CAIE S2 2012 November Q7
10 marks Standard +0.3
7 The number of workers, \(X\), absent from a factory on a particular day has the distribution \(\mathrm { B } ( 80,0.01 )\).
  1. Explain why it is appropriate to use a Poisson distribution as an approximating distribution for \(X\).
  2. Use the Poisson distribution to find the probability that the number of workers absent during 12 randomly chosen days is more than 2 and less than 6 . Following a change in working conditions, the management wishes to test whether the mean number of workers absent per day has decreased.
  3. During 10 randomly chosen days, there were a total of 2 workers absent. Use the Poisson distribution to carry out the test at the \(2 \%\) significance level.
CAIE S2 2013 November Q1
4 marks Moderate -0.3
1 Each computer made in a factory contains 1000 components. On average, 1 in 30000 of these components is defective. Use a suitable approximate distribution to find the probability that a randomly chosen computer contains at least 1 faulty component.
CAIE S2 2013 November Q2
4 marks Standard +0.3
2 Heights of a certain species of animal are known to be normally distributed with standard deviation 0.17 m . A conservationist wishes to obtain a \(99 \%\) confidence interval for the population mean, with total width less than 0.2 m . Find the smallest sample size required.
CAIE S2 2013 November Q3
8 marks Moderate -0.3
3 Following a change in flight schedules, an airline pilot wished to test whether the mean distance that he flies in a week has changed. He noted the distances, \(x \mathrm {~km}\), that he flew in 50 randomly chosen weeks and summarised the results as follows. $$n = 50 \quad \Sigma x = 143300 \quad \Sigma x ^ { 2 } = 410900000$$
  1. Calculate unbiased estimates of the population mean and variance.
  2. In the past, the mean distance that he flew in a week was 2850 km . Test, at the \(5 \%\) significance level, whether the mean distance has changed.
CAIE S2 2013 November Q4
8 marks Standard +0.8
4 The number of radioactive particles emitted per 150-minute period by some material has a Poisson distribution with mean 0.7.
  1. Find the probability that at most 2 particles will be emitted during a randomly chosen 10 -hour period.
  2. Find, in minutes, the longest time period for which the probability that no particles are emitted is at least 0.99 .
CAIE S2 2013 November Q5
8 marks Standard +0.3
5 The volume, in \(\mathrm { cm } ^ { 3 }\), of liquid left in a glass by people when they have finished drinking all they want is modelled by the random variable \(X\) with probability density function given by $$f ( x ) = \begin{cases} k ( x - 2 ) ^ { 2 } & 0 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 3 } { 8 }\).
  2. 20\% of people leave at least \(d \mathrm {~cm} ^ { 3 }\) of liquid in a glass. Find \(d\).
  3. Find \(\mathrm { E } ( X )\).
CAIE S2 2013 November Q6
8 marks Standard +0.3
6 At the last election, 70\% of people in Apoli supported the president. Luigi believes that the same proportion support the president now. Maria believes that the proportion who support the president now is \(35 \%\). In order to test who is right, they agree on a hypothesis test, taking Luigi's belief as the null hypothesis. They will ask 6 people from Apoli, chosen at random, and if more than 3 support the president they will accept Luigi's belief.
  1. Calculate the probability of a Type I error.
  2. If Maria's belief is true, calculate the probability of a Type II error.
  3. In fact 2 of the 6 people say that they support the president. State which error, Type I or Type II, might be made. Explain your answer.
CAIE S2 2013 November Q7
10 marks Standard +0.3
7 Kieran and Andreas are long-jumpers. They model the lengths, in metres, that they jump by the independent random variables \(K \sim \mathrm {~N} ( 5.64,0.0576 )\) and \(A \sim \mathrm {~N} ( 4.97,0.0441 )\) respectively. They each make a jump and measure the length. Find the probability that
  1. the sum of the lengths of their jumps is less than 11 m ,
  2. Kieran jumps more than 1.2 times as far as Andreas.
CAIE S2 2013 November Q5
8 marks Standard +0.3
5 The volume, in \(\mathrm { cm } ^ { 3 }\), of liquid left in a glass by people when they have finished drinking all they want is modelled by the random variable \(X\) with probability density function given by $$f ( x ) = \begin{cases} k ( x - 2 ) ^ { 2 } & 0 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 3 } { 8 }\).
  2. \(20 \%\) of people leave at least \(d \mathrm {~cm} ^ { 3 }\) of liquid in a glass. Find \(d\).
  3. Find \(\mathrm { E } ( X )\).
CAIE S2 2013 November Q1
6 marks Moderate -0.8
1 A random sample of 80 values of a variable \(X\) is taken and these values are summarised below. $$n = 80 \quad \Sigma x = 150.2 \quad \Sigma x ^ { 2 } = 820.24$$ Calculate unbiased estimates of the population mean and variance of \(X\) and hence find a \(95 \%\) confidence interval for the population mean of \(X\).
CAIE S2 2013 November Q2
8 marks Standard +0.3
2 A traffic officer notes the speeds of vehicles as they pass a certain point. In the past the mean of these speeds has been \(62.3 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) and the standard deviation has been \(10.4 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). A speed limit is introduced, and following this, the mean of the speeds of 75 randomly chosen vehicles passing the point is found to be \(59.9 \mathrm {~km} \mathrm {~h} ^ { - 1 }\).
  1. Making an assumption that should be stated, test at the \(2 \%\) significance level whether the mean speed has decreased since the introduction of the speed limit.
  2. Explain whether it was necessary to use the Central Limit theorem in part (i).
CAIE S2 2013 November Q3
8 marks Standard +0.3
3 The waiting time, \(T\) weeks, for a particular operation at a hospital has probability density function given by $$f ( t ) = \begin{cases} \frac { 1 } { 2500 } \left( 100 t - t ^ { 3 } \right) & 0 \leqslant t \leqslant 10 \\ 0 & \text { otherwise } \end{cases}$$
  1. Given that \(\mathrm { E } ( T ) = \frac { 16 } { 3 }\), find \(\operatorname { Var } ( T )\).
  2. \(10 \%\) of patients have to wait more than \(n\) weeks for their operation. Find the value of \(n\), giving your answer correct to the nearest integer.