Questions — CAIE S2 (717 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 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 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 Stats 1 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 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 2012 June Q4
4 Bacteria of a certain type are randomly distributed in the water in two ponds, \(A\) and \(B\). The average numbers of bacteria per \(\mathrm { cm } ^ { 3 }\) in \(A\) and \(B\) are 0.32 and 0.45 respectively.
  1. Samples of \(8 \mathrm {~cm} ^ { 3 }\) of water from \(A\) and \(12 \mathrm {~cm} ^ { 3 }\) of water from \(B\) are taken at random. Find the probability that the total number of bacteria in these samples is at least 3 .
  2. Find the probability that in a random sample of \(155 \mathrm {~cm} ^ { 3 }\) of water from \(A\), the number of bacteria is less than 35 .
CAIE S2 2012 June Q5
5 Fiona and Jhoti each take one shower per day. The times, in minutes, taken by Fiona and Jhoti to take a shower are represented by the independent variables \(F \sim \mathrm {~N} \left( 12.2,2.8 ^ { 2 } \right)\) and \(J \sim \mathrm {~N} \left( 11.8,2.6 ^ { 2 } \right)\) respectively. Find the probability that, on a randomly chosen day,
  1. the total time taken to shower by Fiona and Jhoti is less than 30 minutes,
  2. Fiona takes at least twice as long as Jhoti to take a shower.
CAIE S2 2012 June Q6
6 At a certain shop the weekly demand, in kilograms, for flour is modelled by the random variable \(X\) with probability density function given by $$f ( x ) = \begin{cases} k x ^ { - \frac { 1 } { 2 } } & 4 \leqslant x \leqslant 25
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 1 } { 6 }\).
  2. Calculate the mean weekly demand for flour at the shop.
  3. At the beginning of one week, the shop has 20 kg of flour in stock. Find the probability that this will not be enough to meet the demand for that week.
  4. Give a reason why the model may not be realistic.
CAIE S2 2012 June Q7
7 The weights, \(X\) kilograms, of bags of carrots are normally distributed. The mean of \(X\) is \(\mu\). An inspector wishes to test whether \(\mu = 2.0\). He weighs a random sample of 200 bags and his results are summarised as follows. $$\Sigma x = 430 \quad \Sigma x ^ { 2 } = 1290$$
  1. Carry out the test, at the \(10 \%\) significance level.
  2. You may now assume that the population variance of \(X\) is 1.85 . The inspector weighs another random sample of 200 bags and carries out the same test at the \(10 \%\) significance level.
    (a) State the meaning of a Type II error in this context.
    (b) Given that \(\mu = 2.12\), show that the probability of a Type II error is 0.652 , correct to 3 significant figures.
CAIE S2 2012 June Q1
1 Leaves from a certain type of tree have lengths that are distributed with standard deviation 3.2 cm . A random sample of 250 of these leaves is taken and the mean length of this sample is found to be 12.5 cm .
  1. Calculate a \(99 \%\) confidence interval for the population mean length.
  2. Write down the probability that the whole of a \(99 \%\) confidence interval will lie below the population mean.
CAIE S2 2012 June Q2
2 The independent random variables \(X\) and \(Y\) have the distributions \(\mathrm { N } ( 6.5,14 )\) and \(\mathrm { N } ( 7.4,15 )\) respectively. Find \(\mathrm { P } ( 3 X - Y < 20 )\).
CAIE S2 2012 June Q3
3 The lengths, \(x \mathrm {~mm}\), of a random sample of 150 insects of a certain kind were found. The results are summarised by \(\Sigma x = 7520\) and \(\Sigma x ^ { 2 } = 413540\).
  1. Calculate unbiased estimates of the population mean and variance of the lengths of insects of this kind.
  2. Using the values found in part (i), calculate an estimate of the probability that the mean length of a further random sample of 80 insects of this kind is greater than 53 mm .
CAIE S2 2012 June Q4
4 The number of lions seen per day during a standard safari has the distribution \(\operatorname { Po } ( 0.8 )\). The number of lions seen per day during an off-road safari has the distribution \(\operatorname { Po } ( 2.7 )\). The two distributions are independent.
  1. Susan goes on a standard safari for one day. Find the probability that she sees at least 2 lions.
  2. Deena goes on a standard safari for 3 days and then on an off-road safari for 2 days. Find the probability that she sees a total of fewer than 5 lions.
  3. Khaled goes on a standard safari for \(n\) days, where \(n\) is an integer. He wants to ensure that his chance of not seeing any lions is less than \(10 \%\). Find the smallest possible value of \(n\).
CAIE S2 2012 June Q5
5
  1. Deng wishes to test whether a certain coin is biased so that it is more likely to show Heads than Tails. He throws it 12 times. If it shows Heads more than 9 times, he will conclude that the coin is biased. Calculate the significance level of the test.
  2. Deng throws another coin 100 times in order to test, at the \(5 \%\) significance level, whether it is biased towards Heads. Find the rejection region for this test.
CAIE S2 2012 June Q6
6 Last year Samir found that the time for his journey to work had mean 45.7 minutes and standard deviation 3.2 minutes. Samir wishes to test whether his journey times have increased this year. He notes the times, in minutes, for a random sample of 8 journeys this year with the following results. $$\begin{array} { l l l l l l l l } 46.2 & 41.7 & 49.2 & 47.1 & 47.2 & 48.4 & 53.7 & 45.5 \end{array}$$ It may be assumed that the population of this year's journey times is normally distributed with standard deviation 3.2 minutes.
  1. State, with a reason, whether Samir should use a one-tail or a two-tail test.
  2. Show that there is no evidence at the \(5 \%\) significance level that Samir's mean journey time has increased.
  3. State, with a reason, which one of the errors, Type I or Type II, might have been made in carrying out the test in part (ii).
CAIE S2 2012 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{7333c047-edad-4385-b3f8-248e8725cfcb-3_412_718_1037_715} A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} k \sin x & 0 \leqslant x \leqslant \frac { 2 } { 3 } \pi
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant, as shown in the diagram.
  1. Show that \(k = \frac { 2 } { 3 }\).
  2. Show that the median of \(X\) is 1.32 , correct to 3 significant figures.
  3. Find \(\mathrm { E } ( X )\).
CAIE S2 2013 June Q1
1 Marie wants to choose one student at random from Anthea, Bill and Charlie. She throws two fair coins. If both coins show tails she will choose Anthea. If both coins show heads she will choose Bill. If the coins show one of each she will choose Charlie.
  1. Explain why this is not a fair method for choosing the student.
  2. Describe how Marie could use the two coins to give a fair method for choosing the student.
CAIE S2 2013 June Q2
2 The times taken by students to complete a task are normally distributed with standard deviation 2.4 minutes. A lecturer claims that the mean time is 17.0 minutes. The times taken by a random sample of 5 students were 17.8, 22.4, 16.3, 23.1 and 11.4 minutes. Carry out a hypothesis test at the \(5 \%\) significance level to determine whether the lecturer's claim should be accepted.
CAIE S2 2013 June Q3
3 Weights of cups have a normal distribution with mean 91 g and standard deviation 3.2 g . Weights of saucers have an independent normal distribution with mean 72 g and standard deviation 2.6 g . Cups and saucers are chosen at random to be packed in boxes, with 6 cups and 6 saucers in each box. Given that each empty box weighs 550 g , find the probability that the total weight of a box containing 6 cups and 6 saucers exceeds 1550 g .
CAIE S2 2013 June Q4
4 The lengths, \(x \mathrm {~m}\), of a random sample of 200 balls of string are found and the results are summarised by \(\Sigma x = 2005\) and \(\Sigma x ^ { 2 } = 20175\).
  1. Calculate unbiased estimates of the population mean and variance of the lengths.
  2. Use the values from part (i) to estimate the probability that the mean length of a random sample of 50 balls of string is less than 10 m .
  3. Explain whether or not it was necessary to use the Central Limit theorem in your calculation in part (ii).
CAIE S2 2013 June Q5
5 The probability that a new car of a certain type has faulty brakes is 0.008 . A random sample of 520 new cars of this type is chosen, and the number, \(X\), having faulty brakes is noted.
  1. Describe fully the distribution of \(X\) and describe also a suitable approximating distribution. Justify this approximating distribution.
  2. Use your approximating distribution to find
    (a) \(\mathrm { P } ( X > 3 )\),
    (b) the smallest value of \(n\) such that \(\mathrm { P } ( X = n ) > \mathrm { P } ( X = n + 1 )\).
CAIE S2 2013 June Q6
6 The time in minutes taken by people to read a certain booklet is modelled by the random variable \(T\) with probability density function given by $$f ( t ) = \begin{cases} \frac { 1 } { 2 \sqrt { } t } & 4 \leqslant t \leqslant 9
0 & \text { otherwise } \end{cases}$$
  1. Find the time within which \(90 \%\) of people finish reading the booklet.
  2. Find \(\mathrm { E } ( T )\) and \(\operatorname { Var } ( T )\).
CAIE S2 2013 June Q7
7 Leila suspects that a particular six-sided die is biased so that the probability, \(p\), that it will show a six is greater than \(\frac { 1 } { 6 }\). She tests the die by throwing it 5 times. If it shows a six on 3 or more throws she will conclude that it is biased.
  1. State what is meant by a Type I error in this situation and calculate the probability of a Type I error.
  2. Assuming that the value of \(p\) is actually \(\frac { 2 } { 3 }\), calculate the probability of a Type II error. Leila now throws the die 80 times and it shows a six on 50 throws.
  3. Calculate an approximate \(96 \%\) confidence interval for \(p\).
CAIE S2 2013 June Q1
1 It is known that \(1.2 \%\) of rods made by a certain machine are bent. The random variable \(X\) denotes the number of bent rods in a random sample of 400 rods.
  1. State the distribution of \(X\).
  2. State, with a reason, a suitable approximate distribution for \(X\).
  3. Use your approximate distribution to find the probability that the sample will include more than 2 bent rods.
CAIE S2 2013 June Q2
2 A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 2 } { 3 } x & 1 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$
  1. Find \(\mathrm { E } ( X )\).
  2. Find \(\mathrm { P } ( X < \mathrm { E } ( X ) )\).
  3. Hence explain whether the mean of \(X\) is less than, equal to or greater than the median of \(X\).
CAIE S2 2013 June Q3
3 The heights of a certain variety of plant have been found to be normally distributed with mean 75.2 cm and standard deviation 5.7 cm . A biologist suspects that pollution in a certain region is causing the plants to be shorter than usual. He takes a random sample of \(n\) plants of this variety from this region and finds that their mean height is 73.1 cm . He then carries out an appropriate hypothesis test.
  1. He finds that the value of the test statistic \(z\) is - 1.563 , correct to 3 decimal places. Calculate the value of \(n\). State an assumption necessary for your calculation.
  2. Use this value of the test statistic to carry out the hypothesis test at the 6\% significance level.
CAIE S2 2013 June Q4
4 The masses, in grams, of a certain type of plum are normally distributed with mean \(\mu\) and variance \(\sigma ^ { 2 }\). The masses, \(m\) grams, of a random sample of 150 plums of this type were found and the results are summarised by \(\Sigma m = 9750\) and \(\Sigma m ^ { 2 } = 647500\).
  1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
  2. Calculate a 98\% confidence interval for \(\mu\). Two more random samples of plums of this type are taken and a \(98 \%\) confidence interval for \(\mu\) is calculated from each sample.
  3. Find the probability that neither of these two intervals contains \(\mu\).
CAIE S2 2013 June Q5
5 Packets of cereal are packed in boxes, each containing 6 packets. The masses of the packets are normally distributed with mean 510 g and standard deviation 12 g . The masses of the empty boxes are normally distributed with mean 70 g and standard deviation 4 g .
  1. Find the probability that the total mass of a full box containing 6 packets is between 3050 g and 3150 g .
  2. A packet and an empty box are chosen at random. Find the probability that the mass of the packet is at least 8 times the mass of the empty box.
CAIE S2 2013 June Q6
6 The number of cases of asthma per month at a clinic has a Poisson distribution. In the past the mean has been 5.3 cases per month. A new treatment is introduced. In order to test at the \(5 \%\) significance level whether the mean has decreased, the number of cases in a randomly chosen month is noted.
  1. Find the critical region for the test and, given that the number of cases is 2 , carry out the test.
  2. Explain the meaning of a Type I error in this context and state the probability of a Type I error.
  3. At another clinic the mean number of cases of asthma per month has the independent distribution \(\mathrm { Po } ( 13.1 )\). Assuming that the mean for the first clinic is still 5.3, use a suitable approximating distribution to estimate the probability that the total number of cases in the two clinics in a particular month is more than 20.
CAIE S2 2013 June Q1
1 The mean and variance of the random variable \(X\) are 5.8 and 3.1 respectively. The random variable \(S\) is the sum of three independent values of \(X\). The independent random variable \(T\) is defined by \(T = 3 X + 2\).
  1. Find the variance of \(S\).
  2. Find the variance of \(T\).
  3. Find the mean and variance of \(S - T\).