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 2023 June Q8
8 A new light was installed on a certain footpath. A town councillor decided to use a hypothesis test to investigate whether the number of people using the path in the evening had increased. Before the light was installed, the mean number of people using the path during any 20 -minute period during the evening was 1.01. After the light was installed, the total number, \(n\), of people using the path during 3 randomly chosen 20 -minute periods during the evening was noted.
  1. Given that the value of \(n\) was 6 , use a Poisson distribution to carry out the test at the \(5 \%\) significance level.
  2. Later a similar test, at the \(5 \%\) significance level, was carried out using another 3 randomly chosen 20 -minute periods during the evening. Find the probability of a Type I error.
  3. State what is meant by a Type I error in this context.
  4. State, in context, what further information would be needed in order to find the probability of a Type II error. Do not carry out any further calculation.
    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 2024 June Q1
1 A bus station has exactly four entrances. In the morning the numbers of passengers arriving at these entrances during a 10 -second period have the independent distributions \(\operatorname { Po } ( 0.4 ) , \operatorname { Po } ( 0.1 ) , \operatorname { Po } ( 0.2 )\) and \(\mathrm { Po } ( 0.5 )\). Find the probability that the total number of passengers arriving at the four entrances to the bus station during a randomly chosen 1 -minute period in the morning is more than 3 .
CAIE S2 2024 June Q2
2 The random variable \(X\) has the distribution \(\mathrm { N } \left( 31.2,10.4 ^ { 2 } \right)\). Two independent random values of \(X\), denoted by \(X _ { 1 }\) and \(X _ { 2 }\), are chosen. Find \(\mathrm { P } \left( X _ { 1 } > 3 X _ { 2 } \right)\).
CAIE S2 2024 June Q3
3 The time taken in minutes for a certain daily train journey has a normal distribution with standard deviation 5.8. For a random sample of 20 days the journey times were noted and the mean journey time was found to be 81.5 minutes.
  1. Calculate a \(98 \%\) confidence interval for the population mean journey time.
    A student was asked for the meaning of this confidence interval. The student replied as follows.
    'The times for \(98 \%\) of these journeys are likely to be within the confidence interval.'
  2. Explain briefly whether this statement is true or not.
    Two independent 98\% confidence intervals are found.
  3. Given that at least one of these intervals contains the population mean, find the probability that both intervals contain the population mean.
CAIE S2 2024 June Q4
4
  1. A random sample of 8 boxes of cereal from a certain supplier was taken. Each box was weighed and the masses in grams were as follows. $$\begin{array} { l l l l l l l l } 261 & 249 & 259 & 252 & 255 & 256 & 258 & 254 \end{array}$$ Find unbiased estimates of the population mean and variance.
  2. The supplier claims that the mean mass of boxes of cereal is 253 g . A quality control officer suspects that the mean mass is actually more than 253 g . In order to test this claim, he weighs a random sample of 100 boxes of cereal and finds that the total mass is 25360 g .
    1. Given that the population standard deviation of the masses is 3.5 g , test at the \(5 \%\) significance level whether the population mean mass is more than 253 g .
      An employee says, 'This test is invalid because it uses the normal distribution, but we do not know whether the masses of the boxes are normally distributed.’
    2. Explain briefly whether this statement is true or not.
CAIE S2 2024 June Q5
5 Sales of cell phones at a certain shop occur singly, randomly and independently.
  1. State one further condition that must be satisfied for the number of sales in a certain time period to be well modelled by a Poisson distribution.
    The average number of sales per hour is 1.2 .
    Assume now that a Poisson distribution is a suitable model.
  2. Find the probability that the number of sales during a randomly chosen 12 -hour period will be more than 12 and less than 16 .
  3. Use a suitable approximating distribution to find the probability that the number of sales during a randomly chosen 1-month period (140 hours) will be less than 150 .
CAIE S2 2024 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{5dfbc896-528c-40a6-a296-8a0aae90add4-10_451_469_255_799} The diagram shows the graph of the probability density function, f , of a random variable \(X\). The graph is a quarter circle entirely in the first quadrant with centre \(( 0,0 )\) and radius \(a\), where \(a\) is a positive constant. Elsewhere \(\mathrm { f } ( x ) = 0\).
  1. Show that \(a = \frac { 2 } { \sqrt { \pi } }\).
  2. Show that \(\mathrm { f } ( x ) = \sqrt { \frac { 4 } { \pi } - x ^ { 2 } }\).
  3. Show that \(\mathrm { E } ( X ) = \frac { 8 } { 3 \sqrt { \pi ^ { 3 } } }\).
CAIE S2 2024 June Q7
7 Every July, as part of a research project, Rita collects data about sightings of a particular kind of bird. Each day in July she notes whether she sees this kind of bird or not, and she records the number \(X\) of days on which she sees it. She models the distribution of \(X\) by \(\mathrm { B } ( 31 , p )\), where \(p\) is the probability of seeing this kind of bird on a randomly chosen day in July. Data from previous years suggests that \(p = 0.3\), but in 2022 Rita suspected that the value of \(p\) had been reduced. She decided to carry out a hypothesis test. In July 2022, she saw this kind of bird on 4 days.
  1. Use the binomial distribution to test at the \(5 \%\) significance level whether Rita's suspicion is justified.
    In July 2023, she noted the value of \(X\) and carried out another test at the \(5 \%\) significance level using the same hypotheses.
  2. Calculate the probability of a Type I error.
    Rita models the number of sightings, \(Y\), per year of a different, very rare, kind of bird by the distribution \(B ( 365,0.01 )\).
    1. Use a suitable approximating distribution to find \(\mathrm { P } ( Y = 4 )\).
    2. Justify your approximating distribution in this context.
      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 2024 June Q1
1 A random variable \(X\) has the distribution \(\mathrm { Po } ( 145 )\).
  1. Use a suitable approximating distribution to calculate \(\mathrm { P } ( X \leqslant 150 )\).
  2. Justify the use of your approximating distribution in this case.
CAIE S2 2024 June Q2
2 Henri wants to choose a random sample from the 804 students at his college. He numbers the students from 1 to 804 and then uses random numbers generated by his calculator. The first 20 random digits produced by his calculator are as follows. $$\begin{array} { l l l l l l l l l l l l l l l l l l l l } 5 & 6 & 7 & 1 & 0 & 9 & 8 & 4 & 3 & 1 & 0 & 9 & 6 & 6 & 5 & 0 & 2 & 1 & 7 & 6 \end{array}$$ Henri's first two student numbers are 567 and 109.
  1. Use Henri's digits to find the numbers of the next two students in the sample.
    There were 30 students in Henri's sample. He asked each of them how much time, \(X\) hours, they spent on social media each week, on average. He summarised the results as follows. $$n = 30 \quad \Sigma x = 610 \quad \Sigma x ^ { 2 } = 12405$$
  2. Use this information to calculate an unbiased estimate of the mean of \(X\) and show that an unbiased estimate of the variance of \(X\) is less than 0.1 .
  3. Henri's friend claims that Henri has probably made a mistake in his calculation of \(\Sigma x\) or \(\Sigma x ^ { 2 }\). Use your answer to part (b) to comment on this claim.
CAIE S2 2024 June Q3
3 A student wishes to estimate the proportion, \(p\), of students at her college who have exactly one brother. She surveys a random sample of 50 students at her college and finds that 18 of them have exactly one brother. She calculates an approximate \(\alpha \%\) confidence interval for \(p\) and finds that the lower limit of the confidence interval is 0.244 correct to 3 significant figures. Find \(\alpha\) correct to the nearest integer.
CAIE S2 2024 June Q4
4 A random variable \(X\) has the distribution \(\mathrm { N } ( 10,12 )\). Two independent values of \(X\), denoted by \(X _ { 1 }\) and \(X _ { 2 }\), are chosen at random.
  1. Write down the value of \(\mathrm { P } \left( X _ { 1 } > X _ { 2 } \right)\).
  2. Find \(\mathrm { P } \left( X _ { 1 } > 2 X _ { 2 } - 3 \right)\).
CAIE S2 2024 June Q5
5 The number of goals scored by a sports team in the first half of any match has the distribution \(X \sim \mathrm { Po }\) (3.1). The number of goals scored by the same team in the second half of any match has the distribution \(Y \sim \operatorname { Po } ( 2.4 )\). You may assume that the distributions of \(X\) and \(Y\) are independent.
  1. Find \(\mathrm { P } ( X < 4 )\).
  2. Find the probability that, in a randomly chosen match, the team scores at least 5 goals.
  3. Given that the team scores a total of 5 goals in a randomly chosen match, find the probability that they score exactly 3 goals in the first half.
CAIE S2 2024 June Q6
5 marks
6 The masses of cereal boxes filled by a certain machine have mean 510 grams. An adjustment is made to the machine and an inspector wishes to test whether the mean mass of cereal boxes filled by the machine has decreased. After the adjustment is made, he chooses a random sample of 120 cereal boxes. The mean mass of these boxes is found to be 508 grams. Assume that the standard deviation of the masses is 10 grams.
  1. Test at the \(2.5 \%\) significance level whether the mean mass of cereal boxes filled by the machine has decreased.
    Later the inspector carries out a similar test at the \(2.5 \%\) significance level, using the same hypotheses and another 120 randomly chosen cereal boxes.
    [0pt]
  2. Given that the mean mass is now actually 506 grams, find the probability of a Type II error. [5]
CAIE S2 2024 June Q7
7 The probability density function, f , of a random variable \(X\) is given by $$f ( x ) = \begin{cases} k ( 1 + \cos x ) & 0 \leqslant x \leqslant \pi
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 1 } { \pi }\).
  2. Verify that the median of \(X\) lies between 0.83 and 0.84 .
  3. Find the exact value of \(\mathrm { E } ( X )\).
    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 2024 June Q1
1 The random variable \(X\) has the distribution \(\mathrm { B } ( 4000,0.001 )\).
  1. Use a suitable approximating distribution to find \(\mathrm { P } ( 2 \leqslant X < 5 )\).
  2. Justify your approximating distribution in this case.
CAIE S2 2024 June Q2
1 marks
2 The widths, \(w \mathrm {~cm}\), of a random sample of 150 leaves of a certain kind were measured. The sample mean of \(w\) was found to be 3.12 cm . Using this sample, an approximate \(95 \%\) confidence interval for the population mean of the widths in centimetres was found to be [3.01, 3.23].
  1. Calculate an estimate of the population standard deviation.
  2. Explain whether it was necessary to use the Central Limit theorem in your answer to part (a). [1]
CAIE S2 2024 June Q3
3 The masses in kilograms of large and small bags of cement have the independent distributions \(\mathrm { N } ( 50,2.4 )\) and \(\mathrm { N } ( 26,1.8 )\) respectively. Find the probability that the total mass of 5 randomly chosen large bags of cement is greater than the total mass of 10 randomly chosen small bags of cement.
\includegraphics[max width=\textwidth, alt={}, center]{7c078a14-98f9-4292-ae76-a2642238176f-04_2714_34_143_2012}
\includegraphics[max width=\textwidth, alt={}, center]{7c078a14-98f9-4292-ae76-a2642238176f-05_2724_35_136_20}
CAIE S2 2024 June Q4
4 In this question you should not use an approximating distribution.
At an election in Menham last year, \(24 \%\) of voters supported the Today Party. A student wishes to test whether support for the Today Party has decreased since last year. He chooses a random sample of 25 voters in Menham and finds that exactly 2 of them say that they support the Today Party. Test at the 5\% significance level whether support for the Today Party has decreased.
CAIE S2 2024 June Q5
5 A random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} a x - x ^ { 3 } & 0 \leqslant x \leqslant \sqrt { 2 }
0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.
  1. Show that \(a = 2\) .
  2. Find the median of \(X\) .
  3. Find the exact value of \(\mathrm { E } ( X )\).
CAIE S2 2024 June Q6
6 The numbers of green sweets in 200 randomly chosen packets of Frutos are summarised in the table.
Number of green sweets0123\(> 3\)
Number of packets325097210
  1. Calculate an unbiased estimate for the population mean of the number of green sweets in a packet of Frutos, and show that an unbiased estimate of the population variance is 0.783 correct to 3 significant figures.
    The manufacturers of Frutos claim that the mean number of green sweets in a packet is 1.65 .
    Anji believes that the true value of the mean, \(\mu\), is less than 1.65 . She uses the results from the 200 randomly chosen packets to test the manufacturers’ claim.
  2. State suitable null and alternative hypotheses for the test.
    \includegraphics[max width=\textwidth, alt={}, center]{7c078a14-98f9-4292-ae76-a2642238176f-08_2714_37_143_2008}
  3. Show that the result of Anji's test is significant at the \(5 \%\) level but not at the \(1 \%\) level.
  4. It is given that Anji made a Type I error. Explain how this shows that the significance level that Anji used in her test was not \(1 \%\).
CAIE S2 2024 June Q7
7 The independent random variables \(X\) and \(Y\) have the distributions \(\operatorname { Po } ( 1.9 )\) and \(\operatorname { Po } ( 2.2 )\) respectively.
  1. Find \(\mathrm { P } ( X + Y < 4 )\).
    \includegraphics[max width=\textwidth, alt={}, center]{7c078a14-98f9-4292-ae76-a2642238176f-10_74_1581_406_322}
    \includegraphics[max width=\textwidth, alt={}, center]{7c078a14-98f9-4292-ae76-a2642238176f-10_75_1581_497_322}
  2. Find the probability that \(X = 2\) given that \(X + Y < 4\).
    \includegraphics[max width=\textwidth, alt={}, center]{7c078a14-98f9-4292-ae76-a2642238176f-10_2715_35_144_2012}
  3. A sample of 60 randomly chosen pairs of values of \(X\) and \(Y\) is taken,and the value of \(X + Y\) is calculated for each pair.The sample mean of these 60 values is found. Find the probability that the sample mean of \(X + Y\) is less than 4.0 .
    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 2020 March Q1
1 The booklets produced by a certain publisher contain, on average, 1 incorrect letter per 30000 letters, and these errors occur randomly. A randomly chosen booklet from this publisher contains 12500 letters. Use a suitable approximating distribution to find the probability that this booklet contains at least 2 errors.
CAIE S2 2020 March Q2
2 Lengths of a certain species of lizard are known to be normally distributed with standard deviation 3.2 cm . A naturalist measures the lengths of a random sample of 100 lizards of this species and obtains an \(\alpha \%\) confidence interval for the population mean. He finds that the total width of this interval is 1.25 cm . Find \(\alpha\).
CAIE S2 2020 March Q3
3 In the past, the mean time taken by Freda for a particular daily journey was 39.2 minutes. Following the introduction of a one-way system, Freda wishes to test whether the mean time for the journey has decreased. She notes the times, \(t\) minutes, for 40 randomly chosen journeys and summarises the results as follows. $$n = 40 \quad \Sigma t = 1504 \quad \Sigma t ^ { 2 } = 57760$$
  1. Calculate unbiased estimates of the population mean and variance of the new journey time.
  2. Test, at the \(5 \%\) significance level, whether the population mean time has decreased.