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 2021 June Q5
5 The time, in minutes, spent by customers at a particular gym has the distribution \(\mathrm { N } ( \mu , 38.2 )\). In the past the value of \(\mu\) has been 42.4. Following the installation of some new equipment the management wishes to test whether the value of \(\mu\) has changed.
  1. State what is meant by a Type I error in this context.
  2. The mean time for a sample of 20 customers is found to be 45.6 minutes. Test at the \(2.5 \%\) significance level whether the value of \(\mu\) has changed.
CAIE S2 2021 June Q6
6 The heights, \(h\) centimetres, of a random sample of 100 fully grown animals of a certain species were measured. The results are summarised below. $$n = 100 \quad \Sigma h = 7570 \quad \Sigma h ^ { 2 } = 588050$$
  1. Find unbiased estimates of the population mean and variance.
  2. Calculate a \(99 \%\) confidence interval for the mean height of animals of this species.
    Four random samples were taken and a \(99 \%\) confidence interval for the population mean, \(\mu\), was found from each sample.
  3. Find the probability that all four of these confidence intervals contain the true value of \(\mu\).
CAIE S2 2021 June Q7
7 Customers arrive at a particular shop at random times. It has been found that the mean number of customers who arrive during a 5 -minute interval is 2.1 .
  1. Find the probability that exactly 4 customers arrive during a 10 -minute interval.
  2. Find the probability that at least 4 customers arrive during a 20 -minute interval.
  3. Use a suitable approximating distribution to find the probability that fewer than 40 customers arrive during a 2-hour interval.
    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 June Q1
1 The number of goals scored by a team in a match is independent of other matches, and is denoted by the random variable \(X\), which has a Poisson distribution with mean 1.36. A supporter offers to make a donation of \(
) 5$ to the team for each goal that they score in the next 10 matches. Find the expectation and standard deviation of the amount that the supporter will pay.
CAIE S2 2021 June Q2
2 In the past, the time, in hours, for a particular train journey has had mean 1.40 and standard deviation 0.12 . Following the introduction of some new signals, it is required to test whether the mean journey time has decreased.
  1. State what is meant by a Type II error in this context.
  2. The mean time for a random sample of 50 journeys is found to be 1.36 hours. Assuming that the standard deviation of journey times is still 0.12 hours, test at the \(2.5 \%\) significance level whether the population mean journey time has decreased.
  3. State, with a reason, which of the errors, Type I or Type II, might have been made in the test in part (b).
CAIE S2 2021 June Q3
3 The local council claims that the average number of accidents per year on a particular road is 0.8 . Jane claims that the true average is greater than 0.8 . She looks at the records for a random sample of 3 recent years and finds that the total number of accidents during those 3 years was 5 .
  1. Assume that the number of accidents per year follows a Poisson distribution.
    1. State null and alternative hypotheses for a test of Jane's claim.
    2. Test at the \(5 \%\) significance level whether Jane's claim is justified.
  2. Jane finds that the number of accidents per year has been gradually increasing over recent years. State how this might affect the validity of the test carried out in part (a)(ii).
CAIE S2 2021 June Q4
4 The masses, \(m\) kilograms, of flour in a random sample of 90 sacks of flour are summarised as follows. $$n = 90 \quad \Sigma m = 4509 \quad \Sigma m ^ { 2 } = 225950$$
  1. Find unbiased estimates of the population mean and variance.
  2. Calculate a \(98 \%\) confidence interval for the population mean.
  3. Explain why it was necessary to use the Central Limit theorem in answering part (b).
  4. Find the probability that the confidence interval found in part (b) is wholly above the true value of the population mean.
CAIE S2 2021 June Q5
5 Most plants of a certain type have three leaves. However, it is known that, on average, 1 in 10000 of these plants have four leaves, and plants with four leaves are called 'lucky'. The number of lucky plants in a random sample of 25000 plants is denoted by \(X\).
  1. State, with a justification, an approximating distribution for \(X\), giving the values of any parameters.
    Use your approximating distribution to answer parts (b) and (c).
  2. Find \(\mathrm { P } ( X \leqslant 3 )\).
  3. Given that \(\mathrm { P } ( X = k ) = 2 \mathrm { P } ( X = k + 1 )\), find \(k\).
    The number of lucky plants in a random sample of \(n\) plants, where \(n\) is large, is denoted by \(Y\).
  4. Given that \(\mathrm { P } ( Y \geqslant 1 ) = 0.963\), correct to 3 significant figures, use a suitable approximating distribution to find the value of \(n\).
CAIE S2 2021 June Q6
6 Alethia models the length of time, in minutes, by which her train is late on any day by the random variable \(X\) with probability density function given by $$f ( x ) = \begin{cases} \frac { 3 } { 8000 } ( x - 20 ) ^ { 2 } & 0 \leqslant x \leqslant 20
0 & \text { otherwise } \end{cases}$$
  1. Find the probability that the train is more than 10 minutes late on each of two randomly chosen days.
  2. Find \(\mathrm { E } ( X )\).
  3. The median of \(X\) is denoted by \(m\). Show that \(m\) satisfies the equation \(( m - 20 ) ^ { 3 } = - 4000\), and hence find \(m\) correct to 3 significant figures.
  4. State one way in which Alethia's model may be unrealistic.
    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 June Q1
1 The diameters, \(x\) millimetres, of a random sample of 200 discs made by a certain machine were recorded. The results are summarised below. $$n = 200 \quad \Sigma x = 2520 \quad \Sigma x ^ { 2 } = 31852$$
  1. Calculate a 95\% confidence interval for the population mean diameter.
  2. Jean chose 40 random samples and used each sample to calculate a 95\% confidence interval for the population mean diameter. How many of these 40 confidence intervals would be expected to include the true value of the population mean diameter?
CAIE S2 2022 June Q2
2 Arvind uses an ordinary fair 6-sided die to play a game. He believes he has a system to predict the score when the die is thrown. Before each throw of the die, he writes down what he thinks the score will be. He claims that he can write the correct score more often than he would if he were just guessing. His friend Laxmi tests his claim by asking him to write down the score before each of 15 throws of the die. Arvind writes the correct score on exactly 5 out of 15 throws. Test Arvind's claim at the \(10 \%\) significance level.
CAIE S2 2022 June Q3
3 The lengths, in centimetres, of two types of insect, \(A\) and \(B\), are modelled by the random variables \(X \sim \mathrm {~N} ( 6.2,0.36 )\) and \(Y \sim \mathrm {~N} ( 2.4,0.25 )\) respectively. Find the probability that the length of a randomly chosen type \(A\) insect is greater than the sum of the lengths of 3 randomly chosen type \(B\) insects.
CAIE S2 2022 June Q4
4 The independent random variables \(X\) and \(Y\) have distributions \(\operatorname { Po } ( 2 )\) and \(\mathrm { B } \left( 20 , \frac { 1 } { 4 } \right)\) respectively.
  1. Find the mean and standard deviation of \(X - 3 Y\).
  2. Find \(\mathrm { P } ( Y = 15 X )\).
CAIE S2 2022 June Q5
5 Cars arrive at a fuel station at random and at a constant average rate of 13.5 per hour.
  1. Find the probability that more than 4 cars arrive during a 20-minute period.
  2. Use an approximating distribution to find the probability that the number of cars that arrive during a 12-hour period is between 150 and 160 inclusive.
    Independently of cars, trucks arrive at the fuel station at random and at a constant average rate of 3.6 per 15-minute period.
  3. Find the probability that the total number of cars and trucks arriving at the fuel station during a 10 -minute period is more than 3 and less than 7 .
CAIE S2 2022 June Q6
6 A random variable \(X\) has probability density function f . The graph of \(\mathrm { f } ( x )\) is a straight line segment parallel to the \(x\)-axis from \(x = 0\) to \(x = a\), where \(a\) is a positive constant.
  1. State, in terms of \(a\), the median of \(X\).
  2. Find \(\mathrm { P } \left( X > \frac { 3 } { 4 } a \right)\).
  3. Show that \(\operatorname { Var } ( X ) = \frac { 1 } { 12 } a ^ { 2 }\).
  4. Given that \(\mathrm { P } ( X < b ) = p\), where \(0 < b < \frac { 1 } { 2 } a\), find \(\mathrm { P } \left( \frac { 1 } { 3 } b < X < a - \frac { 1 } { 3 } b \right)\) in terms of \(p\).
CAIE S2 2022 June Q7
7 In the past, the mean time for Jenny's morning run was 28.2 minutes. She does some extra training and she wishes to test whether her mean time has been reduced. After the training Jenny takes a random sample of 40 morning runs. She decides that if the sample mean run time is less than 27 minutes she will conclude that the training has been effective. You may assume that, after the training, Jenny's run time has a standard deviation of 4.0 minutes.
  1. State suitable null and alternative hypotheses for Jenny's test.
  2. Find the probability that Jenny will make a Type I error.
  3. Jenny found that the sample mean run time was 27.2 minutes. Explain briefly whether it is possible for her to make a Type I error or a Type II error or both.
    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 June Q1
1
  1. A javelin thrower noted the lengths of a random sample of 50 of her throws. The sample mean was 72.3 m and an unbiased estimate of the population variance was \(64.3 \mathrm {~m} ^ { 2 }\). Find a \(92 \%\) confidence interval for the population mean length of throws by this athlete.
  2. A discus thrower wishes to calculate a \(92 \%\) confidence interval for the population mean length of his throws. He bases his calculation on his first 50 throws in a week. Comment on this method.
CAIE S2 2022 June Q2
2 In the past, the mean height of plants of a particular species has been 2.3 m . A random sample of 60 plants of this species was treated with fertiliser and the mean height of these 60 plants was found to be 2.4 m . Assume that the standard deviation of the heights of plants treated with fertiliser is 0.4 m . Carry out a test at the \(2.5 \%\) significance level of whether the mean height of plants treated with fertiliser is greater than 2.3 m .
CAIE S2 2022 June Q3
3 It is known that \(1.8 \%\) of children in a certain country have not been vaccinated against measles. A random sample of 200 children in this country is chosen.
  1. Use a suitable approximating distribution to find the probability that there are fewer than 3 children in the sample who have not been vaccinated against measles.
  2. Justify your approximating distribution.
CAIE S2 2022 June Q4
4 The number of cars arriving at a certain road junction on a weekday morning has a Poisson distribution with mean 4.6 per minute. Traffic lights are installed at the junction and council officer wishes to test at the \(2 \%\) significance level whether there are now fewer cars arriving. He notes the number of cars arriving during a randomly chosen 2 -minute period.
  1. State suitable null and alternative hypotheses for the test.
  2. Find the critical region for the test.
    The officer notes that, during a randomly chosen 2 -minute period on a weekday morning, exactly 5 cars arrive at the junction.
  3. Carry out the test.
  4. State, with a reason, whether it is possible that a Type I error has been made in carrying out the test in part (c).
    The number of cars arriving at another junction on a weekday morning also has a Poisson distribution with mean 4.6 per minute.
  5. Use a suitable approximating distribution to find the probability that more than 300 cars will arrive at this junction in an hour.
CAIE S2 2022 June Q5
5 A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 3 } { 16 } \left( 4 x - x ^ { 2 } \right) & 2 \leqslant x \leqslant 4
0 & \text { otherwise } \end{cases}$$
  1. Show that \(\mathrm { E } ( X ) = \frac { 11 } { 4 }\).
  2. Find \(\operatorname { Var } ( X )\).
  3. Given that the median of \(X\) is \(m\), find \(\mathrm { P } ( m < X < 3 )\).
CAIE S2 2022 June Q6
6 The masses, in kilograms, of large and small sacks of grain have the distributions \(\mathrm { N } ( 53,11 )\) and \(\mathrm { N } ( 14,3 )\) respectively.
  1. Find the probability that the mass of a randomly chosen large sack is greater than four times the mass of a randomly chosen small sack.
  2. A lift can safely carry a maximum mass of 1000 kg . Find the probability that the lift can safely carry 12 randomly chosen large sacks and 25 randomly chosen small sacks.
    \(7 X\) is a random variable with distribution \(\operatorname { Po } ( 2.90 )\). A random sample of 100 values of \(X\) is taken. Find the probability that the sample mean is less than 2.88 .
    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 June Q1
1 The number of characters in emails sent by a particular company is modelled by the distribution \(\mathrm { N } \left( 1250,480 ^ { 2 } \right)\). Find the probability that the mean number of characters in a random sample of 100 emails sent by the company is more than 1300 .
CAIE S2 2022 June Q2
2 Anton believes that \(10 \%\) of students at his college are left-handed. Aliya believes that this is an underestimate. She plans to carry out a hypothesis test of the null hypothesis \(p = 0.1\) against the alternative hypothesis \(p > 0.1\), where \(p\) is the actual proportion of students at the college that are left-handed. She chooses a random sample of 20 students from the college. She will reject the null hypothesis if at least 5 of these students are left-handed.
  1. Explain what is meant by a Type I error in this context.
  2. Find the probability of a Type I error in the test.
  3. Given that the true value of \(p\) is 0.3 , find the probability of a Type II error in the test.
CAIE S2 2022 June Q3
3 Batteries of type \(A\) are known to have a mean life of 150 hours. It is required to test whether a new type of battery, type \(B\), has a shorter mean life than type \(A\) batteries.
  1. Give a reason for using a sample rather than the whole population in carrying out this test.
    A random sample of 120 type \(B\) batteries are tested and it is found that their mean life is 147 hours, and an unbiased estimate of the population variance is 225 hours \(^ { 2 }\).
  2. Test, at the \(2 \%\) significance level, whether type \(B\) batteries have a shorter mean life than type \(A\) batteries.
  3. Calculate a \(94 \%\) confidence interval for the population mean life of type \(B\) batteries.