Questions — CAIE S2 (737 questions)

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CAIE S2 2019 November Q7
10 marks Standard +0.8
7 Bob is a self-employed builder. In the past his weekly income had mean \(\\) 546\( and standard deviation \)\\( 120\). Following a change in Bob's working pattern, his mean weekly income for 40 randomly chosen weeks was \(\\) 581\(. You should assume that the standard deviation remains unchanged at \)\\( 120\).
  1. Test at the \(2.5 \%\) significance level whether Bob's mean weekly income has increased.
    Bob finds his mean weekly income for another random sample of 40 weeks and carries out a similar test at the \(2.5 \%\) significance level.
  2. Given that Bob's mean weekly income is now in fact \(\\) 595$, 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 2020 November Q3
6 marks Standard +0.8
3 The masses, in kilograms, of female and male animals of a certain species have the distributions \(\mathrm { N } \left( 102,27 ^ { 2 } \right)\) and \(\mathrm { N } \left( 170,55 ^ { 2 } \right)\) respectively. Find the probability that a randomly chosen female has a mass that is less than half the mass of a randomly chosen male. \includegraphics[max width=\textwidth, alt={}, center]{fb305858-2d96-4a5d-b1a9-a965c248fb8d-06_76_1659_484_244}
CAIE S2 2020 November Q4
5 marks Moderate -0.5
4 \includegraphics[max width=\textwidth, alt={}, center]{fb305858-2d96-4a5d-b1a9-a965c248fb8d-07_316_984_260_577} The diagram shows the probability density function, \(\mathrm { f } ( x )\), of a random variable \(X\). For \(0 \leqslant x \leqslant a\), \(\mathrm { f } ( x ) = k\); elsewhere \(\mathrm { f } ( x ) = 0\).
  1. Express \(k\) in terms of \(a\).
  2. Given that \(\operatorname { Var } ( X ) = 3\), find \(a\).
CAIE S2 2020 November Q3
7 marks Moderate -0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{4a5f9f7e-b045-4c6f-8bda-6c4067668da2-04_332_1100_260_520} A random variable \(X\) takes values between 0 and 3 only and has probability density function as shown in the diagram, where \(c\) is a constant.
  1. Show that \(c = \frac { 2 } { 3 }\).
  2. Find \(\mathrm { P } ( X > 2 )\).
  3. Calculate \(\mathrm { E } ( X )\).
CAIE S2 2020 November Q3
6 marks Standard +0.8
3 The masses, in kilograms, of female and male animals of a certain species have the distributions \(\mathrm { N } \left( 102,27 ^ { 2 } \right)\) and \(\mathrm { N } \left( 170,55 ^ { 2 } \right)\) respectively. Find the probability that a randomly chosen female has a mass that is less than half the mass of a randomly chosen male. \includegraphics[max width=\textwidth, alt={}, center]{937c15d2-fb12-4af8-96d3-c54c81d771ba-06_76_1659_484_244}
CAIE S2 2020 November Q4
5 marks Moderate -0.5
4 \includegraphics[max width=\textwidth, alt={}, center]{937c15d2-fb12-4af8-96d3-c54c81d771ba-07_316_984_260_577} The diagram shows the probability density function, \(\mathrm { f } ( x )\), of a random variable \(X\). For \(0 \leqslant x \leqslant a\), \(\mathrm { f } ( x ) = k\); elsewhere \(\mathrm { f } ( x ) = 0\).
  1. Express \(k\) in terms of \(a\).
  2. Given that \(\operatorname { Var } ( X ) = 3\), find \(a\).
CAIE S2 Specimen Q1
4 marks Standard +0.3
1 Failures of two computers occur at random and independently. On average the first computer fails 1.2 times per year and the second computer fails 2.3 times per year. Find the probability that the total number of failures by the two computers in a 6-month period is more than 1 and less than 4 .
CAIE S2 Specimen Q2
6 marks Moderate -0.5
2 The mean and standard deviation of the time spent by people in a certain library are 29 minutes and 6 minutes respectively.
  1. Find the probability that the mean time spent in the library by a random sample of 120 people is more than 30 minutes.
  2. Explain whether it was necessary to assume that the time spent by people in the library is normally distributed in the solution to part (i).
CAIE S2 Specimen Q3
6 marks Moderate -0.3
3 Jagdeesh measured the lengths, \(x\) minutes, of 60 randomly chosen lectures. His results are summarised below.
  1. Calculate unbiased estimates of the population mean and variance.
  2. Calculate a \(98 \%\) confidence interval for the population mean.
CAIE S2 Specimen Q4
7 marks Moderate -0.8
4 A random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} k ( 3 - x ) & 1 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 2 } { 3 }\).
  2. Find the median of \(X\).
CAIE S2 Specimen Q5
7 marks Standard +0.3
5 On average, 1 in 2500 adults has a certain medical condition.
  1. Use a suitable approximation to find the probability that, in a random sample of 4000 people, more than 3 have this condition.
  2. In a random sample of \(n\) people, where \(n\) is large, the probability that none has the condition is less than 0.05 . Find the smallest possible value of \(n\).
CAIE S2 Specimen Q6
9 marks Standard +0.3
6 The weights, in kilograms, of men and women have the distributions \(\mathrm { N } \left( 78,7 ^ { 2 } \right)\) and \(\mathrm { N } \left( 66,5 ^ { 2 } \right)\) respectively.
  1. The maximum load that a certain cable car can carry safely is 1200 kg . If 9 randomly chosen men and 7 randomly chosen women enter the cable car, find the probability that the cable car can operate safely.
  2. Find the probability that a randomly chosen woman weighs more than a randomly chosen man. [4]
CAIE S2 Specimen Q7
11 marks Moderate -0.8
7 At a certain hospital it was found that the probability that a patient did not arrive for an appointment was 0.2 . The hospital carries out some publicity in the hope that this probability will be reduced. They wish to test whether the publicity has worked.
  1. It is suggested that the first 30 appointments on a Monday should be used for the test. Give a reason why this is not an appropriate sample.
    A suitable sample of 30 appointments is selected and the number of patients that do not arrive is noted. This figure is used to carry out a test at the 5\% significance level.
  2. Explain why the test is one-tail and state suitable null and alternative hypotheses.
  3. State what is meant by a Type I error in this context.
  4. Use the binomial distribution to find the critical region, and find the probability of a Type I error.
  5. In fact 3 patients out of the 30 do not arrive. State the conclusion of the test, explaining your answer.
CAIE S2 2019 March Q1
4 marks Moderate -0.8
1 The masses of a certain variety of plums are known to have standard deviation 13.2 g . A random sample of 200 of these plums is taken and the mean mass of the plums in the sample is found to be 62.3 g .
  1. Calculate a \(98 \%\) confidence interval for the population mean mass.
  2. State with a reason whether it was necessary to use the Central Limit theorem in the calculation in part (i).
CAIE S2 2019 March Q2
5 marks Standard +0.8
2 The independent random variables \(X\) and \(Y\) have the distributions \(\mathrm { N } ( 9.2,12.1 )\) and \(\mathrm { N } ( 3.0,8.6 )\) respectively. Find \(\mathrm { P } ( X > 3 Y )\).
CAIE S2 2019 March Q3
6 marks Standard +0.3
3 At factory \(A\) the mean number of accidents per year is 32 . At factory \(B\) the records of numbers of accidents before 2018 have been lost, but the number of accidents during 2018 was 21. It is known that the number of accidents per year can be well modelled by a Poisson distribution. Use an approximating distribution to test at the \(2 \%\) significance level whether the mean number of accidents at factory \(B\) is less than at factory \(A\).
CAIE S2 2019 March Q4
7 marks Moderate -0.8
4 The lifetimes, \(X\) hours, of a random sample of 50 batteries of a certain kind were found. The results are summarised by \(\Sigma x = 420\) and \(\Sigma x ^ { 2 } = 27530\).
  1. Calculate an unbiased estimate of the population mean of \(X\) and show that an unbiased estimate of the population variance is 490 , correct to 3 significant figures.
  2. The lifetimes of a further large sample of \(n\) batteries of this kind were noted, and the sample mean, \(\bar { X }\), was found. Use your estimates from part (i) to find the value of \(n\) such that \(\mathrm { P } ( \bar { X } > 5 ) = 0.9377\).
    [0pt] [4]
CAIE S2 2019 March Q5
8 marks Standard +0.8
5 The number of eagles seen per hour in a certain location has the distribution \(\operatorname { Po } ( 1.8 )\). The number of vultures seen per hour in the same location has the independent distribution \(\operatorname { Po } ( 2.6 )\).
  1. Find the probability that, in a randomly chosen hour, at least 2 eagles are seen.
  2. Find the probability that, in a randomly chosen half-hour period, the total number of eagles and vultures seen is less than 5 .
    Alex wants to be at least \(99 \%\) certain of seeing at least 1 eagle.
  3. Find the minimum time for which she should watch for eagles.
CAIE S2 2019 March Q6
10 marks Standard +0.3
6 The time taken by volunteers to complete a certain task is normally distributed. In the past the time, in minutes, has had mean 91.4 and standard deviation 6.4. A new, similar task is introduced and the times, \(t\) minutes, taken by a random sample of 6 volunteers to complete the new task are summarised by \(\Sigma t = 568.5\). Andrea plans to carry out a test, at the \(5 \%\) significance level, of whether the mean time for the new task is different from the mean time for the old task.
  1. Give a reason why Andrea should use a two-tail test.
  2. State the probability that a Type I error is made, and explain the meaning of a Type I error in this context.
    You may assume that the times taken for the new task are normally distributed.
  3. Stating another necessary assumption, carry out the test.
CAIE S2 2019 March Q7
10 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{a93e5413-6ad8-4957-8efd-470cf79792e2-12_428_693_260_724} A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} ( \sqrt { } 2 ) \cos x & 0 \leqslant x \leqslant \frac { 1 } { 4 } \pi \\ 0 & \text { otherwise } \end{cases}$$ as shown in the diagram.
  1. Find \(\mathrm { P } \left( X > \frac { 1 } { 6 } \pi \right)\).
  2. Find the median of \(X\).
  3. Find \(\mathrm { E } ( X )\).
    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 2020 March Q5
9 marks Standard +0.3
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 2021 March Q2
9 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{2fefee17-50bb-4375-80f6-7e4bc2606492-04_405_789_260_676} 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 2003 November Q1
4 marks Standard +0.3
1 The result of a memory test is known to be normally distributed with mean \(\mu\) and standard deviation 1.9. It is required to have a \(95 \%\) confidence interval for \(\mu\) with a total width of less than 2.0 . Find the least possible number of tests needed to achieve this.
CAIE S2 2003 November Q2
5 marks Standard +0.3
2 A certain machine makes matches. One match in 10000 on average is defective. Using a suitable approximation, calculate the probability that a random sample of 45000 matches will include 2,3 or 4 defective matches.
CAIE S2 2003 November Q3
5 marks Standard +0.3
3 Tien throws a ball. The distance it travels can be modelled by a normal distribution with mean 20 m and variance \(9 \mathrm {~m} ^ { 2 }\). His younger sister Su Chen also throws a ball and the distance her ball travels can be modelled by a normal distribution with mean 14 m and variance \(12 \mathrm {~m} ^ { 2 }\). Su Chen is allowed to add 5 metres on to her distance and call it her 'upgraded distance'. Find the probability that Tien's distance is larger than Su Chen's upgraded distance.