Questions — CAIE S2 (717 questions)

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CAIE S2 2020 June Q1
1 The lengths, \(X\) centimetres, of a random sample of 7 leaves from a certain variety of tree are as follows.
3.9
4.8
4.8
4.4
CAIE S2 2021 June Q6
6 The probability density function, f, of a random variable \(X\) is given by $$f ( x ) = \begin{cases} k \left( 6 x - x ^ { 2 } \right) & 0 \leqslant x \leqslant 6
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
State the value of \(\mathrm { E } ( X )\) and show that \(\operatorname { Var } ( X ) = \frac { 9 } { 5 }\).
CAIE S2 2021 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{189bcf7b-279f-457b-8232-ace7f0c9797f-05_456_668_260_735} The random variable \(X\) takes values in the range \(1 \leqslant x \leqslant p\), where \(p\) is a constant. The graph of the probability density function of \(X\) is shown in the diagram.
  1. Show that \(p = 2\).
  2. Find \(\mathrm { E } ( X )\).
CAIE S2 2016 March Q1
1 A fair six-sided die is thrown 20 times and the number of sixes, \(X\), is recorded. Another fair six-sided die is thrown 20 times and the number of odd-numbered scores, \(Y\), is recorded. Find the mean and standard deviation of \(X + Y\).
CAIE S2 2016 March Q2
2 Jill shoots arrows at a target. Last week, \(65 \%\) of her shots hit the target. This week Jill claims that she has improved. Out of her first 20 shots this week, she hits the target with 18 shots. Assuming shots are independent, test Jill's claim at the \(1 \%\) significance level.
CAIE S2 2016 March Q3
3 In the past, Arvinder has found that the mean time for his journey to work is 35.2 minutes. He tries a different route to work, hoping that this will reduce his journey time. Arvinder decides to take a random sample of 25 journeys using the new route. If the sample mean is less than 34.7 minutes he will conclude that the new route is quicker. Assume that, for the new route, the journey time has a normal distribution with standard deviation 5.6 minutes.
  1. Find the probability that a Type I error occurs.
  2. Arvinder finds that the sample mean is 34.5 minutes. Explain briefly why it is impossible for him to make a Type II error.
CAIE S2 2016 March Q4
4 The masses, in grams, of large bags of sugar and small bags of sugar are denoted by \(X\) and \(Y\) respectively, where \(X \sim \mathrm {~N} \left( 5.1,0.2 ^ { 2 } \right)\) and \(Y \sim \mathrm {~N} \left( 2.5,0.1 ^ { 2 } \right)\). Find the probability that the mass of a randomly chosen large bag is less than twice the mass of a randomly chosen small bag.
CAIE S2 2016 March Q5
5 The 150 oranges in a random sample from a certain supplier were weighed and the masses, \(X\) grams, were recorded. The results are summarised below. $$n = 150 \quad \Sigma x = 14910 \quad \Sigma x ^ { 2 } = 1525000$$
  1. Calculate a \(99 \%\) confidence interval for the population mean of \(X\).
  2. The supplier claims that the mean mass of his oranges is 100 grams. Use your answer to part (i) to explain whether this claim should be accepted.
  3. State briefly why the sample should be random.
CAIE S2 2016 March Q6
6 The battery in Sue's phone runs out at random moments. Over a long period, she has found that the battery runs out, on average, 3.3 times in a 30-day period.
  1. Find the probability that the battery runs out fewer than 3 times in a 25-day period.
  2. (a) Use an approximating distribution to find the probability that the battery runs out more than 50 times in a year ( 365 days).
    (b) Justify the approximating distribution used in part (ii)(a).
  3. Independently of her phone battery, Sue's computer battery also runs out at random moments. On average, it runs out twice in a 15-day period. Find the probability that the total number of times that her phone battery and her computer battery run out in a 10-day period is at least 4 .
CAIE S2 2016 March Q7
7

  1. \includegraphics[max width=\textwidth, alt={}, center]{3f1a0c67-03a4-4b4f-99c0-4336ba7d56b0-3_255_643_264_790} The diagram shows the graph of the probability density function, f , of a random variable \(X\), where $$f ( x ) = \begin{cases} \frac { 2 } { 9 } \left( 3 x - x ^ { 2 } \right) & 0 \leqslant x \leqslant 3
    0 & \text { otherwise } \end{cases}$$
    1. State the value of \(\mathrm { E } ( X )\) and find \(\operatorname { Var } ( X )\).
    2. State the value of \(\mathrm { P } ( 1.5 \leqslant X \leqslant 4 )\).
    3. Given that \(\mathrm { P } ( 1 \leqslant X \leqslant 2 ) = \frac { 13 } { 27 }\), find \(\mathrm { P } ( X > 2 )\).
  2. A random variable, \(W\), has probability density function given by $$\mathrm { g } ( w ) = \begin{cases} a w & 0 \leqslant w \leqslant b
    0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are constants. Given that the median of \(W\) is 2 , find \(a\) and \(b\).
CAIE S2 2017 March Q1
1 In a survey, 36 out of 120 randomly selected voters in Hungton said that if there were an election next week they would vote for the Alpha party. Calculate an approximate \(90 \%\) confidence interval for the proportion of voters in Hungton who would vote for the Alpha party.
CAIE S2 2017 March Q2
2 Karim has noted the lifespans, in weeks, of a large random sample of certain insects. He carries out a test, at the \(1 \%\) significance level, for the population mean, \(\mu\). Karim's null hypothesis is \(\mu = 6.4\).
  1. Given that Karim's test is two-tail, state the alternative hypothesis.
    Karim finds that the value of the test statistic is \(z = 2.43\).
  2. Explain what conclusion he should draw.
  3. Explain briefly when a one-tail test is appropriate, rather than a two-tail test.
CAIE S2 2017 March Q3
3 The length, in centimetres, of a certain type of snake is modelled by the random variable \(X\) with mean 52 and standard deviation 6.1. A random sample of 75 snakes is selected, and the sample mean, \(\bar { X }\), is found.
  1. Find \(\mathrm { P } ( 51 < \bar { X } < 53 )\).
  2. Explain why it was necessary to use the Central Limit theorem in the solution to part (i).
CAIE S2 2017 March Q4
4 At a doctors' surgery, the number of missed appointments per day has a Poisson distribution. In the past the mean number of missed appointments per day has been 0.9 . Following some publicity, the manager carries out a hypothesis test to determine whether this mean has decreased. If there are fewer than 3 missed appointments in a randomly chosen 5-day period, she will conclude that the mean has decreased.
  1. Find the probability of a Type I error.
  2. State what is meant by a Type I error in this context.
  3. Find the probability of a Type II error if the mean number of missed appointments per day is 0.2 .
CAIE S2 2017 March Q5
5

  1. \includegraphics[max width=\textwidth, alt={}, center]{61ba010c-d6a2-4c19-9998-0ae048244a32-06_292_517_264_338}
    \includegraphics[max width=\textwidth, alt={}, center]{61ba010c-d6a2-4c19-9998-0ae048244a32-06_289_518_264_858}
    \includegraphics[max width=\textwidth, alt={}, center]{61ba010c-d6a2-4c19-9998-0ae048244a32-06_273_510_365_1377} The diagram shows the graphs of three functions, \(f _ { 1 } , f _ { 2 }\) and \(f _ { 3 }\). The function \(f _ { 1 }\) is a probability density function.
    1. State the value of \(k\).
    2. For each of the functions \(\mathrm { f } _ { 2 }\) and \(\mathrm { f } _ { 3 }\), state why it cannot be a probability density function.
  2. The probability density function g is defined by $$g ( x ) = \begin{cases} 6 \left( a ^ { 2 } - x ^ { 2 } \right) & - a \leqslant x \leqslant a
    0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.
    1. Show that \(a = \frac { 1 } { 2 }\).
    2. State the value of \(\mathrm { E } ( X )\).
    3. Find \(\operatorname { Var } ( X )\).
CAIE S2 2017 March Q6
6 The masses, in kilograms, of cartons of sugar and cartons of flour have the distributions \(\mathrm { N } \left( 78.8,12.6 ^ { 2 } \right)\) and \(\mathrm { N } \left( 62.0,10.0 ^ { 2 } \right)\) respectively.
  1. The standard load for a certain crane is 8 cartons of sugar and 3 cartons of flour. The maximum load that can be carried safely by the crane is 900 kg . Stating a necessary assumption, find the percentage of standard loads that will exceed the maximum safe load.
  2. Find the probability that a randomly chosen carton of sugar has a smaller mass than a randomly chosen carton of flour.
CAIE S2 2017 March Q7
7 The number of planes arriving at an airport every hour during daytime is modelled by the random variable \(X\) with distribution \(\operatorname { Po } ( 5.2 )\).
  1. State two assumptions required for the Poisson model to be valid in this context.
  2. (a) Find the probability that the number of planes arriving in a 15 -minute period is greater than 1 and less than 4,
    (b) Find the probability that more than 3 planes will arrive in a 40-minute period.
  3. The airport has enough staff to deal with a maximum of 60 planes landing during a 10-hour day. Use a suitable approximation to find the probability that, on a randomly chosen 10-hour day, staff will be able to deal with all the planes that land.
CAIE S2 2024 March Q1
1 The lengths, \(X \mathrm {~cm}\), of a sample of 100 insects of a certain type were summarised as follows. $$n = 100 \quad \sum x = 36.8 \quad \sum x ^ { 2 } = 17.34$$
  1. Calculate unbiased estimates for the population mean and variance of \(X\).
  2. State a necessary condition for the estimates found in part (a) to be reliable.
CAIE S2 2024 March Q2
2 A random sample of 250 people living in Barapet was chosen. It was found that 78 of these people owned a BETEC phone.
  1. Calculate an approximate \(98 \%\) confidence interval for the proportion of people living in Barapet who own a BETEC phone.
  2. Manjit claims that more than \(40 \%\) of the people living in Barapet own a BETEC phone. Use your answer to part (a) to comment on this claim.
CAIE S2 2024 March Q3
3 In a certain lottery, on average 1 in every 10000 tickets is a prize-winning ticket. An agent sells 6000 tickets.
  1. Use a suitable approximating distribution to find the probability that at least 3 of the tickets sold by the agent are prize-winning tickets.
  2. Justify the use of your approximating distribution in this context.
CAIE S2 2024 March Q4
4 Each year a transport firm uses \(X\) litres of gasoline and \(Y\) litres of diesel fuel, where \(X\) and \(Y\) have the independent distributions \(X \sim \mathrm {~N} ( 10700,950 ) ^ { 2 }\) and \(Y \sim \mathrm {~N} \left( 13400,1210 ^ { 2 } \right)\).
  1. Find the probability that in a randomly chosen year the firm uses more gasoline than diesel fuel.
    The costs per litre of gasoline and diesel fuel are \\(0.80 and
    )0.85 respectively.
  2. Find the probability that the total cost of gasoline and diesel fuel in a randomly chosen year is between \(
    ) 20000\( and \)\\( 22000\).
CAIE S2 2024 March Q5
5 A teacher models the numbers of girls and boys who arrive late for her class on any day by the independent random variables \(G \sim \operatorname { Po } ( 0.10 )\) and \(B \sim \operatorname { Po } ( 0.15 )\) respectively.
  1. Find the probability that during a randomly chosen 2-day period no girls arrive late.
  2. Find the probability that during a randomly chosen 5-day period the total number of students who arrive late is less than 3 .
  3. It is given that the values of \(\mathrm { P } ( G = r )\) and \(\mathrm { P } ( B = r )\) for \(r \geqslant 3\) are very small and can be ignored. Find the probability that on a randomly chosen day more girls arrive late than boys.
    Following a timetable change the teacher claims that on average more students arrive late than before the change. During a randomly chosen 5-day period a total of 4 students are late.
  4. Test the teacher's claim at the \(5 \%\) significance level.
CAIE S2 2024 March Q6
6 The graph of the probability density function f of a random variable \(X\) is symmetrical about the line \(x = 2\). It is given that \(\mathrm { P } ( 2 < X < 5 ) = \frac { 117 } { 256 }\).
  1. Using only this information show that \(\mathrm { P } ( X > - 1 ) = \frac { 245 } { 256 }\).
    It is now given that, for \(x\) in a suitable domain, $$f ( x ) = k \left( 12 + 4 x - x ^ { 2 } \right) , \text { where } k \text { is a constant. }$$
  2. Find the value of \(k\).
  3. A different random variable \(X\) has probability density function \(\mathbf { g } ( x ) = \frac { 2 } { 9 } \left( 2 + x - x ^ { 2 } \right)\). The domain of \(X\) is all values of \(x\) for which \(\mathrm { g } ( x ) \geqslant 0\). Find \(\operatorname { Var } ( X )\).
    \includegraphics[max width=\textwidth, alt={}, center]{ff3433b0-baab-45e3-845e-56a794739bba-11_63_1547_447_347}
CAIE S2 2024 March Q7
7 The heights, in centimetres, of adult females in Litania have mean \(\mu\) and standard deviation \(\sigma\). It is known that in 2004 the values of \(\mu\) and \(\sigma\) were 163.21 and 6.95 respectively. The government claims that the value of \(\mu\) this year is greater than it was in 2004. In order to test this claim a researcher plans to carry out a hypothesis test at the \(1 \%\) significance level. He records the heights of a random sample of 300 adult females in Litania this year and finds the value of the sample mean.
  1. State the probability of a Type I error.
    \includegraphics[max width=\textwidth, alt={}]{ff3433b0-baab-45e3-845e-56a794739bba-12_74_1577_557_322} ........................................................................................................................................ You should assume that the value of \(\sigma\) after 2004 remains at 6.95 .
  2. Given that the value of \(\mu\) this year is actually 164.91 , find the probability of a Type II error.
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE S2 2020 November Q1
1 It is known that, on average, 1 in 300 flowers of a certain kind are white. A random sample of 200 flowers of this kind is selected.
  1. Use an appropriate approximating distribution to find the probability that more than 1 flower in the sample is white.
  2. Justify the approximating distribution used in part (a).
    The probability that a randomly chosen flower of another kind is white is 0.02 . A random sample of 150 of these flowers is selected.
  3. Use an appropriate approximating distribution to find the probability that the total number of white flowers in the two samples is less than 4 .