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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 2015 November Q1
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 2015 November Q2
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 2015 November Q3
3 Jagdeesh measured the lengths, \(x\) minutes, of 60 randomly chosen lectures. His results are summarised below. $$n = 60 \quad \Sigma x = 3420 \quad \Sigma x ^ { 2 } = 195200$$
  1. Calculate unbiased estimates of the population mean and variance.
  2. Calculate a \(98 \%\) confidence interval for the population mean.
CAIE S2 2015 November Q4
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 2015 November Q5
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 2015 November Q6
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.
CAIE S2 2015 November Q7
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 2017 November Q1
1 A random variable, \(X\), has the distribution \(\operatorname { Po } ( 31 )\). Use the normal approximation to the Poisson distribution to find \(\mathrm { P } ( X > 40 )\).
CAIE S2 2017 November Q2
2 An airline has found that, on average, 1 in 100 passengers do not arrive for each flight, and that this occurs randomly. For one particular flight the airline always sells 403 seats. The plane only has room for 400 passengers, so the flight is overbooked if the number of passengers who do not arrive is less than 3 . Use a suitable approximation to find the probability that the flight is overbooked.
CAIE S2 2017 November Q3
3 After an election 153 adults, from a random sample of 200 adults, said that they had voted. Using this information, an \(\alpha \%\) confidence interval for the proportion of all adults who voted in the election was found to be 0.695 to 0.835 , both correct to 3 significant figures. Find the value of \(\alpha\), correct to the nearest integer.
CAIE S2 2017 November Q4
4 The lengths, in millimetres, of rods produced by a machine are normally distributed with mean \(\mu\) and standard deviation 0.9. A random sample of 75 rods produced by the machine has mean length 300.1 mm .
  1. Find a \(99 \%\) confidence interval for \(\mu\), giving your answer correct to 2 decimal places.
    The manufacturer claims that the machine produces rods with mean length 300 mm .
  2. Use the confidence interval found in part (i) to comment on this claim.
CAIE S2 2017 November Q5
5 A continuous random variable, \(X\), has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 4 } ( x + 1 ) & 0 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$
  1. Find \(\mathrm { E } ( X )\).
    ................................................................................................................................. .
  2. Find the median of \(X\).
CAIE S2 2017 November Q6
3 marks
6 The numbers of barrels of oil, in millions, extracted per day in two oil fields \(A\) and \(B\) are modelled by the independent random variables \(X\) and \(Y\) respectively, where \(X \sim \mathrm {~N} \left( 3.2,0.4 ^ { 2 } \right)\) and \(Y \sim \mathrm {~N} \left( 4.3,0.6 ^ { 2 } \right)\). The income generated by the oil from the two fields is \(\\) 90\( per barrel for \)A\( and \)\\( 95\) per barrel for \(B\).
  1. Find the mean and variance of the daily income, in millions of dollars, generated by field \(A\). [3]
  2. Find the probability that the total income produced by the two fields in a day is at least \(\\) 670$ million.
CAIE S2 2017 November Q7
7 In the past the number of cars sold per day at a showroom has been modelled by a random variable with distribution \(\operatorname { Po } ( 0.7 )\). Following an advertising campaign, it is hoped that the mean number of sales per day will increase. In order to test at the \(10 \%\) significance level whether this is the case, the total number of sales during the first 5 days after the campaign is noted. You should assume that a Poisson model is still appropriate.
  1. Given that the total number of cars sold during the 5 days is 5 , carry out the test.
    The number of cars sold per day at another showroom has the independent distribution \(\operatorname { Po } ( 0.6 )\). Assume that the distribution for the first showroom is still \(\operatorname { Po } ( 0.7 )\).
  2. Find the probability that the total number of cars sold in the two showrooms during 3 days is exactly 2 .
CAIE S2 2017 November Q8
8 In order to test the effect of a drug, a researcher monitors the concentration, \(X\), of a certain protein in the blood stream of patients. For patients who are not taking the drug the mean value of \(X\) is 0.185 . A random sample of 150 patients taking the drug was selected and the values of \(X\) were found. The results are summarised below. $$n = 150 \quad \Sigma x = 27.0 \quad \Sigma x ^ { 2 } = 5.01$$ The researcher wishes to test at the \(1 \%\) significance level whether the mean concentration of the protein in the blood stream of patients taking the drug is less than 0.185 .
  1. Carry out the test.
  2. Given that, in fact, the mean concentration for patients taking the drug is 0.175 , find the probability of a Type II error occurring in the test.
CAIE S2 2018 November Q1
1 The standard deviation of the heights of adult males is 7.2 cm . The mean height of a sample of 200 adult males is found to be 176 cm .
  1. Calculate a \(97.5 \%\) confidence interval for the mean height of adult males.
  2. State a necessary condition for the calculation in part (i) to be valid.
CAIE S2 2018 November Q2
2 A headteacher models the number of children who bring a 'healthy' packed lunch to school on any day by the distribution \(\mathrm { B } ( 150 , p )\). In the past, she has found that \(p = \frac { 1 } { 3 }\). Following the opening of a fast food outlet near the school, she wishes to test, at the \(1 \%\) significance level, whether the value of \(p\) has decreased.
  1. State the null and alternative hypotheses for this test.
    On a randomly chosen day she notes the number, \(N\), of children who bring a 'healthy' packed lunch to school. She finds that \(N = 36\) and then, assuming that the null hypothesis is true, she calculates that \(\mathrm { P } ( N \leqslant 36 ) = 0.0084\).
  2. State, with a reason, the conclusion that the headteacher should draw from the test.
  3. According to the model, what is the largest number of children who might bring a packed lunch to school?
CAIE S2 2018 November Q3
3 A population has mean 12 and standard deviation 2.5. A large random sample of size \(n\) is chosen from this population and the sample mean is denoted by \(\bar { X }\). Given that \(\mathrm { P } ( \bar { X } < 12.2 ) = 0.975\), correct to 3 significant figures, find the value of \(n\).
CAIE S2 2018 November Q4
4 Small drops of two liquids, \(A\) and \(B\), are randomly and independently distributed in the air. The average numbers of drops of \(A\) and \(B\) per cubic centimetre of air are 0.25 and 0.36 respectively.
  1. A sample of \(10 \mathrm {~cm} ^ { 3 }\) of air is taken at random. Find the probability that the total number of drops of \(A\) and \(B\) in this sample is at least 4 .
  2. A sample of \(100 \mathrm {~cm} ^ { 3 }\) of air is taken at random. Use an approximating distribution to find the probability that the total number of drops of \(A\) and \(B\) in this sample is less than 60 .
CAIE S2 2018 November Q5
5 The times, in months, taken by a builder to build two types of house, \(P\) and \(Q\), are represented by the independent variables \(T _ { 1 } \sim \mathrm {~N} \left( 2.2,0.4 ^ { 2 } \right)\) and \(T _ { 2 } \sim \mathrm {~N} \left( 2.8,0.5 ^ { 2 } \right)\) respectively.
  1. Find the probability that the total time taken to build one house of each type is less than 6 months.
  2. Find the probability that the time taken to build a type \(Q\) house is more than 1.2 times the time taken to build a type \(P\) house.