Questions — CAIE (7646 questions)

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CAIE S2 2024 June Q3
4 marks Challenging +1.2
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
6 marks Standard +0.3
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
9 marks Standard +0.3
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
10 marks Standard +0.3
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
10 marks Standard +0.3
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
4 marks Moderate -0.8
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
4 marks Moderate -0.5
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
5 marks Standard +0.8
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
5 marks Moderate -0.3
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
10 marks Standard +0.3
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
9 marks Standard +0.3
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
13 marks Standard +0.3
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
3 marks Moderate -0.3
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
5 marks Standard +0.3
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
8 marks Standard +0.3
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.
CAIE S2 2020 March Q4
7 marks Standard +0.3
4 The number of accidents on a certain road has a Poisson distribution with mean 0.4 per 50-day period.
  1. Find the probability that there will be fewer than 3 accidents during a year (365 days).
  2. The probability that there will be no accidents during a period of \(n\) days is greater than 0.95 . Find the largest possible value of \(n\).
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 2020 March Q6
10 marks Standard +0.3
6 The volumes, in millilitres, of large and small cups of tea are modelled by the distributions \(\mathrm { N } ( 200,30 )\) and \(\mathrm { N } ( 110,20 )\) respectively.
  1. Find the probability that the total volume of a randomly chosen large cup of tea and a randomly chosen small cup of tea is less than 300 ml .
  2. Find the probability that the volume of a randomly chosen large cup of tea is more than twice the volume of a randomly chosen small cup of tea.
CAIE S2 2020 March Q7
8 marks Standard +0.3
7 A national survey shows that \(95 \%\) of year 12 students use social media. Arvin suspects that the percentage of year 12 students at his college who use social media is less than the national percentage. He chooses a random sample of 20 students at his college and notes the number who use social media. He then carries out a test at the \(2 \%\) significance level.
  1. Find the rejection region for the test.
  2. Find the probability of a Type I error.
  3. Jimmy believes that the true percentage at Arvin's college is \(70 \%\). Assuming that Jimmy is correct, 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 2021 March Q1
7 marks Moderate -0.5
1 A construction company notes the time, \(t\) days, that it takes to build each house of a certain design. The results for a random sample of 60 such houses are summarised as follows. $$\Sigma t = 4820 \quad \Sigma t ^ { 2 } = 392050$$
  1. Calculate a 98\% confidence interval for the population mean time.
  2. Explain why it was necessary to use the Central Limit theorem in part (a).
CAIE S2 2021 March Q2
9 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{image-not-found} 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 2021 March Q3
4 marks Moderate -0.8
3 An architect wishes to investigate whether the buildings in a certain city are higher, on average, than buildings in other cities. He takes a large random sample of buildings from the city and finds the mean height of the buildings in the sample. He calculates the value of the test statistic, \(z\), and finds that \(z = 2.41\).
  1. Explain briefly whether he should use a one-tail test or a two-tail test.
  2. Carry out the test at the \(1 \%\) significance level.
CAIE S2 2021 March Q4
10 marks Moderate -0.3
4 On average, 1 in 400 microchips made at a certain factory are faulty. The number of faulty microchips in a random sample of 1000 is denoted by \(X\).
  1. State the distribution of \(X\), giving the values of any parameters.
  2. State an approximating distribution for \(X\), giving the values of any parameters.
  3. Use this approximating distribution to find each of the following.
    1. \(\mathrm { P } ( X = 4 )\).
    2. \(\mathrm { P } ( 2 \leqslant X \leqslant 4 )\).
  4. Use a suitable approximating distribution to find the probability that, in a random sample of 700 microchips, there will be at least 1 faulty one.
CAIE S2 2021 March Q5
10 marks Standard +0.3
5 The volumes, in litres, of juice in large and small bottles have the distributions \(\mathrm { N } ( 5.10,0.0102 )\) and \(\mathrm { N } ( 2.51,0.0036 )\) respectively.
  1. Find the probability that the total volume of juice in 3 randomly chosen large bottles and 4 randomly chosen small bottles is less than 25.5 litres.
  2. Find the probability that the volume of juice in a randomly chosen large bottle is at least twice the volume of juice in a randomly chosen small bottle.
CAIE S2 2021 March Q6
10 marks Standard +0.3
6 It is known that \(8 \%\) of adults in a certain town own a Chantor car. After an advertising campaign, a car dealer wishes to investigate whether this proportion has increased. He chooses a random sample of 25 adults from the town and notes how many of them own a Chantor car.
  1. He finds that 4 of the 25 adults own a Chantor car. Carry out a hypothesis test at the 5\% significance level.
  2. Explain which of the errors, Type I or Type II, might have been made in carrying out the test in part (a).
    Later, the car dealer takes another random sample of 25 adults from the town and carries out a similar hypothesis test at the 5\% significance level.
  3. Find the probability of a Type I 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.