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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.
CAIE S2 2022 March Q1
4 marks Easy -1.2
1 The lengths, in millimetres, of a random sample of 12 rods made by a certain machine are as follows.
200
201
198
202
200
199
199
201
197
202
200
199
  1. Find unbiased estimates of the population mean and variance.
  2. Give a statistical reason why these estimates may not be reliable.
CAIE S2 2022 March Q2
5 marks Moderate -0.3
2 Harry has a five-sided spinner with sectors coloured blue, green, red, yellow and black. Harry thinks the spinner may be biased. He plans to carry out a hypothesis test with the following hypotheses. $$\begin{aligned} & \mathrm { H } _ { 0 } : \mathrm { P } ( \text { the spinner lands on blue } ) = \frac { 1 } { 5 } \\ & \mathrm { H } _ { 1 } : \mathrm { P } ( \text { the spinner lands on blue } ) \neq \frac { 1 } { 5 } \end{aligned}$$ Harry spins the spinner 300 times. It lands on blue on 45 spins.
Use a suitable approximation to carry out Harry's test at the \(5 \%\) significance level.
CAIE S2 2022 March Q3
5 marks Standard +0.3
3 A random sample of 500 households in a certain town was chosen. Using this sample, a confidence interval for the proportion, \(p\), of all households in that town that owned two or more cars was found to be \(0.355 < p < 0.445\). Find the confidence level of this confidence interval. Give your answer correct to the nearest integer.
CAIE S2 2022 March Q4
6 marks Standard +0.3
4 In the past the time, in minutes, taken by students to complete a certain challenge had mean 25.5 and standard deviation 5.2. A new challenge is devised and it is expected that students will take, on average, less than 25.5 minutes to complete this challenge. A random sample of 40 students is chosen and their mean time for the new challenge is found to be 23.7 minutes.
  1. Assuming that the standard deviation of the time for the new challenge is 5.2 minutes, test at the \(1 \%\) significance level whether the population mean time for the new challenge is less than 25.5 minutes.
  2. State, with a reason, whether it is possible that a Type I error was made in the test in part (a).
CAIE S2 2022 March Q5
9 marks Standard +0.3
5 The heights of buildings in a large city are normally distributed with mean 18.3 m and standard deviation 2.5 m .
  1. Find the probability that the total height of 5 randomly chosen buildings in the city is more than 95 m .
  2. Find the probability that the difference between the heights of two randomly chosen buildings in the city is less than 1 m .
CAIE S2 2022 March Q6
11 marks Standard +0.3
6 In a game a ball is rolled down a slope and along a track until it stops. The distance, in metres, travelled by the ball is modelled by the random variable \(X\) with probability density function $$f ( x ) = \begin{cases} - k ( x - 1 ) ( x - 3 ) & 1 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Without calculation, explain why \(\mathrm { E } ( X ) = 2\).
  2. Show that \(k = \frac { 3 } { 4 }\).
  3. Find \(\operatorname { Var } ( X )\).
    One turn consists of rolling the ball 3 times and noting the largest value of \(X\) obtained. If this largest value is greater than 2.5, the player scores a point.
  4. Find the probability that on a particular turn the player scores a point.
CAIE S2 2022 March Q7
10 marks Standard +0.8
7
  1. Two ponds, \(A\) and \(B\), each contain a large number of fish. It is known that \(2.4 \%\) of fish in pond \(A\) are carp and \(1.8 \%\) of fish in pond \(B\) are carp. Random samples of 50 fish from pond \(A\) and 60 fish from pond \(B\) are selected. Use appropriate Poisson approximations to find the following probabilities.
    1. The samples contain at least 2 carp from pond \(A\) and at least 2 carp from pond \(B\).
    2. The samples contain at least 4 carp altogether.
  2. The random variables \(X\) and \(Y\) have the distributions \(\operatorname { Po } ( \lambda )\) and \(\operatorname { Po } ( \mu )\) respectively. It is given that
    • \(\mathrm { P } ( X = 0 ) = [ \mathrm { P } ( Y = 0 ) ] ^ { 2 }\),
    • \(\mathrm { P } ( X = 2 ) = k [ \mathrm { P } ( Y = 1 ) ] ^ { 2 }\), where \(k\) is a non-zero constant.
    Find the value of \(k\).
    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 S1 2010 November Q1
2 marks Easy -1.8
1 Name the distribution and suggest suitable numerical parameters that you could use to model the weights in kilograms of female 18-year-old students.
CAIE S1 2010 November Q2
5 marks Easy -1.2
2 In a probability distribution the random variable \(X\) takes the value \(x\) with probability \(k x\), where \(x\) takes values \(1,2,3,4,5\) only.
  1. Draw up a probability distribution table for \(X\), in terms of \(k\), and find the value of \(k\).
  2. Find \(\mathrm { E } ( X )\).
CAIE S1 2010 November Q3
5 marks Moderate -0.3
3 It was found that \(68 \%\) of the passengers on a train used a cell phone during their train journey. Of those using a cell phone, \(70 \%\) were under 30 years old, \(25 \%\) were between 30 and 65 years old and the rest were over 65 years old. Of those not using a cell phone, \(26 \%\) were under 30 years old and \(64 \%\) were over 65 years old.
  1. Draw a tree diagram to represent this information, giving all probabilities as decimals.
  2. Given that one of the passengers is 45 years old, find the probability of this passenger using a cell phone during the journey.
CAIE S1 2010 November Q4
6 marks Easy -1.2
4 Delip measured the speeds, \(x \mathrm {~km}\) per hour, of 70 cars on a road where the speed limit is 60 km per hour. His results are summarised by \(\Sigma ( x - 60 ) = 245\).
  1. Calculate the mean speed of these 70 cars. His friend Sachim used values of \(( x - 50 )\) to calculate the mean.
  2. Find \(\Sigma ( x - 50 )\).
  3. The standard deviation of the speeds is 10.6 km per hour. Calculate \(\Sigma ( x - 50 ) ^ { 2 }\).
CAIE S1 2010 November Q5
8 marks Moderate -0.8
5 The following histogram illustrates the distribution of times, in minutes, that some students spent taking a shower. \includegraphics[max width=\textwidth, alt={}, center]{ec425eaf-8afc-4671-9ef3-ba2477b884ef-3_1031_1326_372_406}
  1. Copy and complete the following frequency table for the data.
    Time \(( t\) minutes \()\)\(2 < t \leqslant 4\)\(4 < t \leqslant 6\)\(6 < t \leqslant 7\)\(7 < t \leqslant 8\)\(8 < t \leqslant 10\)\(10 < t \leqslant 16\)
    Frequency
  2. Calculate an estimate of the mean time to take a shower.
  3. Two of these students are chosen at random. Find the probability that exactly one takes between 7 and 10 minutes to take a shower.
CAIE S1 2010 November Q6
10 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{ec425eaf-8afc-4671-9ef3-ba2477b884ef-4_387_899_255_623} A small aeroplane has 14 seats for passengers. The seats are arranged in 4 rows of 3 seats and a back row of 2 seats (see diagram). 12 passengers board the aeroplane.
  1. How many possible seating arrangements are there for the 12 passengers? Give your answer correct to 3 significant figures. These 12 passengers consist of 2 married couples (Mr and Mrs Lin and Mr and Mrs Brown), 5 students and 3 business people.
  2. The 3 business people sit in the front row. The 5 students each sit at a window seat. Mr and Mrs Lin sit in the same row on the same side of the aisle. Mr and Mrs Brown sit in another row on the same side of the aisle. How many possible seating arrangements are there?
  3. If, instead, the 12 passengers are seated randomly, find the probability that Mrs Lin sits directly behind a student and Mrs Brown sits in the front row.
CAIE S1 2010 November Q7
14 marks Standard +0.3
7 The times spent by people visiting a certain dentist are independent and normally distributed with a mean of 8.2 minutes. \(79 \%\) of people who visit this dentist have visits lasting less than 10 minutes.
  1. Find the standard deviation of the times spent by people visiting this dentist.
  2. Find the probability that the time spent visiting this dentist by a randomly chosen person deviates from the mean by more than 1 minute.
  3. Find the probability that, of 6 randomly chosen people, more than 2 have visits lasting longer than 10 minutes.
  4. Find the probability that, of 35 randomly chosen people, fewer than 16 have visits lasting less than 8.2 minutes. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
CAIE S1 2011 November Q1
5 marks Moderate -0.8
1 When a butternut squash seed is sown the probability that it will germinate is 0.86 , independently of any other seeds. A market gardener sows 250 of these seeds. Use a suitable approximation to find the probability that more than 210 germinate.
CAIE S1 2011 November Q2
6 marks Moderate -0.8
2 The values, \(x\), in a particular set of data are summarised by $$\Sigma ( x - 25 ) = 133 , \quad \Sigma ( x - 25 ) ^ { 2 } = 3762 .$$ The mean, \(\bar { x }\), is 28.325 .
  1. Find the standard deviation of \(x\).
  2. Find \(\Sigma x ^ { 2 }\).
CAIE S1 2011 November Q3
6 marks Moderate -0.3
3 A team of 4 is to be randomly chosen from 3 boys and 5 girls. The random variable \(X\) is the number of girls in the team.
  1. Draw up a probability distribution table for \(X\).
  2. Given that \(\mathrm { E } ( X ) = \frac { 5 } { 2 }\), calculate \(\operatorname { Var } ( X )\).
CAIE S1 2011 November Q4
6 marks Easy -1.8
4 The marks of the pupils in a certain class in a History examination are as follows. $$\begin{array} { l l l l l l l l l l l l l } 28 & 33 & 55 & 38 & 42 & 39 & 27 & 48 & 51 & 37 & 57 & 49 & 33 \end{array}$$ The marks of the pupils in a Physics examination are summarised as follows.
Lower quartile: 28 , Median: 39, Upper quartile: 67.
The lowest mark was 17 and the highest mark was 74 .
  1. Draw box-and-whisker plots in a single diagram on graph paper to illustrate the marks for History and Physics.
  2. State one difference, which can be seen from the diagram, between the marks for History and Physics.
CAIE S1 2011 November Q5
9 marks Standard +0.3
5 The weights of letters posted by a certain business are normally distributed with mean 20 g . It is found that the weights of \(94 \%\) of the letters are within 12 g of the mean.
  1. Find the standard deviation of the weights of the letters.
  2. Find the probability that a randomly chosen letter weighs more than 13 g .
  3. Find the probability that at least 2 of a random sample of 7 letters have weights which are more than 12 g above the mean.