Questions — CAIE S2 (737 questions)

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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
    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 S2 2014 June Q1
3 marks Easy -1.2
1 The weights, in grams, of a random sample of 8 packets of cereal are as follows. $$\begin{array} { l l l l l l l l } 250 & 248 & 255 & 244 & 259 & 250 & 242 & 258 \end{array}$$ Calculate unbiased estimates of the population mean and variance.
CAIE S2 2014 June Q2
5 marks Standard +0.3
2 Each day Samuel travels from \(A\) to \(B\) and from \(B\) to \(C\). He then returns directly from \(C\) to \(A\). The times, in minutes, for these three journeys have the independent distributions \(\mathrm { N } \left( 20,2 ^ { 2 } \right) , \mathrm { N } \left( 18,1.5 ^ { 2 } \right)\) and \(\mathrm { N } \left( 30,1.8 ^ { 2 } \right)\), respectively. Find the probability that, on a randomly chosen day, the total time for his two journeys from \(A\) to \(B\) and \(B\) to \(C\) is less than the time for his return journey from \(C\) to \(A\). [5]
CAIE S2 2014 June Q3
5 marks Standard +0.8
3 The number of calls per day to an enquiry desk has a Poisson distribution. In the past the mean has been 5 . In order to test whether the mean has changed, the number of calls on a random sample of 10 days was recorded. The total number of calls was found to be 61 . Use an approximate distribution to test at the 10\% significance level whether the mean has changed.
CAIE S2 2014 June Q4
8 marks Standard +0.3
4
  1. The random variable \(W\) has the distribution \(\operatorname { Po } ( 1.5 )\). Find the probability that the sum of 3 independent values of \(W\) is greater than 2 .
  2. The random variable \(X\) has the distribution \(\operatorname { Po } ( \lambda )\). Given that \(\mathrm { P } ( X = 0 ) = 0.523\), find the value of \(\lambda\) correct to 3 significant figures.
  3. The random variable \(Y\) has the distribution \(\operatorname { Po } ( \mu )\), where \(\mu \neq 0\). Given that $$\mathrm { P } ( Y = 3 ) = 24 \times \mathrm { P } ( Y = 1 )$$ find \(\mu\).
CAIE S2 2014 June Q5
9 marks Moderate -0.3
5 Mahmoud throws a coin 400 times and finds that it shows heads 184 times. The probability that the coin shows heads on any throw is denoted by \(p\).
  1. Calculate an approximate \(95 \%\) confidence interval for \(p\).
  2. Mahmoud claims that the coin is not fair. Use your answer to part (i) to comment on this claim.
  3. Mahmoud's result of 184 heads in 400 throws gives an \(\alpha \%\) confidence interval for \(p\) with width 0.1 . Calculate the value of \(\alpha\).
CAIE S2 2014 June Q6
10 marks Standard +0.3
6 The time, \(T\) hours, spent by people on a visit to a museum has probability density function $$\mathrm { f } ( t ) = \begin{cases} k t \left( 16 - t ^ { 2 } \right) & 0 \leqslant t \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 1 } { 64 }\).
  2. Calculate the probability that two randomly chosen people each spend less than 1 hour on a visit to the museum.
  3. Find the mean time spent on a visit to the museum.
CAIE S2 2014 June Q7
10 marks Standard +0.3
7 A researcher is investigating the actual lengths of time that patients spend with the doctor at their appointments. He plans to choose a sample of 12 appointments on a particular day.
  1. Which of the following methods is preferable, and why?
    • Choose the first 12 appointments of the day.
    • Choose 12 appointments evenly spaced throughout the day.
    Appointments are scheduled to last 10 minutes. The actual lengths of time, in minutes, that patients spend with the doctor may be assumed to have a normal distribution with mean \(\mu\) and standard deviation 3.4. The researcher suspects that the actual time spent is more than 10 minutes on average. To test this suspicion, he recorded the actual times spent for a random sample of 12 appointments and carried out a hypothesis test at the 1\% significance level.
  2. State the probability of making a Type I error and explain what is meant by a Type I error in this context.
  3. Given that the total length of time spent for the 12 appointments was 147 minutes, carry out the test.
  4. Give a reason why the Central Limit theorem was not needed in part (iii).
CAIE S2 2015 June Q1
5 marks Easy -1.8
1 Jyothi wishes to choose a representative sample of 5 students from the 82 members of her school year.
  1. She considers going into the canteen and choosing a table with five students from her year sitting at it, and using these five people as her sample. Give two reasons why this method is unsatisfactory.
  2. Jyothi decides to use another method. She numbers all the students in her year from 1 to 82 . Then she uses her calculator and generates the following random numbers. $$231492 \quad 762305 \quad 346280$$ From these numbers, she obtains the student numbers \(23,14,76,5,34\) and 62 . Explain how Jyothi obtained these student numbers from the list of random numbers.
CAIE S2 2015 June Q2
5 marks Moderate -0.5
2 Marie claims that she can predict the winning horse at the local races. There are 8 horses in each race. Nadine thinks that Marie is just guessing, so she proposes a test. She asks Marie to predict the winners of the next 10 races and, if she is correct in 3 or more races, Nadine will accept Marie's claim.
  1. State suitable null and alternative hypotheses.
  2. Calculate the probability of a Type I error.
  3. State the significance level of the test.
CAIE S2 2015 June Q3
6 marks Challenging +1.2
3 A die is biased so that the probability that it shows a six on any throw is \(p\).
  1. In an experiment, the die shows a six on 22 out of 100 throws. Find an approximate \(97 \%\) confidence interval for \(p\).
  2. The experiment is repeated and another \(97 \%\) confidence interval is found. Find the probability that exactly one of the two confidence intervals includes the true value of \(p\).
CAIE S2 2015 June Q4
6 marks Moderate -0.8
4 The marks, \(x\), of a random sample of 50 students in a test were summarised as follows. $$n = 50 \quad \Sigma x = 1508 \quad \Sigma x ^ { 2 } = 51825$$
  1. Calculate unbiased estimates of the population mean and variance.
  2. Each student's mark is scaled using the formula \(y = 1.5 x + 10\). Find estimates of the population mean and variance of the scaled marks, \(y\).
CAIE S2 2015 June Q5
7 marks Moderate -0.8
5 The mean breaking strength of cables made at a certain factory is supposed to be 5 tonnes. The quality control department wishes to test whether the mean breaking strength of cables made by a particular machine is actually less than it should be. They take a random sample of 60 cables. For each cable they find the breaking strength by gradually increasing the tension in the cable and noting the tension when the cable breaks.
  1. Give a reason why it is necessary to take a sample rather then testing all the cables produced by the machine.
  2. The mean breaking strength of the 60 cables in the sample is found to be 4.95 tonnes. Given that the population standard deviation of breaking strengths is 0.15 tonnes, test at the \(1 \%\) significance level whether the population mean breaking strength is less than it should be.
  3. Explain whether it was necessary to use the Central Limit theorem in the solution to part (ii).
CAIE S2 2015 June Q6
10 marks Standard +0.3
6 People arrive at a checkout in a store at random, and at a constant mean rate of 0.7 per minute. Find the probability that
  1. exactly 3 people arrive at the checkout during a 5 -minute period,
  2. at least 30 people arrive at the checkout during a 1-hour period. People arrive independently at another checkout in the store at random, and at a constant mean rate of 0.5 per minute.
  3. Find the probability that a total of more than 3 people arrive at this pair of checkouts during a 2-minute period.
CAIE S2 2015 June Q7
11 marks Moderate -0.3
7 The probability density function of the random variable \(X\) is given by $$f ( x ) = \begin{cases} \frac { 3 } { 4 } x ( c - x ) & 0 \leqslant x \leqslant c \\ 0 & \text { otherwise } \end{cases}$$ where \(c\) is a constant.
  1. Show that \(c = 2\).
  2. Sketch the graph of \(y = \mathrm { f } ( x )\) and state the median of \(X\).
  3. Find \(\mathrm { P } ( X < 1.5 )\).
  4. Hence write down the value of \(\mathrm { P } ( 0.5 < X < 1 )\).
CAIE S2 2016 June Q1
5 marks Moderate -0.8
1 A six-sided die shows a six on 25 throws out of 200 throws. Test at the \(10 \%\) significance level the null hypothesis: P (throwing a six) \(= \frac { 1 } { 6 }\), against the alternative hypothesis: P (throwing a six) \(< \frac { 1 } { 6 }\).