Questions — CAIE S2 (717 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
CAIE S2 2006 June Q3
3 Random samples of size 120 are taken from the distribution \(\mathrm { B } ( 15,0.4 )\).
  1. Describe fully the distribution of the sample mean.
  2. Find the probability that the mean of a random sample of size 120 is greater than 6.1.
CAIE S2 2006 June Q4
4 A certain make of washing machine has a wash-time with mean 56.9 minutes and standard deviation 4.8 minutes. A certain make of tumble dryer has a drying-time with mean 61.1 minutes and standard deviation 6.3 minutes. Both times are normally distributed and are independent of each other. Find the probability that a randomly chosen wash-time differs by more than 3 minutes from a randomly chosen drying-time.
CAIE S2 2006 June Q5
5 The random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} 4 x ^ { k } & 0 \leqslant x \leqslant 1
0 & \text { otherwise } \end{cases}$$ where \(k\) is a positive constant.
  1. Show that \(k = 3\).
  2. Show that the mean of \(X\) is 0.8 and find the variance of \(X\).
  3. Find the upper quartile of \(X\).
  4. Find the interquartile range of \(X\).
CAIE S2 2006 June Q6
6 A dressmaker makes dresses for Easifit Fashions. Each dress requires \(2.5 \mathrm {~m} ^ { 2 }\) of material. Faults occur randomly in the material at an average rate of 4.8 per \(20 \mathrm {~m} ^ { 2 }\).
  1. Find the probability that a randomly chosen dress contains at least 2 faults. Each dress has a belt attached to it to make an outfit. Independently of faults in the material, the probability that a belt is faulty is 0.03 . Find the probability that, in an outfit,
  2. neither the dress nor its belt is faulty,
  3. the dress has at least one fault and its belt is faulty. The dressmaker attaches 300 randomly chosen belts to 300 randomly chosen dresses. An outfit in which the dress has at least one fault and its belt is faulty is rejected.
  4. Use a suitable approximation to find the probability that fewer than 3 outfits are rejected.
CAIE S2 2006 June Q7
7 The number of cars caught speeding on a certain length of motorway is 7.2 per day, on average. Speed cameras are introduced and the results shown in the following table are those from a random selection of 40 days after this.
Number of cars caught speeding45678910
Number of days57810523
  1. Calculate unbiased estimates of the population mean and variance of the number of cars per day caught speeding after the speed cameras were introduced.
  2. Taking the null hypothesis \(\mathrm { H } _ { 0 }\) to be \(\mu = 7.2\), test at the \(5 \%\) level whether there is evidence that the introduction of speed cameras has resulted in a reduction in the number of cars caught speeding.
  3. State what is meant by a Type I error in words relating to the context of the test in part (ii). Without further calculation, illustrate on a suitable diagram the region representing the probability of this Type I error.
CAIE S2 2007 June Q1
1 The random variable \(X\) has the distribution \(\mathrm { B } ( 10,0.15 )\). Find the probability that the mean of a random sample of 50 observations of \(X\) is greater than 1.4.
CAIE S2 2007 June Q2
2 The random variable \(X\) has the distribution \(\mathrm { N } \left( 3.2,1.2 ^ { 2 } \right)\). The sum of 60 independent observations of \(X\) is denoted by \(S\). Find \(\mathrm { P } ( S > 200 )\).
CAIE S2 2007 June Q3
3 A machine has produced nails over a long period of time, where the length in millimetres was distributed as \(\mathrm { N } ( 22.0,0.19 )\). It is believed that recently the mean length has changed. To test this belief a random sample of 8 nails is taken and the mean length is found to be 21.7 mm . Carry out a hypothesis test at the \(5 \%\) significance level to test whether the population mean has changed, assuming that the variance remains the same.
CAIE S2 2007 June Q4
4 At a certain airport 20\% of people take longer than an hour to check in. A new computer system is installed, and it is claimed that this will reduce the time to check in. It is decided to accept the claim if, from a random sample of 22 people, the number taking longer than an hour to check in is either 0 or 1 .
  1. Calculate the significance level of the test.
  2. State the probability that a Type I error occurs.
  3. Calculate the probability that a Type II error occurs if the probability that a person takes longer than an hour to check in is now 0.09 .
CAIE S2 2007 June Q5
5 It is proposed to model the number of people per hour calling a car breakdown service between the times 0900 and 2100 by a Poisson distribution.
  1. Explain why a Poisson distribution may be appropriate for this situation. People call the car breakdown service at an average rate of 20 per hour, and a Poisson distribution may be assumed to be a suitable model.
  2. Find the probability that exactly 8 people call in any half hour.
  3. By using a suitable approximation, find the probability that exactly 250 people call in the 12 hours between 0900 and 2100.
CAIE S2 2007 June Q6
6 The daily takings, \(
) x\(, for a shop were noted on 30 randomly chosen days. The takings are summarised by \)\Sigma x = 31500 , \Sigma x ^ { 2 } = 33141816$.
  1. Calculate unbiased estimates of the population mean and variance of the shop's daily takings.
  2. Calculate a \(98 \%\) confidence interval for the mean daily takings. The mean daily takings for a random sample of \(n\) days is found.
  3. Estimate the value of \(n\) for which it is approximately \(95 \%\) certain that the sample mean does not differ from the population mean by more than \(
    ) 6$.
CAIE S2 2007 June Q7
7 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 3 } { 4 } \left( x ^ { 2 } - 1 \right) & 1 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$
  1. Sketch the probability density function of \(X\).
  2. Show that the mean, \(\mu\), of \(X\) is 1.6875 .
  3. Show that the standard deviation, \(\sigma\), of \(X\) is 0.2288 , correct to 4 decimal places.
  4. Find \(\mathrm { P } ( 1 \leqslant X \leqslant \mu + \sigma )\).
CAIE S2 2008 June Q1
1 A magazine conducted a survey about the sleeping time of adults. A random sample of 12 adults was chosen from the adults travelling to work on a train.
  1. Give a reason why this is an unsatisfactory sample for the purposes of the survey.
  2. State a population for which this sample would be satisfactory. A satisfactory sample of 12 adults gave numbers of hours of sleep as shown below.
    \(4.6 \quad 6.8\)
    5.2
    6.2
    5.7
    \(\begin{array} { l l } 7.1 & 6.3 \end{array}\)
    5.6
    7.0
    \(5.8 \quad 6.5\)
    7.2
  3. Calculate unbiased estimates of the mean and variance of the sleeping times of adults.
CAIE S2 2008 June Q2
2 The lengths of time people take to complete a certain type of puzzle are normally distributed with mean 48.8 minutes and standard deviation 15.6 minutes. The random variable \(X\) represents the time taken in minutes by a randomly chosen person to solve this type of puzzle. The times taken by random samples of 5 people are noted. The mean time \(\bar { X }\) is calculated for each sample.
  1. State the distribution of \(\bar { X }\), giving the values of any parameters.
  2. Find \(\mathrm { P } ( \bar { X } < 50 )\).
CAIE S2 2008 June Q3
3 The lengths of red pencils are normally distributed with mean 6.5 cm and standard deviation 0.23 cm .
  1. Two red pencils are chosen at random. Find the probability that their total length is greater than 12.5 cm . The lengths of black pencils are normally distributed with mean 11.3 cm and standard deviation 0.46 cm .
  2. Find the probability that the total length of 3 red pencils is more than 6.7 cm greater than the length of 1 black pencil.
CAIE S2 2008 June Q4
4 People who diet can expect to lose an average of 3 kg in a month. In a book, the authors claim that people who follow a new diet will lose an average of more than 3 kg in a month. The weight losses of the 180 people in a random sample who had followed the new diet for a month were noted. The mean was 3.3 kg and the standard deviation was 2.8 kg .
  1. Test the authors' claim at the \(5 \%\) significance level, stating your null and alternative hypotheses.
  2. State what is meant by a Type II error in words relating to the context of the test in part (i).
CAIE S2 2008 June Q5
5 When a guitar is played regularly, a string breaks on average once every 15 months. Broken strings occur at random times and independently of each other.
  1. Show that the mean number of broken strings in a 5 -year period is 4 . A guitar is fitted with a new type of string which, it is claimed, breaks less frequently. The number of broken strings of the new type was noted after a period of 5 years.
  2. The mean number of broken strings of the new type in a 5 -year period is denoted by \(\lambda\). Find the rejection region for a test at the \(10 \%\) significance level when the null hypothesis \(\lambda = 4\) is tested against the alternative hypothesis \(\lambda < 4\).
  3. Hence calculate the probability of making a Type I error. The number of broken guitar strings of the new type, in a 5 -year period, was in fact 1 .
  4. State, with a reason, whether there is evidence at the \(10 \%\) significance level that guitar strings of the new type break less frequently.
CAIE S2 2008 June Q6
6 People arrive randomly and independently at the elevator in a block of flats at an average rate of 4 people every 5 minutes.
  1. Find the probability that exactly two people arrive in a 1-minute period.
  2. Find the probability that nobody arrives in a 15 -second period.
  3. The probability that at least one person arrives in the next \(t\) minutes is 0.9 . Find the value of \(t\).
CAIE S2 2008 June Q7
7 If Usha is stung by a bee she always develops an allergic reaction. The time taken in minutes for Usha to develop the reaction can be modelled using the probability density function given by $$\mathrm { f } ( t ) = \begin{cases} \frac { k } { t + 1 } & 0 \leqslant t \leqslant 4
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 1 } { \ln 5 }\).
  2. Find the probability that it takes more than 3 minutes for Usha to develop a reaction.
  3. Find the median time for Usha to develop a reaction.
CAIE S2 2009 June Q1
1 In Europe the diameters of women's rings have mean 18.5 mm . Researchers claim that women in Jakarta have smaller fingers than women in Europe. The researchers took a random sample of 20 women in Jakarta and measured the diameters of their rings. The mean diameter was found to be 18.1 mm . Assuming that the diameters of women's rings in Jakarta have a normal distribution with standard deviation 1.1 mm , carry out a hypothesis test at the \(2 \frac { 1 } { 2 } \%\) level to determine whether the researchers' claim is justified.
CAIE S2 2009 June Q2
2 The weights in grams of oranges grown in a certain area are normally distributed with mean \(\mu\) and standard deviation \(\sigma\). A random sample of 50 of these oranges was taken, and a \(97 \%\) confidence interval for \(\mu\) based on this sample was (222.1, 232.1).
  1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
  2. Estimate the sample size that would be required in order for a \(97 \%\) confidence interval for \(\mu\) to have width 8 .
CAIE S2 2009 June Q3
3 Major avalanches can be regarded as randomly occurring events. They occur at a uniform average rate of 8 per year.
  1. Find the probability that more than 3 major avalanches occur in a 3-month period.
  2. Find the probability that any two separate 4 -month periods have a total of 7 major avalanches.
  3. Find the probability that a total of fewer than 137 major avalanches occur in a 20 -year period.
CAIE S2 2009 June Q4
4 In a certain city it is necessary to pass a driving test in order to be allowed to drive a car. The probability of passing the driving test at the first attempt is 0.36 on average. A particular driving instructor claims that the probability of his pupils passing at the first attempt is higher than 0.36 . A random sample of 8 of his pupils showed that 7 passed at the first attempt.
  1. Carry out an appropriate hypothesis test to test the driving instructor's claim, using a significance level of \(5 \%\).
  2. In fact, most of this random sample happened to be careful and sensible drivers. State which type of error in the hypothesis test (Type I or Type II) could have been made in these circumstances and find the probability of this type of error when a sample of size 8 is used for the test.
CAIE S2 2009 June Q5
5 The time in minutes taken by candidates to answer a question in an examination has probability density function given by $$\mathrm { f } ( t ) = \begin{cases} k \left( 6 t - t ^ { 2 } \right) & 3 \leqslant t \leqslant 6
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 1 } { 18 }\).
  2. Find the mean time.
  3. Find the probability that a candidate, chosen at random, takes longer than 5 minutes to answer the question.
  4. Is the upper quartile of the times greater than 5 minutes, equal to 5 minutes or less than 5 minutes? Give a reason for your answer.
CAIE S2 2009 June Q6
6 When Sunil travels from his home in England to visit his relatives in India, his journey is in four stages. The times, in hours, for the stages have independent normal distributions as follows. Bus from home to the airport: \(\quad \mathrm { N } ( 3.75,1.45 )\)
Waiting in the airport: \(\quad \mathrm { N } ( 3.1,0.785 )\)
Flight from England to India: \(\quad \mathrm { N } ( 11,1.3 )\)
Car in India to relatives: \(\quad \mathrm { N } ( 3.2,0.81 )\)
  1. Find the probability that the flight time is shorter than the total time for the other three stages.
  2. Find the probability that, for 6 journeys to India, the mean time waiting in the airport is less than 4 hours.