Questions — CAIE S2 (737 questions)

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CAIE S2 2003 November Q4
8 marks Standard +0.3
4 The number of emergency telephone calls to the electricity board office in a certain area in \(t\) minutes is known to follow a Poisson distribution with mean \(\frac { 1 } { 80 } t\).
  1. Find the probability that there will be at least 3 emergency telephone calls to the office in any 20-minute period.
  2. The probability that no emergency telephone call is made to the office in a period of \(k\) minutes is 0.9 . Find \(k\).
CAIE S2 2003 November Q5
8 marks Moderate -0.3
5 The distance driven in a week by a long-distance lorry driver is a normally distributed random variable with mean 1850 km and standard deviation 117 km .
  1. Find the probability that in a random sample of 26 weeks his average distance driven per week is more than 1800 km .
  2. New driving regulations are introduced and in a random sample of 26 weeks after their introduction the lorry driver drives a total of 47658 km . Assuming the standard deviation remains unchanged, test at the \(10 \%\) level whether his mean weekly driving distance has changed.
CAIE S2 2003 November Q6
9 marks Moderate -0.8
6
  1. Explain what is meant by
    1. a Type I error,
    2. a Type II error.
    3. Roger thinks that a box contains 6 screws and 94 nails. Felix thinks that the box contains 30 screws and 70 nails. In order to test these assumptions they decide to take 5 items at random from the box and inspect them, replacing each item after it has been inspected, and accept Roger's hypothesis (the null hypothesis) if all 5 items are nails.
      (a) Calculate the probability of a Type I error.
      (b) If Felix's hypothesis (the alternative hypothesis) is true, calculate the probability of a Type II error.
CAIE S2 2003 November Q7
11 marks Moderate -0.8
7 The lifetime, \(x\) years, of the power light on a freezer, which is left on continuously, can be modelled by the continuous random variable with density function given by $$\mathrm { f } ( x ) = \begin{cases} k \mathrm { e } ^ { - 3 x } & x > 0 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = 3\).
  2. Find the lower quartile.
  3. Find the mean lifetime.
CAIE S2 2004 November Q1
4 marks Standard +0.3
1 The number of radioactive particles emitted per second by a certain metal is random and has mean 1.7. The radioactive metal is placed next to an object which independently emits particles at random such that the mean number of particles emitted per second is 0.6 . Find the probability that the total number of particles emitted in the next 3 seconds is 6, 7 or 8 .
CAIE S2 2004 November Q2
5 marks Moderate -0.5
2 Over a long period of time it is found that the amount of sunshine on any day in a particular town in Spain has mean 6.7 hours and standard deviation 3.1 hours.
  1. Find the probability that the mean amount of sunshine over a random sample of 300 days is between 6.5 and 6.8 hours.
  2. Give a reason why it is not necessary to assume that the daily amount of sunshine is normally distributed in order to carry out the calculation in part (i).
CAIE S2 2004 November Q3
7 marks Moderate -0.8
3 A random sample of 150 students attending a college is taken, and their travel times, \(t\) minutes, are measured. The data are summarised by \(\Sigma t = 4080\) and \(\Sigma t ^ { 2 } = 159252\).
  1. Calculate unbiased estimates of the population mean and variance.
  2. Calculate a \(94 \%\) confidence interval for the population mean travel time.
CAIE S2 2004 November Q4
7 marks Standard +0.3
4 The weights of men follow a normal distribution with mean 71 kg and standard deviation 7 kg . The weights of women follow a normal distribution with mean 57 kg and standard deviation 5 kg . The total weight of 5 men and 2 women chosen randomly is denoted by \(X \mathrm {~kg}\).
  1. Show that \(\mathrm { E } ( X ) = 469\) and \(\operatorname { Var } ( X ) = 295\).
  2. The total weight of 4 men and 3 women chosen randomly is denoted by \(Y \mathrm {~kg}\). Find the mean and standard deviation of \(X - Y\) and hence find \(\mathrm { P } ( X - Y > 22 )\).
CAIE S2 2004 November Q5
7 marks Standard +0.3
5 Of people who wear contact lenses, 1 in 1500 on average have laser treatment for short sight.
  1. Use a suitable approximation to find the probability that, of a random sample of 2700 contact lens wearers, more than 2 people have laser treatment.
  2. In a random sample of \(n\) contact lens wearers the probability that no one has laser treatment is less than 0.01 . Find the least possible value of \(n\).
CAIE S2 2004 November Q6
9 marks Moderate -0.3
6 A continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} 3 ( 1 - x ) ^ { 2 } & 0 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$ Find
  1. \(\mathrm { P } ( X > 0.5 )\),
  2. the mean and variance of \(X\).
CAIE S2 2004 November Q7
11 marks Standard +0.3
7 In a research laboratory where plants are studied, the probability of a certain type of plant surviving was 0.35 . The laboratory manager changed the growing conditions and wished to test whether the probability of a plant surviving had increased.
  1. The plants were grown in rows, and when the manager requested a random sample of 8 plants to be taken, the technician took all 8 plants from the front row. Explain what was wrong with the technician's sample.
  2. A suitable sample of 8 plants was taken and 4 of these 8 plants survived. State whether the manager's test is one-tailed or two-tailed and also state the null and alternative hypotheses. Using a \(5 \%\) significance level, find the critical region and carry out the test.
  3. State the meaning of a Type II error in the context of the test in part (ii).
  4. Find the probability of a Type II error for the test in part (ii) if the probability of a plant surviving is now 0.4.
CAIE S2 2005 November Q1
4 marks Moderate -0.8
1 The number of words on a page of a book can be modelled by a normal distribution with mean 403 and standard deviation 26.8. Find the probability that the average number of words per page in a random sample of 6 pages is less than 410.
CAIE S2 2005 November Q2
4 marks Moderate -0.3
2 A manufacturer claims that \(20 \%\) of sugar-coated chocolate beans are red. George suspects that this percentage is actually less than \(20 \%\) and so he takes a random sample of 15 chocolate beans and performs a hypothesis test with the null hypothesis \(p = 0.2\) against the alternative hypothesis \(p < 0.2\). He decides to reject the null hypothesis in favour of the alternative hypothesis if there are 0 or 1 red beans in the sample.
  1. With reference to this situation, explain what is meant by a Type I error.
  2. Find the probability of a Type I error in George's test.
CAIE S2 2005 November Q3
5 marks Standard +0.3
3 Flies stick to wet paint at random points. The average number of flies is 2 per square metre. A wall with area \(22 \mathrm {~m} ^ { 2 }\) is painted with a new type of paint which the manufacturer claims is fly-repellent. It is found that 27 flies stick to this wall. Use a suitable approximation to test the manufacturer's claim at the \(1 \%\) significance level. Take the null hypothesis to be \(\mu = 44\), where \(\mu\) is the population mean.
CAIE S2 2005 November Q4
7 marks Moderate -0.3
4
  1. Give a reason why, in carrying out a statistical investigation, a sample rather than a complete population may be used.
  2. Rose wishes to investigate whether men in her town have a different life-span from the national average of 71.2 years. She looks at government records for her town and takes a random sample of the ages of 110 men who have died recently. Their mean age in years was 69.3 and the unbiased estimate of the population variance was 65.61.
    1. Calculate a \(90 \%\) confidence interval for the population mean and explain what you understand by this confidence interval.
    2. State with a reason what conclusion about the life-span of men in her town Rose could draw from this confidence interval.
CAIE S2 2005 November Q5
8 marks Moderate -0.8
5 A continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} a + \frac { 1 } { 3 } x & 1 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.
  1. Show that the value of \(a\) is \(\frac { 1 } { 2 }\).
  2. Find \(\mathrm { P } ( X > 1.8 )\).
  3. Find \(\mathrm { E } ( X )\).
CAIE S2 2005 November Q6
10 marks Standard +0.3
6 A shopkeeper sells electric fans. The demand for fans follows a Poisson distribution with mean 3.2 per week.
  1. Find the probability that the demand is exactly 2 fans in any one week.
  2. The shopkeeper has 4 fans in his shop at the beginning of a week. Find the probability that this will not be enough to satisfy the demand for fans in that week.
  3. Given instead that he has \(n\) fans in his shop at the beginning of a week, find, by trial and error, the least value of \(n\) for which the probability of his not being able to satisfy the demand for fans in that week is less than 0.05 .
CAIE S2 2005 November Q7
12 marks Challenging +1.2
7 A journey in a certain car consists of two stages with a stop for filling up with fuel after the first stage. The length of time, \(T\) minutes, taken for each stage has a normal distribution with mean 74 and standard deviation 7.3. The length of time, \(F\) minutes, it takes to fill up with fuel has a normal distribution with mean 5 and standard deviation 1.7. The length of time it takes to pay for the fuel is exactly 4 minutes. The variables \(T\) and \(F\) are independent and the times for the two stages are independent of each other.
  1. Find the probability that the total time for the journey is less than 154 minutes.
  2. A second car has a fuel tank with exactly twice the capacity of the first car. Find the mean and variance of this car's fuel fill-up time.
  3. This second car's time for each stage of the journey follows a normal distribution with mean 69 minutes and standard deviation 5.2 minutes. The length of time it takes to pay for the fuel for this car is also exactly 4 minutes. Find the probability that the total time for the journey taken by the first car is more than the total time taken by the second car.
CAIE S2 2006 November Q1
3 marks Moderate -0.8
1 The time taken for Samuel to drive home from work is distributed with mean 46 minutes. Samuel discovers a different route and decides to test at the \(5 \%\) level whether the mean time has changed. He tries this route on a large number of different days chosen randomly and calculates the mean time.
  1. State the null and alternative hypotheses for this test.
  2. Samuel calculates the value of his test statistic \(z\) to be - 1.729 . What conclusion can he draw?
CAIE S2 2006 November Q2
4 marks Easy -1.2
2
  1. Write down the mean and variance of the distribution of the means of random samples of size \(n\) taken from a very large population having mean \(\mu\) and variance \(\sigma ^ { 2 }\).
  2. What, if anything, can you say about the distribution of sample means
    1. if \(n\) is large,
    2. if \(n\) is small?
CAIE S2 2006 November Q3
5 marks Moderate -0.8
3 A survey was conducted to find the proportion of people owning DVD players. It was found that 203 out of a random sample of 278 people owned a DVD player.
  1. Calculate a \(97 \%\) confidence interval for the true proportion of people who own a DVD player. A second survey to find the proportion of people owning DVD players was conducted at 10 o'clock on a Thursday morning in a shopping centre.
  2. Give one reason why this is not a satisfactory sample.
CAIE S2 2006 November Q4
7 marks Standard +0.3
4 In summer, wasps' nests occur randomly in the south of England at an average rate of 3 nests for every 500 houses.
  1. Find the probability that two villages in the south of England, with 600 houses and 700 houses, have a total of exactly 3 wasps' nests.
  2. Use a suitable approximation to estimate the probability of there being fewer than 369 wasps' nests in a town with 64000 houses.
CAIE S2 2006 November Q5
10 marks Standard +0.3
5 Climbing ropes produced by a manufacturer have breaking strengths which are normally distributed with mean 160 kg and standard deviation 11.3 kg . A group of climbers have weights which are normally distributed with mean 66.3 kg and standard deviation 7.1 kg .
  1. Find the probability that a rope chosen randomly will break under the combined weight of 2 climbers chosen randomly. Each climber carries, in a rucksack, equipment amounting to half his own weight.
  2. Find the mean and variance of the combined weight of a climber and his rucksack.
  3. Find the probability that the combined weight of a climber and his rucksack is greater than 87 kg .
CAIE S2 2006 November Q6
10 marks Standard +0.8
6 Pieces of metal discovered by people using metal detectors are found randomly in fields in a certain area at an average rate of 0.8 pieces per hectare. People using metal detectors in this area have a theory that ploughing the fields increases the average number of pieces of metal found per hectare. After ploughing, they tested this theory and found that a randomly chosen field of area 3 hectares yielded 5 pieces of metal.
  1. Carry out the test at the \(10 \%\) level of significance.
  2. What would your conclusion have been if you had tested at the \(5 \%\) level of significance? Jack decides that he will reject the null hypothesis that the average number is 0.8 pieces per hectare if he finds 4 or more pieces of metal in another ploughed field of area 3 hectares.
  3. If the true mean after ploughing is 1.4 pieces per hectare, calculate the probability that Jack makes a Type II error.
CAIE S2 2006 November Q7
11 marks Standard +0.3
7 At a town centre car park the length of stay in hours is denoted by the random variable \(X\), which has probability density function given by $$f ( x ) = \begin{cases} k x ^ { - \frac { 3 } { 2 } } & 1 \leqslant x \leqslant 9 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Interpret the inequalities \(1 \leqslant x \leqslant 9\) in the definition of \(\mathrm { f } ( x )\) in the context of the question.
  2. Show that \(k = \frac { 3 } { 4 }\).
  3. Calculate the mean length of stay. The charge for a length of stay of \(x\) hours is \(\left( 1 - \mathrm { e } ^ { - x } \right)\) dollars.
  4. Find the length of stay for the charge to be at least 0.75 dollars
  5. Find the probability of the charge being at least 0.75 dollars.