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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 2021 March Q2
9 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{2fefee17-50bb-4375-80f6-7e4bc2606492-04_405_789_260_676} 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 2002 November Q1
3 marks Moderate -0.3
1 The time taken, \(T\) minutes, for a special anti-rust paint to dry was measured for a random sample of 120 painted pieces of metal. The sample mean was 51.2 minutes and an unbiased estimate of the population variance was 37.4 minutes \(^ { 2 }\). Determine a \(99 \%\) confidence interval for the mean drying time. \(21.5 \%\) of the population of the UK can be classified as 'very tall'.
  1. The random variable \(X\) denotes the number of people in a sample of \(n\) people who are classified as very tall. Given that \(\mathrm { E } ( X ) = 2.55\), find \(n\).
  2. By using the Poisson distribution as an approximation to a binomial distribution, calculate an approximate value for the probability that a sample of size 210 will contain fewer than 3 people who are classified as very tall.
CAIE S2 2002 November Q3
7 marks Standard +0.3
3 From previous years' observations, the lengths of salmon in a river were found to be normally distributed with mean 65 cm . A researcher suspects that pollution in water is restricting growth. To test this theory, she measures the length \(x \mathrm {~cm}\) of a random sample of \(n\) salmon and calculates that \(\bar { x } = 64.3\) and \(s = 4.9\), where \(s ^ { 2 }\) is the unbiased estimate of the population variance. She then carries out an appropriate hypothesis test.
  1. Her test statistic \(z\) has a value of - 1.807 correct to 3 decimal places. Calculate the value of \(n\).
  2. Using this test statistic, carry out the hypothesis test at the \(5 \%\) level of significance and state what her conclusion should be.
CAIE S2 2002 November Q4
7 marks Standard +0.3
4 The number of accidents per month at a certain road junction has a Poisson distribution with mean 4.8. A new road sign is introduced warning drivers of the danger ahead, and in a subsequent month 2 accidents occurred.
  1. A hypothesis test at the \(10 \%\) level is used to determine whether there were fewer accidents after the new road sign was introduced. Find the critical region for this test and carry out the test.
  2. Find the probability of a Type I error. \(5 X\) and \(Y\) are independent random variables each having a Poisson distribution. \(X\) has mean 2.5 and \(Y\) has mean 3.1.
  3. Find \(\mathrm { P } ( X + Y > 3 )\).
  4. A random sample of 80 values of \(X\) is taken. Find the probability that the sample mean is less than 2.4.
CAIE S2 2002 November Q6
10 marks Standard +0.3
6 The average speed of a bus, \(x \mathrm {~km} \mathrm {~h} ^ { - 1 }\), on a certain journey is a continuous random variable \(X\) with probability density function given by $$f ( x ) = \begin{cases} \frac { k } { x ^ { 2 } } & 20 \leqslant x \leqslant 28 \\ 0 & \text { otherwise } \end{cases}$$
  1. Show that \(k = 70\).
  2. Find \(\mathrm { E } ( X )\).
  3. Find \(\mathrm { P } ( X < \mathrm { E } ( X ) )\).
  4. Hence determine whether the mean is greater or less than the median.
CAIE S2 2002 November Q7
10 marks Standard +0.3
7 Bottles of wine are stacked in racks of 12. The weights of these bottles are normally distributed with mean 1.3 kg and standard deviation 0.06 kg . The weights of the empty racks are normally distributed with mean 2 kg and standard deviation 0.3 kg .
  1. Find the probability that the total weight of a full rack of 12 bottles of wine is between 17 kg and 18 kg .
  2. Two bottles of wine are chosen at random. Find the probability that they differ in weight by more than 0.05 kg .
CAIE S2 2003 November Q1
4 marks Standard +0.3
1 The result of a memory test is known to be normally distributed with mean \(\mu\) and standard deviation 1.9. It is required to have a \(95 \%\) confidence interval for \(\mu\) with a total width of less than 2.0 . Find the least possible number of tests needed to achieve this.
CAIE S2 2003 November Q2
5 marks Standard +0.3
2 A certain machine makes matches. One match in 10000 on average is defective. Using a suitable approximation, calculate the probability that a random sample of 45000 matches will include 2,3 or 4 defective matches.
CAIE S2 2003 November Q3
5 marks Standard +0.3
3 Tien throws a ball. The distance it travels can be modelled by a normal distribution with mean 20 m and variance \(9 \mathrm {~m} ^ { 2 }\). His younger sister Su Chen also throws a ball and the distance her ball travels can be modelled by a normal distribution with mean 14 m and variance \(12 \mathrm {~m} ^ { 2 }\). Su Chen is allowed to add 5 metres on to her distance and call it her 'upgraded distance'. Find the probability that Tien's distance is larger than Su Chen's upgraded distance.
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
    (a) a Type I error,
    (b) a Type II error.
  2. 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.
    (a) Calculate a \(90 \%\) confidence interval for the population mean and explain what you understand by this confidence interval.
    (b) State with a reason what conclusion about the life-span of men in her town Rose could draw from this confidence interval.