Questions — CAIE (7646 questions)

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CAIE S2 2014 June Q6
8 marks Standard +0.3
6 A machine is designed to generate random digits between 1 and 5 inclusive. Each digit is supposed to appear with the same probability as the others, but Max claims that the digit 5 is appearing less often than it should. In order to test this claim the manufacturer uses the machine to generate 25 digits and finds that exactly 1 of these digits is a 5 .
  1. Carry out a test of Max's claim at the \(2.5 \%\) significance level.
  2. Max carried out a similar hypothesis test by generating 1000 digits between 1 and 5 inclusive. The digit 5 appeared 180 times. Without carrying out the test, state the distribution that Max should use, including the values of any parameters.
  3. State what is meant by a Type II error in this context.
CAIE S2 2014 June Q7
9 marks Moderate -0.8
7 A Lost Property office is open 7 days a week. It may be assumed that items are handed in to the office randomly, singly and independently.
  1. State another condition for the number of items handed in to have a Poisson distribution. It is now given that the number of items handed in per week has the distribution \(\operatorname { Po } ( 4.0 )\).
  2. Find the probability that exactly 2 items are handed in on a particular day.
  3. Find the probability that at least 4 items are handed in during a 10-day period.
  4. Find the probability that, during a certain week, 5 items are handed in altogether, but no items are handed in on the first day of the week.
CAIE S2 2014 June Q8
10 marks Standard +0.3
8 In an examination, the marks in the theory paper and the marks in the practical paper are denoted by the random variables \(X\) and \(Y\) respectively, where \(X \sim \mathrm {~N} ( 57,13 )\) and \(Y \sim \mathrm {~N} ( 28,5 )\). You may assume that each candidate's marks in the two papers are independent. The final score of each candidate is found by calculating \(X + 2.5 Y\). A candidate is chosen at random. Without using a continuity correction, find the probability that this candidate
  1. has a final score that is greater than 140 ,
  2. obtains at least 20 more marks in the theory paper than in the practical paper.
CAIE S2 2015 June Q1
5 marks Moderate -0.8
1 \includegraphics[max width=\textwidth, alt={}, center]{cfffe79d-91c9-48b8-a3e6-887d7891441d-2_478_691_260_724} The random variable \(X\) has probability density function, f , as shown in the diagram, where \(a\) is a constant. Find the value of \(a\) and hence show that \(\mathrm { E } ( X ) = 0.943\) correct to 3 significant figures. [5]
CAIE S2 2015 June Q2
6 marks Moderate -0.3
2 Sami claims that he can read minds. He asks each of 50 people to choose one of the 5 letters A, B, C, D or E. He then tells each person which letter he believes they have chosen. He gets 13 correct. Sami says "This shows that I can read minds, because 13 is more than I would have got right if I were just guessing."
  1. State null and alternative hypotheses for a test of Sami's claim.
  2. Test at the \(10 \%\) significance level whether Sami's claim is justified.
CAIE S2 2015 June Q3
6 marks Standard +0.3
3 The daily times, in minutes, that Yu Ming takes showering, getting dressed and having breakfast are independent and have the distributions \(\mathrm { N } \left( 9,2.2 ^ { 2 } \right) , \mathrm { N } \left( 8,1.3 ^ { 2 } \right)\) and \(\mathrm { N } \left( 17,2.6 ^ { 2 } \right)\) respectively. The total daily time that Yu Ming takes for all three activities is denoted by \(T\) minutes.
  1. Find the mean and variance of \(T\).
  2. Yu Ming notes the value of \(T\) on each day in a random sample of 70 days and calculates the sample mean. Find the probability that the sample mean is between 33 and 35 .
CAIE S2 2015 June Q4
7 marks Standard +0.3
4 In the past, the time taken by vehicles to drive along a particular stretch of road has had mean 12.4 minutes and standard deviation 2.1 minutes. Some new signs are installed and it is expected that the mean time will increase. In order to test whether this is the case, the mean time for a random sample of 50 vehicles is found. You may assume that the standard deviation is unchanged.
  1. The mean time for the sample of 50 vehicles is found to be 12.9 minutes. Test at the \(2.5 \%\) significance level whether the population mean time has increased.
  2. State what is meant by a Type II error in this context.
  3. State what extra piece of information would be needed in order to find the probability of a Type II error.
CAIE S2 2015 June Q5
7 marks Moderate -0.3
5 The masses, \(m\) grams, of a random sample of 80 strawberries of a certain type were measured and summarised as follows. $$n = 80 \quad \Sigma m = 4200 \quad \Sigma m ^ { 2 } = 229000$$
  1. Find unbiased estimates of the population mean and variance.
  2. Calculate a 98\% confidence interval for the population mean. 50 random samples of size 80 were taken and a \(98 \%\) confidence interval for the population mean, \(\mu\), was found from each sample.
  3. Find the number of these 50 confidence intervals that would be expected to include the true value of \(\mu\).
CAIE S2 2015 June Q6
9 marks Moderate -0.8
6 A publishing firm has found that errors in the first draft of a new book occur at random and that, on average, there is 1 error in every 3 pages of a first draft. Find the probability that in a particular first draft there are
  1. exactly 2 errors in 10 pages,
  2. at least 3 errors in 6 pages,
  3. fewer than 50 errors in 200 pages.
CAIE S2 2015 June Q7
10 marks Moderate -0.8
7 The independent variables \(X\) and \(Y\) are such that \(X \sim \mathrm {~B} ( 10,0.8 )\) and \(Y \sim \mathrm { Po } ( 3 )\). Find
  1. \(\mathrm { E } ( 7 X + 5 Y - 2 )\),
  2. \(\operatorname { Var } ( 4 X - 3 Y + 3 )\),
  3. \(\mathrm { P } ( 2 X - Y = 18 )\).
CAIE S2 2015 June Q1
3 marks Moderate -0.8
1 The independent random variables \(X\) and \(Y\) have standard deviations 3 and 6 respectively. Calculate the standard deviation of \(4 X - 5 Y\).
CAIE S2 2015 June Q2
5 marks Standard +0.3
2 Cloth made at a certain factory has been found to have an average of 0.1 faults per square metre. Suki claims that the cloth made by her machine contains, on average, more than 0.1 faults per square metre. In a random sample of \(5 \mathrm {~m} ^ { 2 }\) of cloth from Suki's machine, it was found that there were 2 faults. Assuming that the number of faults per square metre has a Poisson distribution,
  1. state null and alternative hypotheses for a test of Suki's claim,
  2. test at the \(10 \%\) significance level whether Suki's claim is justified.
CAIE S2 2015 June Q3
5 marks Moderate -0.3
3 In a golf tournament, the number of times in a day that a 'hole-in-one' is scored is denoted by the variable \(X\), which has a Poisson distribution with mean 0.15 . Mr Crump offers to pay \(\\) 200$ each time that a hole-in-one is scored during 5 days of play. Find the expectation and variance of the amount that Mr Crump pays.
CAIE S2 2015 June Q4
8 marks Standard +0.3
4 In the past, the flight time, in hours, for a particular flight has had mean 6.20 and standard deviation 0.80 . Some new regulations are introduced. In order to test whether these new regulations have had any effect upon flight times, the mean flight time for a random sample of 40 of these flights is found.
  1. State what is meant by a Type I error in this context.
  2. The mean time for the sample of 40 flights is found to be 5.98 hours. Assuming that the standard deviation of flight times is still 0.80 hours, test at the \(5 \%\) significance level whether the population mean flight time has changed.
  3. State, with a reason, which of the errors, Type I or Type II, might have been made in your answer to part (ii).
CAIE S2 2015 June Q5
9 marks Standard +0.8
5 The volumes, \(v\) millilitres, of juice in a random sample of 50 bottles of Cooljoos are measured and summarised as follows. $$n = 50 \quad \Sigma v = 14800 \quad \Sigma v ^ { 2 } = 4390000$$
  1. Find unbiased estimates of the population mean and variance.
  2. An \(\alpha \%\) confidence interval for the population mean, based on this sample, is found to have a width of 5.45 millilitres. Find \(\alpha\). Four random samples of size 10 are taken and a \(96 \%\) confidence interval for the population mean is found from each sample.
  3. Find the probability that these 4 confidence intervals all include the true value of the population mean.
CAIE S2 2015 June Q6
10 marks Moderate -0.3
6 The waiting time, \(T\) minutes, for patients at a doctor's surgery has probability density function given by $$\mathrm { f } ( t ) = \begin{cases} k \left( 225 - t ^ { 2 } \right) & 0 \leqslant t \leqslant 15 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 1 } { 2250 }\).
  2. Find the probability that a patient has to wait for more than 10 minutes.
  3. Find the mean waiting time.
CAIE S2 2015 June Q7
10 marks Moderate -0.8
7 In a certain lottery, 10500 tickets have been sold altogether and each ticket has a probability of 0.0002 of winning a prize. The random variable \(X\) denotes the number of prize-winning tickets that have been sold.
  1. State, with a justification, an approximating distribution for \(X\).
  2. Use your approximating distribution to find \(\mathrm { P } ( X < 4 )\).
  3. Use your approximating distribution to find the conditional probability that \(X < 4\), given that \(X \geqslant 1\).
CAIE S2 2018 June Q1
3 marks Easy -1.2
1 A random sample of 75 values of a variable \(X\) gave the following results. $$n = 75 \quad \Sigma x = 153.2 \quad \Sigma x ^ { 2 } = 340.24$$ Find unbiased estimates for the population mean and variance of \(X\).
CAIE S2 2018 June Q2
4 marks Moderate -0.3
2 A six-sided die is suspected of bias. The die is thrown 100 times and it is found that the score is 2 on 20 throws. It is given that the probability of obtaining a score of 2 on any throw is \(p\).
  1. Find an approximate \(94 \%\) confidence interval for \(p\).
  2. Use your answer to part (i) to comment on whether the die may be biased.
CAIE S2 2018 June Q3
4 marks Standard +0.3
3 The number of e-readers sold in a 10-day period in a shop is modelled by the distribution \(\operatorname { Po } ( 5.1 )\). Use an approximating distribution to find the probability that fewer than 140 e-readers are sold in a 300-day period.
CAIE S2 2018 June Q4
7 marks Moderate -0.8
4 The volume, in millilitres, of a small cup of coffee has the distribution \(\mathrm { N } ( 103.4,10.2 )\). The volume of a large cup of coffee is 1.5 times the volume of a small cup of coffee.
  1. Find the mean and standard deviation of the volume of a large cup of coffee.
  2. Find the probability that the total volume of a randomly chosen small cup of coffee and a randomly chosen large cup of coffee is greater than 250 ml .
CAIE S2 2018 June Q5
9 marks Standard +0.3
5 The mass, in kilograms, of rocks in a certain area has mean 14.2 and standard deviation 3.1.
  1. Find the probability that the mean mass of a random sample of 50 of these rocks is less than 14.0 kg .
  2. Explain whether it was necessary to assume that the population of the masses of these rocks is normally distributed.
  3. A geologist suspects that rocks in another area have a mean mass which is less than 14.2 kg . A random sample of 100 rocks in this area has sample mean 13.5 kg . Assuming that the standard deviation for rocks in this area is also 3.1 kg , test at the \(2 \%\) significance level whether the geologist is correct.
CAIE S2 2018 June Q6
11 marks Moderate -0.3
6 The time, in minutes, taken by people to complete a test is modelled by the continuous random variable \(X\) with probability density function given by $$f ( x ) = \begin{cases} \frac { k } { x ^ { 2 } } & 5 \leqslant x \leqslant 10 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = 10\).
  2. Show that \(\mathrm { E } ( X ) = 10 \ln 2\).
  3. Find \(\mathrm { P } ( X > 9 )\).
  4. Given that \(\mathrm { P } ( X < a ) = 0.6\), find \(a\).
CAIE S2 2018 June Q7
12 marks Standard +0.3
7 The number of absences by girls from a certain class on any day is modelled by a random variable with distribution \(\operatorname { Po } ( 0.2 )\). The number of absences by boys from the same class on any day is modelled by an independent random variable with distribution \(\operatorname { Po } ( 0.3 )\).
  1. Find the probability that, during a randomly chosen 2-day period, the total number of absences is less than 3 .
  2. Find the probability that, during a randomly chosen 5-day period, the number of absences by boys is more than 3.
  3. The teacher claims that, during the football season, there are more absences by boys than usual. In order to test this claim at the 5\% significance level, he notes the number of absences by boys during a randomly chosen 5-day period during the football season.
    1. State what is meant by a Type I error in this context.
    2. State appropriate null and alternative hypotheses and find the probability of a Type I error.
    3. In fact there were 4 absences by boys during this period. Test the teacher's claim at the 5\% significance level.
      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 2018 June Q1
3 marks Standard +0.3
1 The numbers of alpha, beta and gamma particles emitted per minute by a certain piece of rock have independent distributions \(\operatorname { Po } ( 0.2 ) , \operatorname { Po } ( 0.3 )\) and \(\operatorname { Po } ( 0.6 )\) respectively. Find the probability that the total number of particles emitted during a 4 -minute period is less than 4.