Questions — CAIE S2 (737 questions)

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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.
CAIE S2 2018 June Q2
3 marks Moderate -0.8
2 The random variable \(X\) has the distribution \(\mathrm { N } ( 3,1.2 )\). The random variable \(A\) is defined by \(A = 2 X\). The random variable \(B\) is defined by \(B = X _ { 1 } + X _ { 2 }\), where \(X _ { 1 }\) and \(X _ { 2 }\) are independent random values of \(X\). Describe fully the distribution of \(A\) and the distribution of \(B\). Distribution of \(A\) : \(\_\_\_\_\) Distribution of \(B\) : \(\_\_\_\_\)
CAIE S2 2018 June Q3
4 marks Moderate -0.8
3 The management of a factory wished to find a range within which the time taken to complete a particular task generally lies. It is given that the times, in minutes, have a normal distribution with mean \(\mu\) and standard deviation 6.5. A random sample of 15 employees was chosen and the mean time taken by these employees was found to be 52 minutes.
  1. Calculate a \(95 \%\) confidence interval for \(\mu\).
    Later another \(95 \%\) confidence interval for \(\mu\) was found, based on a random sample of 30 employees.
  2. State, with a reason, whether the width of this confidence interval was less than, equal to or greater than the width of the previous interval.
CAIE S2 2018 June Q4
9 marks Standard +0.3
4 The mean mass of packets of sugar is supposed to be 505 g . A random sample of 10 packets filled by a certain machine was taken and the masses, in grams, were found to be as follows. $$\begin{array} { l l l l l l l l l l } 500 & 499 & 496 & 495 & 498 & 490 & 492 & 501 & 494 & 494 \end{array}$$
  1. Find unbiased estimates of the population mean and variance.
    The mean mass of packets produced by this machine was found to be less than 505 g , so the machine was adjusted. Following the adjustment, the masses of a random sample of 150 packets from the machine were measured and the total mass was found to be 75660 g .
  2. Given that the population standard deviation is 3.6 g , test at the \(2 \%\) significance level whether the machine is still producing packets with mean mass less than 505 g .
  3. Explain why the use of the normal distribution is justified in carrying out the test in part (ii). [1]
CAIE S2 2018 June Q5
9 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{b054d0a0-01b6-4785-807c-851551b90544-06_382_743_260_699} The diagram shows the probability density function, f , of a random variable \(X\), in terms of the constants \(a\) and \(b\).
  1. Find \(b\) in terms of \(a\).
  2. Show that \(\mathrm { f } ( x ) = \frac { 2 } { a } - \frac { 2 } { a ^ { 2 } } x\).
  3. Given that \(\mathrm { E } ( X ) = 0.5\), find \(a\).
CAIE S2 2018 June Q6
10 marks Standard +0.3
6 Accidents on a particular road occur at a constant average rate of 1 every 4.8 weeks.
  1. State, in context, one condition for the number of accidents in a given period to be modelled by a Poisson distribution.
    Assume now that a Poisson distribution is a suitable model.
  2. Find the probability that exactly 4 accidents will occur during a randomly chosen 12-week period.
  3. Find the probability that more than 3 accidents will occur during a randomly chosen 10 -week period.
  4. Use a suitable approximating distribution to find the probability that fewer than 30 accidents will occur during a randomly chosen 2 -year period ( \(104 \frac { 2 } { 7 }\) weeks).