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CAIE S2 2015 November Q5
9 marks Standard +0.3
5
  1. Narika has a die which is known to be biased so that the probability of throwing a 6 on any throw is \(\frac { 1 } { 100 }\). She uses an approximating distribution to calculate the probability of obtaining no 6s in 450 throws. Find the percentage error in using the approximating distribution for this calculation.
  2. Johan claims that a certain six-sided die is biased so that it shows a 6 less often than it would if the die were fair. In order to test this claim, the die is thrown 25 times and it shows a 6 on only 2 throws. Test at the \(10 \%\) significance level whether Johan's claim is justified.
CAIE S2 2015 November Q6
9 marks Standard +0.8
6 Parcels arriving at a certain office have weights \(W \mathrm {~kg}\), where the random variable \(W\) has mean \(\mu\) and standard deviation 0.2 . The value of \(\mu\) used to be 2.60 , but there is a suspicion that this may no longer be true. In order to test at the 5\% significance level whether the value of \(\mu\) has increased, a random sample of 75 parcels is chosen. You may assume that the standard deviation of \(W\) is unchanged.
  1. The mean weight of the 75 parcels is found to be 2.64 kg . Carry out the test.
  2. Later another test of the same hypotheses at the \(5 \%\) significance level, with another random sample of 75 parcels, is carried out. Given that the value of \(\mu\) is now 2.68 , calculate the probability of a Type II error.
CAIE S2 2015 November Q7
9 marks Standard +0.3
7 The diameter, in cm, of pistons made in a certain factory is denoted by \(X\), where \(X\) is normally distributed with mean \(\mu\) and variance \(\sigma ^ { 2 }\). The diameters of a random sample of 100 pistons were measured, with the following results. $$n = 100 \quad \Sigma x = 208.7 \quad \Sigma x ^ { 2 } = 435.57$$
  1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\). The pistons are designed to fit into cylinders. The internal diameter, in cm , of the cylinders is denoted by \(Y\), where \(Y\) has an independent normal distribution with mean 2.12 and variance 0.000144 . A piston will not fit into a cylinder if \(Y - X < 0.01\).
  2. Using your answers to part (i), find the probability that a randomly chosen piston will not fit into a randomly chosen cylinder.
CAIE S2 2016 November Q1
3 marks Easy -1.3
1 The weights, in kilograms, of a random sample of eight 16-year old males are given below. $$\begin{array} { l l l l l l l l } 58.9 & 63.5 & 62.7 & 59.4 & 66.9 & 68.0 & 60.4 & 68.2 \end{array}$$ Find unbiased estimates of the population mean and variance of the weights of all 16 -year old males.
CAIE S2 2016 November Q2
5 marks Standard +0.3
2 A die has six faces numbered \(1,2,3,4,5,6\). Manjit suspects that the die is biased so that it shows a six on fewer throws than it would if it were fair. In order to test her suspicion, she throws the die a certain number of times and counts the number of sixes.
  1. State suitable null and alternative hypotheses for Manjit's test.
  2. There are no sixes in the first 15 throws. Show that this result is not significant at the \(5 \%\) level.
  3. Find the smallest value of \(n\) such that, if there are no sixes in the first \(n\) throws, this result is significant at the 5\% level.
CAIE S2 2016 November Q3
7 marks Standard +0.3
3 Particles are emitted randomly from a radioactive substance at a constant average rate of 3.6 per minute. Find the probability that
  1. more than 3 particles are emitted during a 20 -second period,
  2. more than 240 particles are emitted during a 1-hour period.
CAIE S2 2016 November Q4
7 marks Standard +0.3
4 Each week a farmer sells \(X\) litres of milk and \(Y \mathrm {~kg}\) of cheese, where \(X\) and \(Y\) have the independent distributions \(\mathrm { N } \left( 1520,53 ^ { 2 } \right)\) and \(\mathrm { N } \left( 175,12 ^ { 2 } \right)\) respectively.
  1. Find the mean and standard deviation of the total amount of milk that the farmer sells in 4 randomly chosen weeks. During a year when milk prices are low, the farmer makes a loss of 2 cents per litre on milk and makes a profit of 21 cents per kg on cheese, so the farmer's overall weekly profit is \(( 21 Y - 2 X )\) cents.
  2. Find the probability that, in a randomly chosen week, the farmer's overall profit is positive.
CAIE S2 2016 November Q5
8 marks Standard +0.3
5
  1. The masses, in grams, of certain tomatoes are normally distributed with standard deviation 9 grams. A random sample of 100 tomatoes has a sample mean of 63 grams. Find a \(90 \%\) confidence interval for the population mean mass of these tomatoes.
  2. The masses, in grams, of certain potatoes are normally distributed with known population standard deviation but unknown population mean. A random sample of potatoes is taken in order to find a confidence interval for the population mean. Using a sample of size 50 , a \(95 \%\) confidence interval is found to have width 8 grams.
    1. Using another sample of size 50 , an \(\alpha \%\) confidence interval has width 4 grams. Find \(\alpha\).
    2. Find the sample size \(n\), such that a \(95 \%\) confidence interval has width 4 grams.
CAIE S2 2016 November Q6
9 marks Moderate -0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{e0ad3268-117e-4a0c-942d-84ee148d8907-3_371_504_260_534} \includegraphics[max width=\textwidth, alt={}, center]{e0ad3268-117e-4a0c-942d-84ee148d8907-3_373_495_260_1123} \includegraphics[max width=\textwidth, alt={}, center]{e0ad3268-117e-4a0c-942d-84ee148d8907-3_371_497_776_534} \includegraphics[max width=\textwidth, alt={}, center]{e0ad3268-117e-4a0c-942d-84ee148d8907-3_367_488_778_1128} The diagrams show the probability density functions of four random variables \(W , X , Y\) and \(Z\). Each of the four variables takes values between 0 and 3 only, and their medians are \(m _ { W } , m _ { X } , m _ { Y }\) and \(m _ { Z }\) respectively.
  1. List \(m _ { W } , m _ { X } , m _ { Y }\) and \(m _ { Z }\) in order of size, starting with the largest.
  2. The probability density function of \(X\) is given by $$f ( x ) = \begin{cases} \frac { 4 } { 81 } x ^ { 3 } & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$
    1. Show that \(\mathrm { E } ( X ) = \frac { 12 } { 5 }\).
    2. Calculate \(\mathrm { P } ( X > \mathrm { E } ( X ) )\).
    3. Write down the value of \(\mathrm { P } ( X < 2 \mathrm { E } ( X ) )\).
CAIE S2 2016 November Q7
11 marks Standard +0.3
7 In the past the time, in minutes, taken for a particular rail journey has been found to have mean 20.5 and standard deviation 1.2. Some new railway signals are installed. In order to test whether the mean time has decreased, a random sample of 100 times for this journey are noted. The sample mean is found to be 20.3 minutes. You should assume that the standard deviation is unchanged.
  1. Carry out a significance test, at the \(4 \%\) level, of whether the population mean time has decreased. Later another significance test of the same hypotheses, using another random sample of size 100 , is carried out at the \(4 \%\) level.
  2. Given that the population mean is now 20.1, find the probability of a Type II error.
  3. State what is meant by a Type II error in this context.
CAIE S2 2016 November Q6
9 marks Moderate -0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{1e20bcc7-a501-4df0-9d49-cca2db4c279a-3_371_504_260_534} \includegraphics[max width=\textwidth, alt={}, center]{1e20bcc7-a501-4df0-9d49-cca2db4c279a-3_373_495_260_1123} \includegraphics[max width=\textwidth, alt={}, center]{1e20bcc7-a501-4df0-9d49-cca2db4c279a-3_371_497_776_534} \includegraphics[max width=\textwidth, alt={}, center]{1e20bcc7-a501-4df0-9d49-cca2db4c279a-3_367_488_778_1128} The diagrams show the probability density functions of four random variables \(W , X , Y\) and \(Z\). Each of the four variables takes values between 0 and 3 only, and their medians are \(m _ { W } , m _ { X } , m _ { Y }\) and \(m _ { Z }\) respectively.
  1. List \(m _ { W } , m _ { X } , m _ { Y }\) and \(m _ { Z }\) in order of size, starting with the largest.
  2. The probability density function of \(X\) is given by $$f ( x ) = \begin{cases} \frac { 4 } { 81 } x ^ { 3 } & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$
    1. Show that \(\mathrm { E } ( X ) = \frac { 12 } { 5 }\).
    2. Calculate \(\mathrm { P } ( X > \mathrm { E } ( X ) )\).
    3. Write down the value of \(\mathrm { P } ( X < 2 \mathrm { E } ( X ) )\).
CAIE S2 2016 November Q1
3 marks Easy -1.2
1 The random variable \(X\) has the distribution \(\operatorname { Po } ( 3.5 )\). Find \(\mathrm { P } ( X < 3 )\).
CAIE S2 2016 November Q2
3 marks Easy -1.8
2 Dominic wishes to choose a random sample of five students from the 150 students in his year. He numbers the students from 1 to 150 . Then he uses his calculator to generate five random numbers between 0 and 1 . He multiplies each random number by 150 and rounds up to the next whole number to give a student number.
  1. Dominic's first random number is 0.392 . Find the student number that is produced by this random number.
  2. Dominic's second student number is 104 . Find a possible random number that would produce this student number.
  3. Explain briefly why five random numbers may not be enough to produce a sample of five student numbers.
CAIE S2 2016 November Q3
5 marks Standard +0.3
3 A men's triathlon consists of three parts: swimming, cycling and running. Competitors' times, in minutes, for the three parts can be modelled by three independent normal variables with means 34.0, 87.1 and 56.9, and standard deviations 3.2, 4.1 and 3.8, respectively. For each competitor, the total of his three times is called the race time. Find the probability that the mean race time of a random sample of 15 competitors is less than 175 minutes.
CAIE S2 2016 November Q4
5 marks Challenging +1.2
4 The manufacturer of a tablet computer claims that the mean battery life is 11 hours. A consumer organisation wished to test whether the mean is actually greater than 11 hours. They invited a random sample of members to report the battery life of their tablets. They then calculated the sample mean. Unfortunately a fire destroyed the records of this test except for the following partial document. \includegraphics[max width=\textwidth, alt={}, center]{c460afa4-1387-421d-87ac-74a64be99714-2_467_593_1612_776} Given that the population of battery lives is normally distributed with standard deviation 1.6 hours, find the set of possible values of the sample size, \(n\).
CAIE S2 2016 November Q5
8 marks Moderate -0.3
5 It is claimed that \(30 \%\) of packets of Froogum contain a free gift. Andre thinks that the actual proportion is less than \(30 \%\) and he decides to carry out a hypothesis test at the \(5 \%\) significance level. He buys 20 packets of Froogum and notes the number of free gifts he obtains.
  1. State null and alternative hypotheses for the test.
  2. Use a binomial distribution to find the probability of a Type I error. Andre finds that 3 of the 20 packets contain free gifts.
  3. Carry out the test.
CAIE S2 2016 November Q6
8 marks Standard +0.3
6 A variable \(X\) takes values \(1,2,3,4,5\), and these values are generated at random by a machine. Each value is supposed to be equally likely, but it is suspected that the machine is not working properly. A random sample of 100 values of \(X\), generated by the machine, gives the following results. $$n = 100 \quad \Sigma x = 340 \quad \Sigma x ^ { 2 } = 1356$$
  1. Find a 95\% confidence interval for the population mean of the values generated by the machine.
  2. Use your answer to part (i) to comment on whether the machine may be working properly.
CAIE S2 2016 November Q7
9 marks Standard +0.3
7 Men arrive at a clinic independently and at random, at a constant mean rate of 0.2 per minute. Women arrive at the same clinic independently and at random, at a constant mean rate of 0.3 per minute.
  1. Find the probability that at least 2 men and at least 3 women arrive at the clinic during a 5 -minute period.
  2. Find the probability that fewer than 36 people arrive at the clinic during a 1-hour period.
CAIE S2 2016 November Q8
9 marks Standard +0.3
8 \includegraphics[max width=\textwidth, alt={}, center]{c460afa4-1387-421d-87ac-74a64be99714-4_302_517_276_427} \includegraphics[max width=\textwidth, alt={}, center]{c460afa4-1387-421d-87ac-74a64be99714-4_304_508_274_1215} \includegraphics[max width=\textwidth, alt={}, center]{c460afa4-1387-421d-87ac-74a64be99714-4_305_506_717_431} \includegraphics[max width=\textwidth, alt={}, center]{c460afa4-1387-421d-87ac-74a64be99714-4_302_504_717_1217} The diagrams show the probability density functions of four random variables \(W , X , Y\) and \(Z\). Each of the four variables takes values between - 3 and 3 only, and their standard deviations are \(\sigma _ { W } , \sigma _ { X } , \sigma _ { Y }\) and \(\sigma _ { Z }\) respectively.
  1. List \(\sigma _ { W } , \sigma _ { X } , \sigma _ { Y }\) and \(\sigma _ { Z }\) in order of size, starting with the largest.
  2. The probability density function of \(X\) is given by $$f ( x ) = \begin{cases} \frac { 1 } { 18 } x ^ { 2 } & - 3 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$
    1. Show that \(\sigma _ { X } = 2.32\) correct to 3 significant figures.
    2. Calculate \(\mathrm { P } \left( X > \sigma _ { X } \right)\).
    3. Write down the value of \(\mathrm { P } \left( X > 2 \sigma _ { X } \right)\).
CAIE Further Paper 4 2022 June Q1
8 marks Standard +0.3
1 A manager is investigating the times taken by employees to complete a particular task as a result of the introduction of new technology. He claims that the mean time taken to complete the task is reduced by more than 0.4 minutes. He chooses a random sample of 10 employees. The times taken, in minutes, before and after the introduction of the new technology are recorded in the table.
Employee\(A\)\(B\)\(C\)D\(E\)\(F\)G\(H\)IJ
Time before new technology10.29.812.411.610.811.214.610.612.311.0
Time after new technology9.68.512.410.910.210.612.810.812.510.6
  1. Test at the 10\% significance level whether the manager's claim is justified.
  2. State an assumption that is necessary for this test to be valid.
CAIE Further Paper 4 2022 June Q2
7 marks Challenging +1.2
2 The probability generating function, \(\mathrm { G } _ { Y } ( t )\), of the random variable \(Y\) is given by $$G _ { Y } ( t ) = 0.04 + 0.2 t + 0.37 t ^ { 2 } + 0.3 t ^ { 3 } + 0.09 t ^ { 4 }$$
  1. Find \(\operatorname { Var } ( Y )\).
    The random variable \(Y\) is the sum of two independent observations of the random variable \(X\).
  2. Find the probability generating function of \(X\), giving your answer as a polynomial in \(t\).
CAIE Further Paper 4 2022 June Q3
8 marks Standard +0.3
3 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} k x ( 4 - x ) & 0 \leqslant x < 2 \\ k ( 6 - x ) & 2 \leqslant x \leqslant 6 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 3 } { 40 }\).
  2. Given that \(\mathrm { E } ( X ) = 2.5\), find \(\operatorname { Var } ( X )\).
  3. Find the median value of \(X\).
CAIE Further Paper 4 2022 June Q4
8 marks Standard +0.3
4 A scientist is investigating the numbers of a particular type of butterfly in a certain region. He claims that the numbers of these butterflies found per square metre can be modelled by a Poisson distribution with mean 2.5. He takes a random sample of 120 areas, each of one square metre, and counts the number of these butterflies in each of these areas. The following table shows the observed frequencies together with some of the expected frequencies using the scientist's Poisson distribution.
Number per square metre0123456\(\geqslant 7\)
Observed frequency1220363213610
Expected frequency9.8524.6330.7825.65\(p\)8.023.34\(q\)
  1. Find the values of \(p\) and \(q\), correct to 2 decimal places.
  2. Carry out a goodness of fit test, at the \(10 \%\) significance level, to test the scientist's claim.
CAIE Further Paper 4 2022 June Q5
9 marks Standard +0.3
5 Raman is researching the heights of male giraffes in a particular region. Raman assumes that the heights of male giraffes in this region are normally distributed. He takes a random sample of 8 male giraffes from the region and measures the height, in metres, of each giraffe. These heights are as follows. $$\begin{array} { c c c c c c c c } 5.2 & 5.8 & 4.9 & 6.1 & 5.5 & 5.9 & 5.4 & 5.6 \end{array}$$
  1. Find a \(90 \%\) confidence interval for the population mean height of male giraffes in this region. [5]
    Raman claims that the population mean height of male giraffes in the region is less than 5.9 metres.
  2. Test at the \(2.5 \%\) significance level whether this sample provides sufficient evidence to support Raman's claim.
CAIE Further Paper 4 2022 June Q6
10 marks Standard +0.3
6 A teacher at a large college gave a mathematical puzzle to all the students. The median time taken by a random sample of 24 students to complete the puzzle was 18.0 minutes. The students were then given practice in solving puzzles. Two weeks later, the students were given another mathematical puzzle of the same type as the first. The times, in minutes, taken by the random sample of 24 students to complete this puzzle are as follows.
18.217.516.415.120.526.519.223.2
17.918.825.819.917.716.217.316.6
17.120.120.312.616.021.422.718.4
The teacher claims that the practice has not made any difference to the average time taken to complete a puzzle of this type. Carry out a Wilcoxon signed-rank test, at the 10\% significance level, to test whether there is sufficient evidence to reject the teacher's claim.
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