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CAIE Further Paper 4 2022 June Q4
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
5 marks
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
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.
If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE Further Paper 4 2023 June Q1
1 The lengths of the leaves of a particular type of tree are normally distributed with mean \(\mu \mathrm { cm }\). The lengths, \(x \mathrm {~cm}\), of a random sample of 12 leaves of this type are recorded. The results are summarised as follows. $$\sum x = 91.2 \quad \sum x ^ { 2 } = 695.8$$ Find a 95\% confidence interval for \(\mu\).
CAIE Further Paper 4 2023 June Q2
2 The children at two large schools, \(P\) and \(Q\), are all given the same puzzle to solve. A random sample of size 10 is taken from the children at school \(P\). Their individual times to complete the puzzle give a sample mean of 9.12 minutes and an unbiased variance estimate of 2.16 minutes \({ } ^ { 2 }\). A random sample of size 12 is taken from the children at school \(Q\). Their individual times, \(x\) minutes, to complete the puzzle are summarised by $$\sum x = 99.6 \quad \sum ( x - \bar { x } ) ^ { 2 } = 21.5$$ where \(\bar { x }\) is the sample mean. Times to complete the puzzle are assumed to be normally distributed with the same population variance. Test at the \(5 \%\) significance level whether the population mean time taken to complete the puzzle by children at school \(P\) is greater than the population mean time taken to complete the puzzle by children at school \(Q\).
CAIE Further Paper 4 2023 June Q3
3 A random sample of 50 values of the continuous random variable \(X\) was taken. These values are summarised in the following table.
Interval\(1 \leqslant x < 1.5\)\(1.5 \leqslant x < 2\)\(2 \leqslant x < 2.5\)\(2.5 \leqslant x < 3\)\(3 \leqslant x < 3.5\)\(3.5 \leqslant x \leqslant 4\)
Observed frequency338111312
It is required to test the goodness of fit of the distribution with probability density function \(f\) given by $$f ( x ) = \begin{cases} \frac { 1 } { 24 } \left( \frac { 4 } { x ^ { 2 } } + x ^ { 2 } \right) & 1 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$ The expected frequencies, correct to 4 decimal places, are given in the following table.
Interval\(1 \leqslant x < 1.5\)\(1.5 \leqslant x < 2\)\(2 \leqslant x < 2.5\)\(2.5 \leqslant x < 3\)\(3 \leqslant x < 3.5\)\(3.5 \leqslant x \leqslant 4\)
Expected frequency4.4271\(a\)6.12858.4549\(b\)14.9678
  1. Show that \(a = 4.6007\) and find the value of \(b\).
  2. Carry out a goodness of fit test, at the \(10 \%\) significance level, to test whether f is a satisfactory model for the data.
CAIE Further Paper 4 2023 June Q4
4 A random sample of 13 technology companies is chosen and the numbers of employees in 2018 and in 2022 are recorded.
CompanyABCD\(E\)\(F\)G\(H\)IJ\(K\)\(L\)M
Number in 2018104191262349705143514942912863041104
Number in 20221062412722810125253215644924782941154
A researcher claims that there has been an increase in the median number of employees at technology companies between 2018 and 2022.
  1. Carry out a Wilcoxon matched-pairs signed-rank test, at the \(5 \%\) significance level, to test whether the data supports this claim.
    The researcher notices that the figures for company \(G\) have been recorded incorrectly. In fact, the number of employees in 2018 was 32 and the number of employees in 2022 was 35.
  2. Explain, with numerical justification, whether or not the conclusion of the test in part (a) remains the same.
CAIE Further Paper 4 2023 June Q5
5 Harry has three coins.
  • One coin is biased so that, when it is thrown, the probability of obtaining a head is \(\frac { 1 } { 3 }\).
  • The second coin is biased so that, when it is thrown, the probability of obtaining a head is \(\frac { 1 } { 4 }\).
  • The third coin is biased so that, when it is thrown, the probability of obtaining a head is \(\frac { 1 } { 5 }\).
The random variable \(X\) is the number of heads that Harry obtains when he throws all three coins together.
  1. Find the probability generating function of \(X\).
    Isaac has two fair coins. The random variable \(Y\) is the number of heads that Isaac obtains when he throws both of his coins together. The random variable \(Z\) is the total number of heads obtained when Harry throws his three coins and Isaac throws his two coins.
  2. Find the probability generating function of \(Z\), expressing your answer as a polynomial in \(t\).
  3. Use the probability generating function of \(Z\) to find \(E ( Z )\).
CAIE Further Paper 4 2023 June Q6
6 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 3 } { 28 } \left( e ^ { \frac { 1 } { 2 } x } + 4 e ^ { - \frac { 1 } { 2 } x } \right) & 0 \leqslant x \leqslant 2 \ln 3 \\ 0 & \text { otherwise } \end{cases}$$
  1. Find the cumulative distribution function of \(X\).
    The random variable \(Y\) is defined by \(Y = e ^ { \frac { 1 } { 2 } ( X ) }\).
  2. Find the probability density function of \(Y\).
  3. Find the 30th percentile of \(Y\).
  4. Find \(\mathrm { E } \left( Y ^ { 4 } \right)\).
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE Further Paper 4 2024 June Q1
1 The times taken by members of a large cycling club to complete a cross-country circuit have a normal distribution with mean \(\mu\) minutes. The times taken, \(x\) minutes, are recorded for a random sample of 14 members of the club. The results are summarised as follows, where \(\bar { x }\) is the sample mean. $$\bar { x } = 42.8 \quad \sum ( x - \bar { x } ) ^ { 2 } = 941.5$$ Find a 95\% confidence interval for \(\mu\).
CAIE Further Paper 4 2024 June Q2
2 A large number of students are taking a Physics course. They are assessed by a practical examination and a written examination. The marks out of 100 obtained by a random sample of 15 students in each of the examinations are as follows.
StudentA\(B\)CD\(E\)\(F\)\(G\)HIJ\(K\)\(L\)\(M\)\(N\)\(O\)
Practical examination666324525976885148369172686760
Written examination635739504771876556397870616270
Use a sign test, at the \(10 \%\) significance level, to test whether, on average, the practical examination marks are higher than the written examination marks.
CAIE Further Paper 4 2024 June Q3
3 A factory produces metal discs. The manager claims that the diameters of these discs have a median of 22.0 mm . The diameters, in mm , of a random sample of 12 discs produced by this factory are as follows. $$\begin{array} { l l l l l l l l l l l l } 22.4 & 20.9 & 22.8 & 21.5 & 23.2 & 22.9 & 23.9 & 21.7 & 19.8 & 23.6 & 22.6 & 23.0 \end{array}$$
  1. Carry out a Wilcoxon signed-rank test, at the \(10 \%\) significance level, to test whether there is any evidence against the manager's claim.
  2. State an assumption that is necessary for this test to be valid.
CAIE Further Paper 4 2024 June Q4
4 The random variable \(Y\) is the sum of two independent observations of the random variable \(X\). The probability generating function \(\mathrm { G } _ { Y } ( \mathrm { t } )\) of \(Y\) is given by $$G _ { Y } ( t ) = \frac { t ^ { 2 } } { ( 4 - 3 t ) ^ { 4 } }$$
  1. Find \(\mathrm { E } ( \mathrm { Y } )\).
  2. Write down an expression for the probability generating function of \(X\).
  3. Find \(\mathrm { P } ( X = 4 )\).
CAIE Further Paper 4 2024 June Q5
5 Two companies, \(P\) and \(Q\), produce a certain type of paint brush. An independent examiner rates the quality of the brushes produced as poor, satisfactory or good. He takes a random sample of brushes from each company. The examiner's ratings are summarised in the table.
CompanyPoorSatisfactoryGood
\(P\)184364
\(Q\)222231
  1. Test, at the \(5 \%\) significance level, whether quality of brushes is independent of company.
  2. Compare the quality of the brushes produced by the two companies.
CAIE Further Paper 4 2024 June Q6
6 Jade is a swimming instructor at a sports college. She claims that, as a result of an intensive training course, the mean time taken by students to swim 50 metres has reduced by more than 1 second. She chooses a random sample of 10 students. The times taken, in seconds, before and after the training course are recorded in the table.
StudentABCD\(E\)\(F\)G\(H\)IJ
Time before course54.247.452.159.055.351.048.952.258.451.4
Time after course50.146.352.558.851.448.449.548.758.351.4
  1. Test, at the 10\% significance level, whether Jade's claim is justified.
  2. State an assumption that is necessary for this test to be valid.
CAIE Further Paper 4 2024 June Q7
7 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \left\{ \begin{array} { c c } \frac { x } { 4 } \left( 4 - x ^ { 2 } \right) & 0 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{array} \right.$$
  1. Find \(\operatorname { Var } ( \sqrt { X } )\).
    The continuous random variable \(Y\) is defined by \(Y = X ^ { 2 }\).
  2. Find the probability density function of \(Y\).
  3. Find the exact value of the median of \(Y\).
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE Further Paper 4 2020 November Q1
1 Kayla is investigating the lengths of the leaves of a certain type of tree found in two forests \(X\) and \(Y\). She chooses a random sample of 40 leaves of this type from forest \(X\) and records their lengths, \(x \mathrm {~cm}\). She also records the lengths, \(y \mathrm {~cm}\), for a random sample of 60 leaves of this type from forest \(Y\). Her results are summarised as follows. $$\sum x = 242.0 \quad \sum x ^ { 2 } = 1587.0 \quad \sum y = 373.2 \quad \sum y ^ { 2 } = 2532.6$$ Find a \(90 \%\) confidence interval for the difference between the population mean lengths of leaves in forests \(X\) and \(Y\).
CAIE Further Paper 4 2020 November Q2
2 Metal rods produced by a certain factory are claimed to have a median breaking strength of 200 tonnes. For a random sample of 9 rods, the breaking strengths, measured in tonnes, were as follows. $$\begin{array} { l l l l l l l l l } 210 & 186 & 188 & 208 & 184 & 191 & 215 & 198 & 196 \end{array}$$ A scientist believes that the median breaking strength of metal rods produced by this factory is less than 200 tonnes.
  1. Use a Wilcoxon signed-rank test, at the \(5 \%\) significance level, to test whether there is evidence to support the scientist’s belief.
  2. Give a reason why a Wilcoxon signed-rank test is preferable to a sign test, when both are valid.
CAIE Further Paper 4 2020 November Q3
3 Apples are sold in bags of 5. Based on her previous experience, Freya claims that the probability of any apple weighing more than 100 grams is 0.35 , independently of other apples in the bag. The apples in a random sample of 150 bags are checked and the number, \(x\), in each bag weighing more than 100 grams is recorded. The results are shown in the following table.
\(x\)012345
Frequency12394637124
Carry out a goodness of fit test at the \(5 \%\) significance level and hence comment on Freya's claim.
CAIE Further Paper 4 2020 November Q4
4 Members of the Sprints athletics club have been taking part in an intense training scheme, aimed at reducing their times taken to run 400 m . For a random sample of 9 athletes from the club, the times taken, in seconds, before and after the training scheme are given in the following table.
Athlete\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)
Time before48.848.250.349.649.448.947.650.348.4
Time after47.947.849.649.149.648.947.749.148.1
The organiser of the training scheme claims that on average an athlete's time will be reduced by at least 0.3 seconds. Test at the 10\% significance level whether the organiser's claim is justified, stating any assumption that you make.
CAIE Further Paper 4 2020 November Q5
5 Keira has two unbiased coins. She tosses both coins. The number of heads obtained by Keira is denoted by \(X\).
  1. Find the probability generating function \(\mathrm { G } _ { \mathrm { X } } ( \mathrm { t } )\) of \(X\).
    Hassan has three coins, two of which are biased so that the probability of obtaining a head when the coin is tossed is \(\frac { 1 } { 3 }\). The corresponding probability for the third coin is \(\frac { 1 } { 4 }\). The number of heads obtained by Hassan when he tosses these three coins is denoted by \(Y\).
  2. Find the probability generating function \(\mathrm { G } _ { Y } ( \mathrm { t } )\) of \(Y\).
    The random variable \(Z\) is the total number of heads obtained by Keira and Hassan.
  3. Find the probability generating function of \(Z\), expressing your answer as a polynomial.
  4. Use the probability generating function of \(Z\) to find \(\mathrm { E } ( Z )\).
  5. Use the probability generating function of \(Z\) to find the most probable value of \(Z\).
CAIE Further Paper 4 2020 November Q6
6 The continuous random variable \(X\) has cumulative distribution function F given by $$F ( x ) = \begin{cases} 0 & x < 0 \\ \frac { 1 } { 60 } \left( 16 x - x ^ { 2 } \right) & 0 \leqslant x \leqslant 6 \\ 1 & x > 6 \end{cases}$$
  1. Find the interquartile range of \(X\).
  2. Find \(\mathrm { E } \left( X ^ { 3 } \right)\).
    The random variable \(Y\) is such that \(Y = \sqrt { X }\).
  3. Find the probability density function of \(Y\).
    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 Further Paper 4 2021 November Q1
1 The times taken for students at a college to run 200 m have a normal distribution with mean \(\mu \mathrm { s }\). The times, \(x\) s, are recorded for a random sample of 10 students from the college. The results are summarised as follows, where \(\bar { x }\) is the sample mean. $$\bar { x } = 25.6 \quad \sum ( x - \bar { x } ) ^ { 2 } = 78.5$$
  1. Find a 90\% confidence interval for \(\mu\).
    A test of the null hypothesis \(\mu = k\) is carried out on this sample, using a \(10 \%\) significance level. The test does not support the alternative hypothesis \(\mu < k\).
  2. Find the greatest possible value of \(k\).
CAIE Further Paper 4 2021 November Q2
2 The continuous random variable \(X\) has cumulative distribution function F given by $$F ( x ) = \left\{ \begin{array} { l c } 0 & x < - 1 \\ \frac { 1 } { 2 } ( 1 + x ) ^ { 2 } & - 1 \leqslant x \leqslant 0 \\ 1 - \frac { 1 } { 2 } ( 1 - x ) ^ { 2 } & 0 < x \leqslant 1 \\ 1 & x > 1 \end{array} \right.$$
  1. Find the probability density function of \(X\).
  2. Find \(\mathrm { P } \left( - \frac { 1 } { 2 } \leqslant X \leqslant \frac { 1 } { 2 } \right)\).
  3. Find \(\mathrm { E } \left( X ^ { 2 } \right)\).
  4. Find \(\operatorname { Var } \left( X ^ { 2 } \right)\).
CAIE Further Paper 4 2021 November Q3
3 A supermarket sells pears in packs of 8 . Some of the pears in a pack may not be ripe, and the supermarket manager claims that the number of unripe pears in a pack can be modelled by the distribution \(\mathrm { B } ( 8,0.15 )\). A random sample of 150 packs was selected and the number of unripe pears in each pack was recorded. The following table shows the observed frequencies together with some of the expected frequencies using the manager's binomial distribution.
Number of unripe pears per pack012345\(\geqslant 6\)
Observed frequency35484315630
Expected frequency40.874\(p\)35.64112.5792.7750.392\(q\)
  1. Find the values of \(p\) and \(q\).
  2. Carry out a goodness of fit test, at the \(5 \%\) significance level, to test whether the manager's claim is justified.