Questions — CAIE Further Paper 4 (131 questions)

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CAIE Further Paper 4 2022 June Q3
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
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 2022 June Q1
1 The times taken by members of a large quiz club to complete a challenge have a normal distribution with mean \(\mu\) minutes. The times, \(x\) minutes, are recorded for a random sample of 8 members of the club. The results are summarised as follows, where \(\bar { x }\) is the sample mean. $$\bar { x } = 33.8 \quad \sum ( x - \bar { x } ) ^ { 2 } = 94.5$$ Find a 95\% confidence interval for \(\mu\).
CAIE Further Paper 4 2022 June Q2
2 A scientist is investigating the size of shells at various beach locations. She selects four beach locations and takes a random sample of shells from each of these beaches. She classifies each shell as large or small. Her results are summarised in the following table.
\multirow{2}{*}{}Beach location
A\(B\)CDTotal
\multirow{2}{*}{Size of shell}Large68699681314
Small28556439186
Total96124160120500
Test, at the 10\% significance level, whether the size of shell is independent of the beach location.
CAIE Further Paper 4 2022 June Q3
3 George throws two coins, \(A\) and \(B\), at the same time. Coin \(A\) is biased so that the probability of obtaining a head is \(a\). Coin \(B\) is biased so that the probability of obtaining a head is \(b\), where \(\mathrm { b } < \mathrm { a }\). The probability generating function of \(X\), the number of heads obtained by George, is \(\mathrm { G } _ { \mathrm { X } } ( \mathrm { t } )\). The coefficients of \(t\) and \(t ^ { 2 }\) in \(\mathrm { G } _ { \mathrm { X } } ( \mathrm { t } )\) are \(\frac { 5 } { 12 }\) and \(\frac { 1 } { 12 }\) respectively.
  1. Find the value of \(a\).
    The random variable \(Y\) is the sum of two independent observations of \(X\).
  2. Find the probability generating function of \(Y\), giving your answer as a polynomial in \(t\).
  3. Find \(\operatorname { Var } ( Y )\).
CAIE Further Paper 4 2022 June Q4
4 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 3 } { 8 } \left( 1 + \frac { 1 } { x ^ { 2 } } \right) & 1 \leqslant x \leqslant 3
0 & \text { otherwise } \end{cases}$$
  1. Find \(\mathrm { E } ( \sqrt { X } )\).
    The random variable \(Y\) is given by \(Y = X ^ { 2 }\).
  2. Find the probability density function of \(Y\).
  3. Find the 40th percentile of \(Y\).
CAIE Further Paper 4 2022 June Q5
5 A manager claims that the lengths of the rubber tubes that his company produces have a median of 5.50 cm . The lengths, in cm , of a random sample of 11 tubes produced by this company are as follows.
5.565.455.475.585.545.525.605.355.595.51
It is required to test at the \(10 \%\) significance level the null hypothesis that the population median length is 5.50 cm against the alternative hypothesis that the population median length is not equal to 5.50 cm . Show that both a sign test and a Wilcoxon signed-rank test give the same conclusion and state this conclusion.
CAIE Further Paper 4 2022 June Q6
6 A company has two machines, \(A\) and \(B\), which independently fill small bottles with a liquid. The volumes of liquid per bottle, in suitable units, filled by machines \(A\) and \(B\) are denoted by \(x\) and \(y\) respectively. A scientist at the company takes a random sample of 40 bottles filled by machine \(A\) and a random sample of 50 bottles filled by machine \(B\). The results are summarised as follows. $$\sum x = 1120 \quad \sum x ^ { 2 } = 31400 \quad \sum y = 1370 \quad \sum y ^ { 2 } = 37600$$ The population means of the volumes of liquid in the bottles filled by machines \(A\) and \(B\) are denoted by \(\mu _ { A }\) and \(\mu _ { B }\).
  1. Test at the \(2 \%\) significance level whether there is any difference between \(\mu _ { A }\) and \(\mu _ { B }\).
  2. Find the set of values of \(\alpha\) for which there would be evidence at the \(\alpha \%\) significance level that \(\mu _ { \mathrm { A } } - \mu _ { \mathrm { B } }\) is greater than 0.25.
    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 2023 June Q1
1 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 1 } { 6 } \left( x ^ { - \frac { 1 } { 3 } } - x ^ { - \frac { 2 } { 3 } } \right) & 1 \leqslant x \leqslant 27
0 & \text { otherwise } \end{cases}$$
  1. Find the cumulative distribution function of \(X\).
    The random variable \(Y\) is defined by \(Y = X ^ { \frac { 1 } { 3 } }\).
  2. Find the probability density function of \(Y\).
  3. Find the exact value of the median of \(Y\).
CAIE Further Paper 4 2023 June Q2
2 Shane is studying the lengths of the tails of male red kangaroos. He takes a random sample of 14 male red kangaroos and measures the length of the tail, \(x \mathrm {~m}\), for each kangaroo. He then calculates a \(90 \%\) confidence interval for the population mean tail length, \(\mu \mathrm { m }\), of male red kangaroos. He assumes that the tail lengths are normally distributed and finds that \(1.11 \leqslant \mu \leqslant 1.14\). Find the values of \(\sum x\) and \(\sum x ^ { 2 }\) for this sample.
CAIE Further Paper 4 2023 June Q3
3 A large number of students took two test papers in mathematics. The teacher believes that the marks obtained in Paper 1 will be higher than the marks obtained in Paper 2. She chooses a random sample of 9 students and compares their marks. The marks are shown in the table.
Student\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)
Paper 1467355648642666860
Paper 2416661639040584270
  1. Carry out a Wilcoxon matched-pairs signed-rank test, at the \(5 \%\) significance level, to test whether the data supports the teacher’s belief.
  2. State an assumption that you have made in carrying out the test in part (a).
CAIE Further Paper 4 2023 June Q4
4 An inspector is checking the lengths of metal rods produced by two machines, \(X\) and \(Y\). These rods should be of the same length, but the inspector suspects that those made by machine \(X\) are shorter, on average, than those made by machine \(Y\). The inspector chooses a random sample of 80 rods made by machine \(X\) and a random sample of 60 rods made by machine \(Y\). The lengths of these rods are \(x \mathrm {~cm}\) and \(y \mathrm {~cm}\) respectively. Her results are summarised as follows. $$\sum x = 164.0 \quad \sum x ^ { 2 } = 338.1 \quad \sum y = 124.8 \quad \sum y ^ { 2 } = 261.1$$
  1. Test at the \(10 \%\) significance level whether the data supports the inspector's suspicion.
  2. Give a reason why it is not necessary to make any assumption about the distributions of the lengths of the rods.
CAIE Further Paper 4 2023 June Q5
5 The random variable \(X\) has probability generating function \(\mathrm { G } _ { X } ( \mathrm { t } )\) given by $$\mathrm { G } _ { \mathrm { X } } ( \mathrm { t } ) = \mathrm { k } \left( 1 + 3 \mathrm { t } + 4 \mathrm { t } ^ { 2 } \right)$$ where \(k\) is a constant.
  1. Show that \(\mathrm { E } ( X ) = \frac { 11 } { 8 }\).
    The random variable \(Y\) has probability generating function \(\mathrm { G } _ { \gamma } ( \mathrm { t } )\) given by $$G _ { \gamma } ( t ) = \frac { 1 } { 3 } t ^ { 2 } ( 1 + 2 t )$$ The random variables \(X\) and \(Y\) are independent and \(\mathrm { Z } = \mathrm { X } + \mathrm { Y }\).
  2. Find the probability generating function of \(Z\), expressing your answer as a polynomial in \(t\).
  3. Use your answer to part (b) to find the value of \(\operatorname { Var } ( Z )\).
  4. Write down the most probable value of \(Z\).
CAIE Further Paper 4 2023 June Q6
6 A scientist is investigating whether the ability to remember depends on age. A random sample of 150 students in different age groups is chosen. Each student is shown a set of 20 objects for thirty seconds and then asked to list as many as they can remember. The students are graded \(A\) or \(B\) according to how many objects they remembered correctly: grade \(A\) for 16 or more correct and grade \(B\) for fewer than 16 correct. The results are shown in the table.
\cline { 2 - 4 } \multicolumn{1}{c|}{}Age of students
\cline { 2 - 4 } \multicolumn{1}{c|}{}\(11 - 12\) years\(13 - 14\) years\(15 - 16\) years
Grade \(A\)251619
Grade \(B\)284517
  1. Carry out a \(\chi ^ { 2 }\)-test at the \(2.5 \%\) significance level to test whether grade is independent of age of student.
    The scientist decides instead to use three grades: grade \(A\) for 16 or more correct, grade \(B\) for 10 to 15 correct and grade \(C\) for fewer than 10 correct. The results are shown in the following table.
    \multirow{2}{*}{}Age of students
    11-12 years13-14 years15-16 years
    Grade \(A\)251619
    Grade \(B\)122711
    Grade \(C\)16186
    With this second set of data, the test statistic is calculated as 10.91.
  2. Complete the \(\chi ^ { 2 }\)-test at the \(2.5 \%\) significance level for this second set of data.
  3. State, with a reason, whether you would prefer to use the result from part (a) or part (b) to investigate whether the ability to remember depends on age.
    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.