Questions — CAIE Further Paper 4 (131 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
CAIE Further Paper 4 2021 November Q4
4 Manet has developed a new training course to help athletes improve their time taken to run 800 m . Manet claims that his course will decrease an athlete's time by more than 2 s on average. For a random sample of 10 athletes the times taken, in seconds, before and after the course are given in the following table.
Athlete\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)\(J\)
Before150146131135126142130129137134
After145138129135122135132128127137
Use a \(t\)-test, at the \(5 \%\) significance level, to test whether Manet's claim is justified, stating any assumption that you make.
CAIE Further Paper 4 2021 November Q5
5 Nine balls labelled \(1,2,3,4,5,6,7,8,9\) are placed in a bag. Kai selects three balls at random from the bag, without replacement. The random variable \(X\) is the number of balls selected by Kai that are labelled with a multiple of 3 .
  1. Find the probability generating function \(\mathrm { G } _ { \mathrm { X } } ( \mathrm { t } )\) of \(X\).
    The balls are replaced in the bag.
    Jacob now selects two balls at random from the bag, without replacement. The random variable \(Y\) is the number of balls selected by Jacob that are labelled with an even number.
  2. Find the probability generating function \(\mathrm { G } _ { Y } ( \mathrm { t } )\) of \(Y\).
    The random variable \(Z\) is the sum of the number of balls that are labelled with a multiple of 3 selected by Kai and the number of balls that are labelled with an even number selected by Jacob.
  3. Find the probability generating function of \(Z\), expressing your answer as a polynomial.
  4. Use the probability generating function of \(Z\) to find \(\mathrm { E } ( Z )\).
CAIE Further Paper 4 2021 November Q6
6 The blood cholesterol levels, measured in suitable units, of a random sample of 11 women and a random sample of 12 men are shown below.
Women51552421671522567513798238235
Men3112621703021753202202607235186333
Carry out a Wilcoxon rank-sum test, at the \(5 \%\) significance level, to test whether, on average, there is a difference in cholesterol levels between women and men.
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 number, \(x\), of pine trees was counted in each of 40 randomly chosen regions of equal size in country \(A\). The number, \(y\), of pine trees was counted in each of 60 randomly chosen regions of the same equal size in country \(B\). The results are summarised as follows. $$\sum x = 752 \quad \sum x ^ { 2 } = 14320 \quad \sum y = 1548 \quad \sum y ^ { 2 } = 40200$$ Find a 95\% confidence interval for the difference between the mean number of pine trees in regions of this size in countries \(A\) and \(B\).
CAIE Further Paper 4 2021 November Q2
2 It is claimed that the heights of a particular age group of boys follow a normal distribution with mean 125 cm and standard deviation 12 cm . Observations for a randomly chosen group of 60 boys in this age group are summarised in the following table. The table also gives the expected frequencies, correct to 2 decimal places, based on the normal distribution with mean 125 cm and standard deviation 12 cm .
Height, \(x \mathrm {~cm}\)\(x < 100\)\(100 \leqslant x < 110\)\(110 \leqslant x < 120\)\(120 \leqslant x < 130\)\(130 \leqslant x < 140\)\(x \geqslant 140\)
Observed frequency031523118
Expected frequency1.125.2213.9719.3813.976.34
  1. Show how the expected frequency for \(130 \leqslant x < 140\) is obtained.
  2. Carry out a goodness of fit test, at the \(5 \%\) significance level, to determine whether the claim is supported by the data.
CAIE Further Paper 4 2021 November Q3
3 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} a + \frac { 1 } { 5 } x & 0 \leqslant x < 1
2 a - \frac { 1 } { 5 } x & 1 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.
  1. Find the value of \(a\).
  2. Find \(\mathrm { E } \left( X ^ { 2 } \right)\).
  3. Find the cumulative distribution function of \(X\).
CAIE Further Paper 4 2021 November Q4
4 Applicants for a particular college take a written test when they attend for interview. There are two different written tests, \(A\) and \(B\), and each applicant takes one or the other. The interviewer wants to determine whether the medians of the distribution of marks obtained in the two tests are equal. The marks obtained by a random sample of 8 applicants who took test \(A\) and a random sample of 8 applicants who took test \(B\) are as follows.
Test \(A\)4632291233182540
Test \(B\)3628493748354131
  1. Carry out a Wilcoxon rank-sum test at the \(5 \%\) significance level to determine whether there is a difference in the population median marks obtained in the two tests.
    The interviewer considers using the given information to carry out a paired sample \(t\)-test to determine whether there is a difference in the population means for the two tests.
  2. Give two reasons why it is not appropriate to use this test.
CAIE Further Paper 4 2021 November Q5
5 The random variable \(X\) is such that \(\mathrm { P } ( \mathrm { X } = \mathrm { r } ) = \mathrm { kr } ^ { 2 }\) for \(r = 1,2,3,4\), where \(k\) is a constant.
  1. Find the value of \(k\).
  2. Find the probability generating function \(\mathrm { G } _ { X } ( \mathrm { t } )\) of \(X\).
    The random variable \(Y\) has probability generating function \(\mathrm { G } _ { Y } ( \mathrm { t } ) = \frac { 1 } { 4 } + \frac { 1 } { 2 } \mathrm { t } + \frac { 1 } { 4 } \mathrm { t } ^ { 2 }\).
    The random variable \(Z\) is the sum of \(X\) and \(Y\).
  3. Assuming that \(X\) and \(Y\) are independent, find the probability generating function \(\mathrm { G } _ { \mathrm { Z } } ( \mathrm { t } )\) of \(Z\) as a polynomial in \(t\).
  4. Given that \(\mathrm { E } ( \mathrm { Z } ) = \frac { 13 } { 3 }\), use \(\mathrm { G } _ { \mathrm { Z } } ( \mathrm { t } )\) to find \(\operatorname { Var } ( \mathrm { Z } )\).
CAIE Further Paper 4 2021 November Q6
10 marks
6 A scientist is investigating the masses of a particular type of fish found in lakes \(A\) and \(B\). He chooses a random sample of 10 fish of this type from lake \(A\) and records their masses, \(x \mathrm {~kg}\), as follows.
0.9
1.8
1.8
1.9
2.1
2.4
2.6
2.2
2.5
3.0 The scientist also chooses a random sample of 12 fish of this type from lake \(B\), but he only has a summary of their masses, \(y \mathrm {~kg}\), as follows. $$\sum y = 24.48 \quad \sum y ^ { 2 } = 53.75$$ Test at the \(10 \%\) significance level whether the mean mass of fish of this type in lake \(A\) is greater than the mean mass of fish of this type in lake \(B\). You should state any assumptions that you need to make for the test to be valid.
[0pt] [10]
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 2022 November Q1
1 Jasmine is researching the heights of pine trees in forests in two regions \(A\) and \(B\). She chooses a random sample of 50 pine trees in region \(A\) and records their heights, \(x \mathrm {~m}\). She also chooses a random sample of 60 pine trees in region \(B\) and records their heights, \(y \mathrm {~m}\). Her results are summarised as follows. $$\sum x = 1625 \quad \sum x ^ { 2 } = 53200 \quad \sum y = 1854 \quad \sum y ^ { 2 } = 57900$$ Find a \(95 \%\) confidence interval for the difference between the population mean heights of pine trees in regions \(A\) and \(B\).
CAIE Further Paper 4 2022 November Q2
2 An organisation runs courses to train students to become engineers. These students are taught in groups of 8 . The director of the organisation claims that on average \(60 \%\) of the students in a group achieve a pass. A random sample of 150 groups of 8 students is chosen. The following table shows the observed frequencies together with some of the expected frequencies using the appropriate binomial distribution.
Number of passes per group012345678
Observed frequency00824453626101
Expected frequency\(p\)1.1806.19318.57934.836\(q\)\(r\)13.4372.519
  1. Find the values of \(p , q\) and \(r\) giving your answers correct to 3 decimal places.
  2. Carry out a goodness of fit test, at the \(10 \%\) significance level, to test whether there is evidence to reject the director's claim.
CAIE Further Paper 4 2022 November Q3
3 A large college is holding a piano competition. Each student has played a particular piece of music and two judges have each awarded a mark out of 80 . The marks awarded to a random sample of 14 students are shown in the following table.
Student\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)\(J\)\(K\)\(L\)\(M\)\(N\)
Judge 17954637469525057554263555648
Judge 27562607376413151455549506536
  1. One of the students claims that on average Judge 1 is awarding higher marks than Judge 2. Carry out a Wilcoxon matched-pairs signed-rank test at the 5\% significance level to test whether the data supports the student's claim.
  2. Give a reason why it is preferable to use a Wilcoxon matched-pairs signed-rank test in this situation rather than a paired sample \(t\)-test.
CAIE Further Paper 4 2022 November Q4
4 Jason has three biased coins. For each coin the probability of obtaining a head when it is thrown is \(\frac { 2 } { 3 }\). Jason throws all three coins. The number of heads obtained is denoted by \(X\).
  1. Find the probability generating function \(\mathrm { G } _ { \mathrm { X } } ( \mathrm { t } )\) of \(X\).
    Jason also has two unbiased coins. He throws all five coins. The number of heads obtained from the two unbiased coins is denoted by \(Y\). It is given that \(G _ { Y } ( t ) = \frac { 1 } { 4 } + \frac { 1 } { 2 } t + \frac { 1 } { 4 } t ^ { 2 }\). The random variable \(Z\) is the total number of heads obtained when Jason throws all five coins.
  2. Find the probability generating function of \(Z\), expressing your answer as a polynomial.
  3. Find \(\mathrm { E } ( \mathrm { Z } )\).
CAIE Further Paper 4 2022 November Q5
5 The continuous random variable \(X\) has cumulative distribution function F given by $$F ( x ) = \begin{cases} 0 & x < 0
1 - \frac { 1 } { 144 } ( 12 - x ) ^ { 2 } & 0 \leqslant x \leqslant 12
1 & x > 12 \end{cases}$$
  1. Find the upper quartile of \(X\).
  2. Find \(\operatorname { Var } \left( X ^ { 2 } \right)\).
    The random variable \(Y\) is given by \(Y = \sqrt { X }\).
  3. Find the probability density function of \(Y\).
CAIE Further Paper 4 2022 November Q6
6 A company manufactures copper pipes. The pipes are produced by two different machines, \(A\) and \(B\). An inspector claims that the mean diameter of the pipes produced by machine \(A\) is greater than the mean diameter of the pipes produced by machine \(B\). He takes a random sample of 12 pipes produced by machine \(A\) and measures their diameters, \(x \mathrm {~cm}\). His results are summarised as follows. $$\sum x = 6.24 \quad \sum x ^ { 2 } = 3.26$$ He also takes a random sample of 10 pipes produced by machine \(B\) and measures their diameters in cm. His results are as follows. $$\begin{array} { l l l l l l l l l l } 0.48 & 0.53 & 0.47 & 0.54 & 0.54 & 0.55 & 0.46 & 0.55 & 0.50 & 0.48 \end{array}$$ The diameters of the pipes produced by each machine are assumed to be normally distributed with equal population variances. Test at the \(2.5 \%\) significance level whether the data supports the inspector'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 November Q1
1 A basketball club has a large number of players. The heights, \(x \mathrm {~m}\), of a random sample of 10 of these players are measured. A \(90 \%\) confidence interval for the population mean height, \(\mu \mathrm { m }\), of players in this club is calculated. It is assumed that heights are normally distributed. The confidence interval is \(1.78 \leqslant \mu \leqslant 2.02\). Find the values of \(\sum x\) and \(\sum x ^ { 2 }\) for this sample.
CAIE Further Paper 4 2022 November Q2
2 In the colleges in three regions of a particular country, students are given individual targets to achieve. Their performance is measured against their individual target and graded as 'above target', 'on target' or 'below target'. For a random sample of students from each of the three regions, the observed frequencies are summarised in the following table.
\multirow{2}{*}{}Region
ABCTotal
\multirow{3}{*}{Performance}Above target624144147
On target1029495291
Below target564561162
Total220180200600
Test, at the 10\% significance level, whether performance is independent of region.
CAIE Further Paper 4 2022 November Q3
3 A scientist is investigating the masses of birds of a certain species in country \(X\) and country \(Y\). She takes a random sample of 50 birds of this species from country \(X\) and a random sample of 80 birds of this species from country \(Y\). She records their masses in \(\mathrm { kg } , x\) and \(y\), respectively. Her results are summarised as follows. $$\sum x = 75.5 \quad \sum x ^ { 2 } = 115.2 \quad \sum y = 116.8 \quad \sum y ^ { 2 } = 172.6$$ The population mean masses of these birds in countries \(X\) and \(Y\) are \(\mu _ { x } \mathrm {~kg}\) and \(\mu _ { y } \mathrm {~kg}\) respectively.
Test, at the \(5 \%\) significance level, the null hypothesis \(\mu _ { \mathrm { x } } = \mu _ { \mathrm { y } }\) against the alternative hypothesis \(\mu _ { \mathrm { x } } > \mu _ { \mathrm { y } }\). State your conclusion in the context of the question.
CAIE Further Paper 4 2022 November Q4
4 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} k & 0 \leqslant x < 1
k x & 1 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 2 } { 5 }\).
  2. Find the interquartile range of \(X\).
  3. Find \(\operatorname { Var } ( X )\).
CAIE Further Paper 4 2022 November Q5
5 A 6 -sided dice, \(A\), with faces numbered \(1,2,3,4,5,6\) is biased so that the probability of throwing a 6 is \(\frac { 1 } { 4 }\). The random variable \(X\) is the number of 6s obtained when dice \(A\) is thrown twice.
  1. Find the probability generating function of \(X\).
    A second dice, \(B\), with faces numbered \(1,2,3,4,5,6\) is unbiased. The random variable \(Y\) is the number of 6s obtained when dice \(B\) is thrown twice. The random variable \(Z\) is the total number of 6s obtained when both dice are thrown twice.
  2. Find the probability generating function of \(Z\), expressing your answer as a polynomial.
  3. Find \(\operatorname { Var } ( Z )\).
  4. Use the probability generating function of \(Z\) to find the most probable value of \(Z\).
CAIE Further Paper 4 2022 November Q6
6 The manager of a technology company \(A\) claims that his employees earn more per year than the employees at technology company \(B\). The amounts earned per year, in hundreds of dollars, by a random sample of 12 employees from company \(A\) and an independent random sample of 12 employees from company \(B\) are shown below.
Company \(A\)461482374512415452502427398545612359
Company \(B\)454506491384361443401472414342355437
  1. Carry out a Wilcoxon rank-sum test at the \(5 \%\) significance level to test whether the manager's claim is supported by the data.
  2. Explain whether a paired sample \(t\)-test would be appropriate to test the manager's claim if earnings are normally distributed.
    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 November Q1
1 Maya is an athlete who competes in 1500-metre races. Last summer her practice run times had mean 4.22 minutes. Over the winter she has done some intense training to try to improve her times. A random sample of 10 of her practice run times, \(x\) minutes, this summer are summarised as follows. $$\sum x = 42.05 \quad \sum x ^ { 2 } = 176.83$$ Maya's new practice run times are normally distributed. She believes that on average her times have improved as a result of her training. Test, at the \(5 \%\) significance level, whether Maya’s belief is supported by the data.
CAIE Further Paper 4 2023 November Q2
2 A town council has published its plans for redeveloping the town centre and residents are being asked whether they approve or disapprove. A random sample of 250 responses has been selected from residents in the four main streets in the town: North, East, South and West Streets. The results are shown in the table.
\cline { 2 - 5 } \multicolumn{1}{c|}{}North StreetEast StreetSouth StreetWest Street
Approve33544226
Disapprove1939289
Test, at the \(5 \%\) significance level, whether the opinions of the residents are independent of the streets on which they live.
CAIE Further Paper 4 2023 November Q3
3 Scientists are studying the effects of exercise on LDL blood cholesterol levels. Over a three-month period, a large group of people exercised for 20 minutes each day. For a randomly chosen sample of 10 of these people, the LDL blood cholesterol levels were measured at the beginning and the end of the three-month period. The results, measured in suitable units, are as follows.
\cline { 2 - 12 } \multicolumn{1}{c|}{}Person\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)
\multirow{2}{*}{
Cholesterol
level
}		Beginning72841209010213564758088
\cline { 2 - 12 }End64761059210511567757584
  1. Test, at the \(2.5 \%\) significance level, whether there is evidence that the population mean LDL blood cholesterol level has reduced by more than 2 units after the three-month period.
  2. State any assumption that you have made in part (a).
    \includegraphics[max width=\textwidth, alt={}, center]{44829994-2ef0-488d-aa3b-99fb0e36d733-06_399_1383_269_324} As shown in the diagram, the continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} m x & 0 \leqslant x \leqslant 2
    \frac { k } { x ^ { 2 } } + c & 2 \leqslant x \leqslant 6
    0 & \text { otherwise } \end{cases}$$ where \(m , k\) and \(c\) are constants.
CAIE Further Paper 4 2023 November Q5
5 The random variable \(X\) has the geometric distribution \(\operatorname { Geo } ( p )\).
  1. Show that the probability generating function of \(X\) is \(\frac { \mathrm { pt } } { 1 - \mathrm { qt } }\), where \(\mathrm { q } = 1 - \mathrm { p }\).
  2. Use the probability generating function of \(X\) to show that \(\operatorname { Var } ( X ) = \frac { \mathrm { q } } { \mathrm { p } ^ { 2 } }\).
    Kenny throws an ordinary fair 6-sided dice repeatedly. The random variable \(X\) is the number of throws that Kenny takes in order to obtain a 6 . The random variable \(Z\) denotes the sum of two independent values of \(X\).
  3. Find the probability generating function of \(Z\).