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CAIE Further Paper 4 2021 November Q4
8 marks Standard +0.3
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
9 marks Standard +0.3
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 Standard +0.8
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
7 marks Standard +0.3
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
8 marks Standard +0.8
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
8 marks Standard +0.3
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
8 marks Standard +0.3
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
10 marks Challenging +1.2
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
9 marks Standard +0.8
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
6 marks Challenging +1.2
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
7 marks Standard +0.3
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
8 marks Standard +0.3
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
10 marks Standard +0.3
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
9 marks Standard +0.8
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
10 marks Standard +0.3
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
6 marks Standard +0.3
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
7 marks Moderate -0.3
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
8 marks Standard +0.3
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
10 marks
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 \(Z\).
CAIE Further Paper 4 2023 November Q6
10 marks Standard +0.3
6 A school is conducting an experiment to see whether the distance that children can throw a ball increases in hot weather. On a cold day, all the children at the school were asked to throw a ball as far as possible. The distances thrown were measured and recorded. The median distance thrown by a random sample of 25 of the children was 22.0 m . The children were asked to throw the ball again on a hot day. The distances thrown by the same 25 children were measured and recorded and these distances, in m , are shown below.
21.223.522.918.619.4
22.126.520.225.720.6
22.317.422.227.023.9
28.222.627.223.023.7
19.822.723.321.524.3
The teacher claims that on average the distances thrown will be further when it is hot.
Carry out a Wilcoxon signed-rank test, at the 5\% significance level, to test whether the data supports the teacher's claim.
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 2023 November Q1
6 marks Standard +0.3
1 A factory produces small bottles of natural spring water. Two different machines, \(X\) and \(Y\), are used to fill empty bottles with the water. A quality control engineer checks the volumes of water in the bottles filled by each of the machines. He chooses a random sample of 60 bottles filled by machine \(X\) and a random sample of 75 bottles filled by machine \(Y\). The volumes of water, \(x\) and \(y\) respectively, in millilitres, are summarised as follows. $$\sum x = 6345 \quad \sum ( x - \bar { x } ) ^ { 2 } = 243.8 \quad \sum y = 7614 \quad \sum ( y - \bar { y } ) ^ { 2 } = 384.9$$ \(\bar { x }\) and \(\bar { y }\) are the sample means of the volume of water in the bottles filled by machines \(X\) and \(Y\) respectively. Find a \(95 \%\) confidence interval for the difference between the mean volume of water in bottles filled by machine \(X\) and the mean volume of water in bottles filled by machine \(Y\).
CAIE Further Paper 4 2023 November Q2
8 marks Standard +0.3
2 The number of breakdowns on a particular section of road is recorded each day over a period of 90 days. It is suggested that the number of breakdowns follows a Poisson distribution with mean 3.5. The data is summarised in the table, together with some of the expected frequencies resulting from the suggested Poisson distribution.
Number of breakdowns per day012345678 or more
Observed frequency0513172116954
Expected frequency2.7189.51216.64616.99311.8953.4692.407
  1. Complete the table.
  2. Carry out a goodness of fit test, at the 10\% significance level, to determine whether or not \(\operatorname { Po } ( 3.5 )\) is a good fit to the data.
CAIE Further Paper 4 2023 November Q3
10 marks Challenging +1.2
3 Toby has a bag which contains 6 red marbles and 3 green marbles. He randomly chooses 3 marbles from the bag, without replacement. The random variable \(X\) is the number of red marbles that Toby obtains.
  1. Find the probability generating function of \(X\).
    Ling also has a bag which contains 6 red marbles and 3 green marbles. He randomly chooses 2 marbles from his bag, without replacement. The random variable \(Y\) is the number of red marbles that Ling obtains. It is given that the probability generating function of \(Y\) is \(\frac { 1 } { 12 } \left( 1 + 6 t + 5 t ^ { 2 } \right)\). The random variable \(Z\) is the total number of red marbles that Toby and Ling obtain.
  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 \(\operatorname { Var } ( Z )\).
CAIE Further Paper 4 2023 November Q4
10 marks Challenging +1.8
4
\includegraphics[max width=\textwidth, alt={}, center]{a9f9cf66-0734-4316-99ae-c57090d08135-08_353_1141_255_463} The diagram shows the continuous random variable \(X\) with probability density function f given by $$f ( x ) = \begin{cases} \frac { 1 } { 128 } \left( 4 a x - b x ^ { 3 } \right) & 0 \leqslant x \leqslant 4 \\ c & 4 \leqslant x \leqslant 6 \\ 0 & \text { otherwise } \end{cases}$$ where \(a , b\) and \(c\) are constants.
The upper quartile of \(X\) is equal to 4 .
  1. Show that \(c = \frac { 1 } { 8 }\) and find the values of \(a\) and \(b\).
  2. Find the exact value of the median of \(X\).
  3. Find \(\mathrm { E } ( \sqrt { X } )\), giving your answer correct to 2 decimal places.
CAIE Further Paper 4 2023 November Q5
16 marks Standard +0.8
5 A company is deciding which of two machines, \(X\) and \(Y\), can make a certain type of electrical component more quickly. The times taken, in minutes, to make one component of this type are recorded for a random sample of 8 components made by machine \(X\) and a random sample of 9 components made by machine \(Y\). These times are as follows.
Machine \(X\)4.04.64.74.85.05.25.65.8
Machine \(Y\)4.54.95.15.35.45.75.96.36.4
The manager claims that on average the time taken by machine \(X\) to make one component is less than that taken by machine \(Y\).
  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. Assuming that the times taken to produce the components by the two machines are normally distributed with equal variances, carry out a \(t\)-test at the \(5 \%\) significance level to test whether the manager's claim is supported by the data.
    \section*{Question 5(c) is printed on the next page.}
  3. In general, would you expect the conclusions from the tests in parts (a) and (b) to be the same? Give a reason for your answer.
    If you use the following page to complete the answer to any question, the question number must be clearly shown.