Questions — CAIE Further Paper 4 (131 questions)

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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 2024 June Q1
1 A college uses two assessments, \(X\) and \(Y\), when interviewing applicants for research posts at the college. These assessments have been used for a large number of applicants this year. The scores for a random sample of 9 applicants who took assessment \(X\) are as follows. $$\begin{array} { l l l l l l l l l } 21.4 & 24.6 & 25.3 & 22.7 & 20.8 & 21.5 & 22.9 & 21.3 & 22.3 \end{array}$$ The scores for a random sample of 10 applicants who took assessment \(Y\) are as follows. $$\begin{array} { l l l l l l l l l l } 20.9 & 23.5 & 24.8 & 21.9 & 23.4 & 24.0 & 23.8 & 24.1 & 25.1 & 25.8 \end{array}$$ The interviewer believes that the population median score from assessment \(X\) is lower than the population median score from assessment \(Y\). Carry out a Wilcoxon rank-sum test, at the \(1 \%\) significance level, to test whether the interviewer's belief is supported by the data.
\includegraphics[max width=\textwidth, alt={}, center]{b5ff998a-fcb6-4a1b-ae86-ec66b0dccc3c-02_2714_37_143_2008}
CAIE Further Paper 4 2024 June Q2
2 A rowing club has a large number of members.A random sample of 12 of these members is taken and the pulse rate,\(x\) beats per minute(bpm),of each is measured after a 30 -minute training session.A \(98 \%\) confidence interval for the population mean pulse rate,\(\mu \mathrm { bpm }\) ,is calculated from the sample as \(64.22 < \mu < 68.66\) .
  1. Find the values of \(\sum x\) and \(\sum x ^ { 2 }\) .
  2. State an assumption that is necessary for the confidence interval to be valid.
    \includegraphics[max width=\textwidth, alt={}, center]{b5ff998a-fcb6-4a1b-ae86-ec66b0dccc3c-04_2718_38_141_2009}
CAIE Further Paper 4 2024 June Q3
3 There are three bus companies in a city. The council is investigating whether the buses reliably arrive at their destination on time. The results from random samples of buses from each company are summarised in the following table.
\multirow{2}{*}{}Bus company
\(A\)\(B\)\(C\)Total
\multirow{3}{*}{Arrival}Early22221054
On time305242124
Late28261872
Total8010070250
Test, at the \(5 \%\) significance level, whether the reliability of buses is independent of bus company.
CAIE Further Paper 4 2024 June Q4
4 The random variable \(X\) has probability generating function \(\mathrm { G } _ { X } ( t )\) given by $$\mathrm { G } _ { X } ( t ) = \operatorname { ct } ( 1 + t ) ^ { 5 }$$ where \(c\) is a constant.
  1. Find the value of \(c\).
  2. Find the value of \(\mathrm { E } ( X )\).
    \includegraphics[max width=\textwidth, alt={}, center]{b5ff998a-fcb6-4a1b-ae86-ec66b0dccc3c-06_2718_33_141_2014} The random variable \(Y\) is the sum of two independent values of \(X\).
  3. Write down the probability generating function of \(Y\) and hence find \(\operatorname { Var } ( Y )\).
  4. Find \(\mathrm { P } ( Y = 5 )\).
CAIE Further Paper 4 2024 June Q5
5 The continuous random variable \(X\) has cumulative distribution function F given by $$F ( x ) = \begin{cases} 0 & x < 2 ,
\frac { ( x - 2 ) ^ { 2 } } { 12 } & 2 \leqslant x < 4 ,
1 - \frac { ( 8 - x ) ^ { 2 } } { 24 } & 4 \leqslant x \leqslant 8 ,
1 & x > 8 . \end{cases}$$
\includegraphics[max width=\textwidth, alt={}]{b5ff998a-fcb6-4a1b-ae86-ec66b0dccc3c-08_2718_35_143_2012}
(c) Find the exact value of the interquartile range of \(X\).
CAIE Further Paper 4 2024 June Q6
6 Seva is investigating the lengths of the tails of adult wallabies in two regions of Australia, \(X\) and \(Y\). He chooses a random sample of 50 adult wallabies from region \(X\) and records the lengths, \(x \mathrm {~cm}\), of their tails. He also chooses a random sample of 40 adult wallabies from region \(Y\) and records the lengths, \(y \mathrm {~cm}\), of their tails. His results are summarised as follows. $$\sum x = 1080 \quad \sum x ^ { 2 } = 23480 \quad \sum y = 940 \quad \sum y ^ { 2 } = 22220$$ It cannot be assumed that the population variances of the two distributions are the same.
  1. Find a \(90 \%\) confidence interval for the difference between the population mean lengths of the tails of adult wallabies in regions \(X\) and \(Y\).
    \includegraphics[max width=\textwidth, alt={}, center]{b5ff998a-fcb6-4a1b-ae86-ec66b0dccc3c-10_2718_38_141_2010} The population mean lengths of the tails of adult wallabies in regions \(X\) and \(Y\) are \(\mu _ { X } \mathrm {~cm}\) and \(\mu _ { Y } \mathrm {~cm}\) respectively.
  2. Test, at the \(10 \%\) significance level, the null hypothesis \(\mu _ { Y } - \mu _ { X } = 1.1\) against the alternative hypothesis \(\mu _ { Y } - \mu _ { X } > 1.1\). State your conclusion in the context of the question.
    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 2020 November Q1
5 marks
1 The heights of the members of a large sports club are normally distributed. A random sample of 11 members of the club is chosen and their heights, \(x \mathrm {~cm}\), are measured. The results are summarised as follows, where \(\bar { x }\) denotes the sample mean of \(x\). $$\bar { x } = 176.2 \quad \sum ( x - \bar { x } ) ^ { 2 } = 313.1$$ Test, at the \(5 \%\) significance level, the null hypothesis that the population mean height for members of this club is equal to 172.5 cm against the alternative hypothesis that the mean differs from 172.5 cm . [5]
CAIE Further Paper 4 2020 November Q2
2 A large school is holding an essay competition and each student has submitted an essay. To ensure fairness, each essay is given a mark out of 100 by two different judges. The marks awarded to the essays submitted by a random sample of 12 students are shown in the following table.
Student\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)\(J\)\(K\)\(L\)
Judge 1627452486855566437708159
Judge 2657047497674675450777275
  1. One of the students claims that Judge 2 is awarding higher marks than Judge 1. Carry out a Wilcoxon matched-pairs signed-rank test at the \(5 \%\) significance level to test whether the data supports the student’s claim.
    It is discovered later that the marks awarded to student \(A\) have been entered incorrectly. In fact, Judge 1 awarded 65 marks and Judge 2 awarded 62 marks.
  2. By considering how this change affects the test statistic, explain why the conclusion of the test carried out in part (a) remains the same.
CAIE Further Paper 4 2020 November Q3
3 A random sample of 200 observations of the continuous random variable \(X\) was taken and the values are summarised in the following table.
Interval\(0 \leqslant x < 0.5\)\(0.5 \leqslant x < 1\)\(1 \leqslant x < 1.5\)\(1.5 \leqslant x < 2\)\(2 \leqslant x < 2.5\)\(2.5 \leqslant x < 3\)
Observed frequency52340414645
It is required to test the goodness of fit of the distribution with probability density function f given by $$f ( x ) = \begin{cases} \frac { 1 } { 9 } x ( 4 - x ) & 0 \leqslant x \leqslant 3
0 & \text { otherwise } \end{cases}$$ Most of the relevant expected frequencies, correct to 2 decimal places, are given in the following table.
Interval\(0 \leqslant x < 0.5\)\(0.5 \leqslant x < 1\)\(1 \leqslant x < 1.5\)\(1.5 \leqslant x < 2\)\(2 \leqslant x < 2.5\)\(2.5 \leqslant x < 3\)
Expected frequency\(p\)\(q\)37.9643.5243.5237.96
  1. Show that \(p = 10.19\) and find the value of \(q\).
  2. Carry out a goodness of fit test, at the \(5 \%\) significance level, to test whether f is a satisfactory model for the data.
CAIE Further Paper 4 2020 November Q4
4 The continuous random variable \(X\) has cumulative distribution function F given by $$F ( x ) = \begin{cases} 0 & x < 2
\frac { 1 } { 60 } x ^ { 2 } - \frac { 1 } { 15 } & 2 \leqslant x \leqslant 8
1 & x > 8 \end{cases}$$
  1. Find \(\mathrm { P } ( 3 \leqslant X \leqslant 6 )\).
  2. Find \(\mathrm { E } ( \sqrt { X } )\).
  3. Find \(\operatorname { Var } ( \sqrt { X } )\).
  4. The random variable \(Y\) is defined by \(Y = X ^ { 3 }\). Find the probability density function of \(Y\).
CAIE Further Paper 4 2020 November Q5
5 marks
5 The random variable \(X\) has the binomial distribution \(\mathrm { B } ( n , p )\).
  1. Write down an expression for \(\mathrm { P } ( \mathrm { X } = \mathrm { r } )\) and hence show that the probability generating function of \(X\) is \(( \mathrm { q } + \mathrm { pt } ) ^ { \mathrm { n } }\), where \(\mathrm { q } = 1 - \mathrm { p }\).
  2. Use the probability generating function of \(X\) to prove that \(\mathrm { E } ( \mathrm { X } ) = \mathrm { np }\) and \(\operatorname { Var } ( \mathrm { X } ) = \mathrm { np } ( 1 - \mathrm { p } )\). [5]
CAIE Further Paper 4 2020 November Q6
6 Nassa is researching the lengths of a particular type of snake in two countries, \(A\) and \(B\).
  1. He takes a random sample of 10 snakes of this type from country \(A\) and measures the length, \(x \mathrm {~m}\), of each snake. He then calculates a \(90 \%\) confidence interval for the population mean length, \(\mu \mathrm { m }\), for snakes of this type, assuming that snake lengths have a normal distribution. This confidence interval is \(3.36 \leqslant \mu \leqslant 4.22\). Find the sample mean and an unbiased estimate for the population variance.
  2. Nassa also measures the lengths, \(y \mathrm {~m}\), of a random sample of 8 snakes of this type taken from country \(B\). His results are summarised as follows. $$\sum y = 27.86 \quad \sum y ^ { 2 } = 98.02$$ Nassa claims that the mean length of snakes of this type in country \(B\) is less than the mean length of snakes of this type in country \(A\). Nassa assumes that his sample from country \(B\) also comes from a normal distribution, with the same variance as the distribution from country \(A\). Test at the \(10 \%\) significance level whether there is evidence to support Nassa’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 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.