Questions — CAIE Further Paper 4 (131 questions)

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CAIE Further Paper 4 2020 June Q1
1 Young children are learning to read using two different reading schemes, \(A\) and \(B\). The standards achieved are measured against the national average standard achieved and classified as above average, average or below average. For two randomly chosen groups of young children, the numbers in each category are shown in the table.
\cline { 2 - 4 } \multicolumn{1}{c|}{}Standard achieved
\cline { 2 - 4 } \multicolumn{1}{c|}{}Above averageAverageBelow average
Scheme \(A\)313522
Scheme \(B\)195043
Test at the \(5 \%\) significance level whether standard achieved is independent of the reading scheme used.
CAIE Further Paper 4 2020 June Q2
2 A random sample of 40 observations of a random variable \(X\) and a random sample of 50 observations of a random variable \(Y\) are taken. The resulting values for the sample means, \(\bar { x }\) and \(\bar { y }\), and the unbiased estimates, \(\mathrm { s } _ { \mathrm { x } } ^ { 2 }\) and \(\mathrm { s } _ { \mathrm { y } } ^ { 2 }\), for the population variances are as follows. $$\bar { x } = 24.4 \quad \bar { y } = 17.2 \quad s _ { x } ^ { 2 } = 10.2 \quad s _ { y } ^ { 2 } = 11.1$$ Find a \(90 \%\) confidence interval for the difference between the population means of \(X\) and \(Y\).
CAIE Further Paper 4 2020 June Q3
3 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 1 } { 5 } x & 0 \leqslant x < 2
\frac { 2 } { 15 } ( 5 - x ) & 2 \leqslant x \leqslant 5
0 & \text { otherwise } \end{cases}$$
  1. Find the cumulative distribution function of \(X\).
  2. Find the median value of \(X\).
  3. Find \(\mathrm { E } \left( X ^ { 2 } \right)\).
  4. Find \(\mathrm { P } ( 1 \leqslant x \leqslant 3 )\).
CAIE Further Paper 4 2020 June Q4
3 marks
4 The discrete random variable \(X\) has probability generating function \(\mathrm { G } _ { \mathrm { X } } ( \mathrm { t } )\) given by $$G _ { X } ( t ) = 0.2 t + 0.5 t ^ { 2 } + 0.3 t ^ { 3 }$$ The random variable \(Y\) is the sum of two independent observations of \(X\).
  1. Find the probability generating function of \(Y\), giving your answer as an expanded polynomial in \(t\). [3]
  2. Use the probability generating function of \(Y\) to find \(\mathrm { E } ( Y )\) and \(\operatorname { Var } ( Y )\).
CAIE Further Paper 4 2020 June Q5
5 Students at two colleges, \(A\) and \(B\), are competing in a computer games challenge.
  1. The time taken for a randomly chosen student from college \(A\) to complete the challenge has a normal distribution with mean \(\mu\) minutes. The times taken, \(x\) minutes, are recorded for a random sample of 10 students chosen from college \(A\). The results are summarised as follows. $$\sum x = 828 \quad \sum x ^ { 2 } = 68622$$ A test is carried out on the data at the \(5 \%\) significance level and the result supports the claim that \(\mu > k\). Find the greatest possible value of \(k\).
  2. A random sample of 8 students is chosen from college \(B\). Their times to complete the same challenge give a sample mean of 79.8 minutes and an unbiased variance estimate of 9.966 minutes \({ } ^ { 2 }\). Use a 2 -sample test at the \(5 \%\) significance level to test whether the mean time for students at college \(B\) to complete the challenge is the same as the mean time for students at college \(A\) to complete the challenge. You should assume that the two distributions are normal and have the same population variance.
CAIE Further Paper 4 2020 June Q6
6 A biologist is studying the effect of nutrients on the heights to which plants grow. A random sample of 24 similar young plants is divided into two equal groups \(A\) and \(B\). The plants in group \(A\) are fed with nutrients and water and the plants in group \(B\) are given only water. After four weeks, the height, in cm, of each plant is measured and the results are as follows.
Group \(A\)12.311.812.113.211.110.613.812.012.212.413.513.9
Group \(B\)11.710.810.911.311.212.611.010.511.912.510.711.6
The biologist decides to carry out a test at the \(5 \%\) significance level to test whether the nutrients have resulted in an increase in growth.
  1. She carries out a Wilcoxon rank-sum test. Give a reason why this is an appropriate choice of test.
  2. Carry out the Wilcoxon rank-sum test for these results.
    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 June Q1
1 A random sample of 7 observations of a variable \(X\) are as follows. $$\begin{array} { l l l l l l l } 8.26 & 7.78 & 7.92 & 8.04 & 8.27 & 7.95 & 8.34 \end{array}$$ The population mean of \(X\) is \(\mu\).
  1. Test, at the \(10 \%\) significance level, the null hypothesis \(\mu = 8.22\) against the alternative hypothesis \(\mu < 8.22\).
  2. State an assumption necessary for the test in part (a) to be valid.
CAIE Further Paper 4 2021 June Q2
2 A driving school employs four instructors to prepare people for their driving test. The allocation of people to instructors is random. For each of the instructors, the following table gives the number of people who passed and the number who failed their driving test last year.
Instructor \(A\)Instructor \(B\)Instructor \(C\)Instructor \(D\)Total
Pass72425268234
Fail33344158166
Total1057693126400
Test at the 10\% significance level whether success in the driving test is independent of the instructor.
CAIE Further Paper 4 2021 June Q3
3 The continuous random variable \(X\) has cumulative distribution function F given by $$F ( x ) = \begin{cases} 0 & x < 0
\frac { 1 } { 81 } x ^ { 2 } & 0 \leqslant x \leqslant 9
1 & x > 9 \end{cases}$$
  1. Find \(\mathrm { E } ( \sqrt { X } )\).
  2. Find \(\operatorname { Var } ( \sqrt { X } )\).
  3. The random variable \(Y\) is given by \(Y ^ { 3 } = X\). Find the probability density function of \(Y\).
CAIE Further Paper 4 2021 June Q4
4 A scientist is investigating the lengths of the leaves of birch trees in different regions. He takes a random sample of 50 leaves from birch trees in region \(A\) and a random sample of 60 leaves from birch trees in region \(B\). He records their lengths in \(\mathrm { cm } , x\) and \(y\), respectively. His results are summarised as follows. $$\sum x = 282 \quad \sum x ^ { 2 } = 1596 \quad \sum y = 328 \quad \sum y ^ { 2 } = 1808$$ The population mean lengths of leaves from birch trees in regions \(A\) and \(B\) are \(\mu _ { A } \mathrm {~cm}\) and \(\mu _ { B } \mathrm {~cm}\) respectively. Carry out a test at the \(5 \%\) significance level to test the null hypothesis \(\mu _ { \mathrm { A } } = \mu _ { \mathrm { B } }\) against the alternative hypothesis \(\mu _ { \mathrm { A } } \neq \mu _ { \mathrm { B } }\).
CAIE Further Paper 4 2021 June Q5
5 Georgio has designed two new uniforms \(X\) and \(Y\) for the employees of an airline company. A random sample of 11 employees are each asked to assess each of the two uniforms for practicality and appearance, and to give a total score out of 100. The scores are given in the table.
Employee\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)\(J\)\(K\)
Uniform \(X\)8274425960739498623650
Uniform \(Y\)7875635667829990724861
  1. Give a reason why a Wilcoxon signed-rank test may be more appropriate than a \(t\)-test for investigating whether there is any evidence of a preference for one of the uniforms.
  2. Carry out a Wilcoxon matched-pairs signed-rank test at the \(10 \%\) significance level.
CAIE Further Paper 4 2021 June Q6
6 Tanji has a bag containing 4 red balls and 2 blue balls. He selects 3 balls at random from the bag, without replacement. The number of red balls selected by Tanji is denoted by \(X\).
  1. Find the probability generating function \(\mathrm { G } _ { \mathrm { X } } ( \mathrm { t } )\) of \(X\).
    Tanji also has two coins, each biased so that the probability of obtaining a head when it is thrown is \(\frac { 1 } { 4 }\). He throws the two coins at the same time. The number of heads obtained is denoted by \(Y\).
  2. Find the probability generating function \(\mathrm { G } _ { Y } ( \mathrm { t } )\) of \(Y\).
    The random variable \(Z\) is the sum of the number of red balls selected by Tanji and the number of heads obtained.
  3. Find the probability generating function of \(Z\), expressing your answer as a polynomial.
  4. Use the probability generating function of \(Z\) to find \(E ( Z )\) and \(\operatorname { Var } ( Z )\).
    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 June Q1
1 Two randomly selected groups of students, with similar ranges of abilities, take the same examination in different rooms. One group of 140 students takes the examination with background music playing. The other group of 210 students takes the examination in silence. Each student is awarded a grade for their performance in the examination and the numbers from each group gaining each grade are shown in the following table.
\cline { 2 - 4 } \multicolumn{1}{c|}{}Grade awarded
\cline { 2 - 4 } \multicolumn{1}{c|}{}ABC
Background music495140
Silence936849
Test at the 10\% significance level whether grades awarded are independent of whether background music is playing during the examination.
CAIE Further Paper 4 2020 June Q2
2 The times, in milliseconds, taken by a computer to perform a certain task were recorded on 10 randomly chosen occasions. The times were as follows. $$\begin{array} { l l l l l l l l l l } 6.44 & 6.16 & 5.62 & 5.82 & 6.51 & 6.62 & 6.19 & 6.42 & 6.34 & 6.28 \end{array}$$ It is claimed that the median time to complete the task is 6.4 milliseconds.
  1. Carry out a Wilcoxon signed-rank test at the \(5 \%\) significance level to test this claim.
  2. State an underlying assumption that is made when using a Wilcoxon signed-rank test.
CAIE Further Paper 4 2020 June Q3
3 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 3 } { 16 } ( 2 - \sqrt { x } ) & 0 \leqslant x < 1
\frac { 3 } { 16 \sqrt { x } } & 1 \leqslant x \leqslant 9
0 & \text { otherwise } \end{cases}$$
  1. Find \(\mathrm { E } ( X )\).
    The random variable \(Y\) is such that \(Y = \sqrt { X }\).
  2. Find the probability density function of \(Y\).
CAIE Further Paper 4 2020 June Q4
4 A company has two different machines, \(X\) and \(Y\), each of which fills empty cups with coffee. The manager is investigating the volumes of coffee, \(x\) and \(y\), measured in appropriate units, in the cups filled by machines \(X\) and \(Y\) respectively. She chooses a random sample of 50 cups filled by machine \(X\) and a random sample of 40 cups filled by machine \(Y\). The volumes are summarised as follows. $$\sum x = 15.2 \quad \sum x ^ { 2 } = 5.1 \quad \sum y = 13.4 \quad \sum y ^ { 2 } = 4.8$$ The manager claims that there is no difference between the mean volume of coffee in cups filled by machine \(X\) and the mean volume of coffee in cups filled by machine \(Y\). Test the manager's claim at the \(10 \%\) significance level.
CAIE Further Paper 4 2020 June Q5
5 A large number of children are competing in a throwing competition. The distances, in metres, thrown by a random sample of 8 children are as follows.
\(\begin{array} { l l l l l l l l } 19.8 & 22.1 & 24.4 & 21.5 & 20.8 & 26.3 & 23.7 & 25.0 \end{array}\)
  1. Assuming that distances are normally distributed, test, at the \(5 \%\) significance level, whether the population mean distance thrown is more than 22.0 metres.
  2. Find a 95\% confidence interval for the population mean distance thrown.
CAIE Further Paper 4 2020 June Q6
6 A bag contains 4 red balls and 6 blue balls. Rassa selects two balls at random, without replacement, from the bag. The number of red balls selected by Rassa is denoted by \(X\).
  1. Find the probability generating function, \(\mathrm { G } _ { \mathrm { X } } ( \mathrm { t } )\), of \(X\).
    Rassa also tosses two coins. One coin is biased so that the probability of a head is \(\frac { 2 } { 3 }\). The other coin is biased so that the probability of a head is \(p\). The probability generating function of \(Y\), the number of heads obtained by Rassa, is \(\mathrm { G } _ { Y } ( \mathrm { t } )\). The coefficient of \(t\) in \(\mathrm { G } _ { Y } ( \mathrm { t } )\) is \(\frac { 7 } { 12 }\).
  2. Find \(\mathrm { G } _ { Y } ( \mathrm { t } )\).
    The random variable \(Z\) is the sum of the number of red balls selected and the number of heads obtained by Rassa.
  3. Find the probability generating function of \(Z\), expressing your answer as a polynomial.
  4. Use the probability generating function of \(Z\) to find \(\mathrm { E } ( Z )\).
    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 June Q1
1 Farmer \(A\) grows apples of a certain variety. Each tree produces 14.8 kg of apples, on average, per year. Farmer \(B\) grows apples of the same variety and claims that his apple trees produce a higher mass of apples per year than Farmer \(A\) 's trees. The masses of apples from Farmer \(B\) 's trees may be assumed to be normally distributed. A random sample of 10 trees from Farmer \(B\) is chosen. The masses, \(x \mathrm {~kg}\), of apples produced in a year are summarised as follows. $$\sum x = 152.0 \quad \sum x ^ { 2 } = 2313.0$$ Test, at the \(5 \%\) significance level, whether Farmer \(B\) 's claim is justified.
CAIE Further Paper 4 2021 June Q2
2 A company is developing a new flavour of chocolate by varying the quantities of the ingredients. A random selection of 9 flavours of chocolate are judged by two tasters who each give marks out of 100 to each flavour of chocolate.
Chocolate\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)
Taster 1728675929879876062
Taster 2847274958587827568
Carry out a Wilcoxon matched-pairs signed-rank test at the \(10 \%\) significance level to investigate whether, on average, there is a difference between marks awarded by the two tasters.
CAIE Further Paper 4 2021 June Q3
3 The heights, \(x \mathrm {~m}\), of a random sample of 50 adult males from country \(A\) were recorded. The heights, \(y \mathrm {~m}\), of a random sample of 40 adult males from country \(B\) were also recorded. The results are summarised as follows. $$\Sigma x = 89.0 \quad \Sigma x ^ { 2 } = 159.4 \quad \Sigma y = 67.2 \quad \Sigma y ^ { 2 } = 113.1$$ Find a 95\% confidence interval for the difference between the mean heights of adult males from country \(A\) and adult males from country \(B\).
\(4 X\) is a discrete random variable which takes the values \(0,2,4 , \ldots\). The probability generating function of \(X\) is given by $$G _ { X } ( t ) = \frac { 1 } { 3 - 2 t ^ { 2 } }$$
  1. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
  2. Find \(\mathrm { P } ( X = 4 )\).
CAIE Further Paper 4 2021 June Q5
5 Chai packs china mugs into cardboard boxes. Chai's manager suspects that breakages occur at random times and that the number of breakages may follow a Poisson distribution. He takes a small sample of observations and finds that the number of breakages in a one-hour period has a mean of 2.4 and a standard deviation of 1.5.
  1. Explain how this information tends to support the manager's suspicion.
    The manager now takes a larger sample and claims that the numbers of breakages in a one-hour period follow a Poisson distribution. The numbers of breakages in a random sample of 180 one-hour periods are summarised in the following table.
    Number of breakages01234567 or more
    Frequency213346312316100
    The mean number of breakages calculated from this sample is 2.5.
  2. Use the data from this larger sample to carry out a goodness of fit test, at the \(10 \%\) significance level, to test the claim.
CAIE Further Paper 4 2021 June Q6
6 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 1 } { 8 } & 0 \leqslant x < 1
\frac { 1 } { 28 } ( 8 - x ) & 1 \leqslant x \leqslant 8
0 & \text { otherwise } \end{cases}$$
  1. Find the cumulative distribution function of \(X\).
  2. Find the value of the constant \(a\) such that \(\mathrm { P } ( \mathrm { X } \leqslant \mathrm { a } ) = \frac { 5 } { 7 }\).
    The random variable \(Y\) is given by \(Y = \sqrt [ 3 ] { X }\).
  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 2022 June Q1
1 A manager is investigating the times taken by employees to complete a particular task as a result of the introduction of new technology. He claims that the mean time taken to complete the task is reduced by more than 0.4 minutes. He chooses a random sample of 10 employees. The times taken, in minutes, before and after the introduction of the new technology are recorded in the table.
Employee\(A\)\(B\)\(C\)D\(E\)\(F\)G\(H\)IJ
Time before new technology10.29.812.411.610.811.214.610.612.311.0
Time after new technology9.68.512.410.910.210.612.810.812.510.6
  1. Test at the 10\% significance level whether the manager's claim is justified.
  2. State an assumption that is necessary for this test to be valid.
CAIE Further Paper 4 2022 June Q2
2 The probability generating function, \(\mathrm { G } _ { Y } ( t )\), of the random variable \(Y\) is given by $$G _ { Y } ( t ) = 0.04 + 0.2 t + 0.37 t ^ { 2 } + 0.3 t ^ { 3 } + 0.09 t ^ { 4 }$$
  1. Find \(\operatorname { Var } ( Y )\).
    The random variable \(Y\) is the sum of two independent observations of the random variable \(X\).
  2. Find the probability generating function of \(X\), giving your answer as a polynomial in \(t\).