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CAIE S2 2019 November Q2
6 marks Standard +0.3
2 Cars arrive at a filling station randomly and at a constant average rate of 2.4 cars per minute.
  1. Calculate the probability that fewer than 4 cars arrive in a 2 -minute period.
  2. Use a suitable approximating distribution to calculate the probability that at least 140 cars arrive in a 1-hour period.
CAIE S2 2019 November Q3
6 marks Standard +0.3
3 The times, in minutes, taken by competitors to complete a puzzle have mean \(\mu\) and standard deviation 3 . The times taken by a random sample of 10 competitors are noted and the results are given below. \(\begin{array} { l l l } 25.2 & 26.8 & 18.5 \end{array}\) 25.5
30.1 \(28.9 \quad 27.0\) \(26.1 \quad 26.0\) 24.9
  1. Stating a necessary assumption, calculate a \(97 \%\) confidence interval for \(\mu\).
  2. Two more random samples, each of 10 competitors, are taken. Their times are used to calculate two more \(97 \%\) confidence intervals for \(\mu\). Find the probability that neither of these intervals contains the true value of \(\mu\).
CAIE S2 2019 November Q4
6 marks Moderate -0.3
4 A train company claims that \(92 \%\) of trains on a particular line arrive on time. Sanjeep suspects that the true percentage is less than \(92 \%\). He chooses a random sample of 20 trains on this line and finds that exactly 16 of them arrive on time. Making an assumption that should be stated, test at the 5\% significance level whether Sanjeep's suspicion is justified.
[0pt] [6]
CAIE S2 2019 November Q5
8 marks Moderate -0.3
5
  1. The random variable \(X\) has the distribution \(\mathrm { B } ( 300,0.01 )\). Use a Poisson approximation to find \(\mathrm { P } ( 2 < X < 6 )\).
  2. The random variable \(Y\) has the distribution \(\mathrm { Po } ( \lambda )\), and \(\mathrm { P } ( Y = 0 ) = \mathrm { P } ( Y = 2 )\). Find \(\lambda\).
  3. The random variable \(Z\) has the distribution \(\mathrm { Po } ( 5.2 )\) and it is given that \(\mathrm { P } ( Z = n ) < \mathrm { P } ( Z = n + 1 )\).
    1. Write down an inequality in \(n\).
    2. Hence or otherwise find the largest possible value of \(n\).
CAIE S2 2019 November Q6
10 marks Moderate -0.3
6 A random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} k \left( 3 x - x ^ { 2 } \right) & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$
  1. Show that \(k = \frac { 2 } { 9 }\).
  2. Find \(\mathrm { P } ( 1 \leqslant X \leqslant 2 )\).
  3. Find \(\operatorname { Var } ( X )\).
CAIE S2 2019 November Q7
10 marks Standard +0.8
7 Bob is a self-employed builder. In the past his weekly income had mean \(\\) 546\( and standard deviation \)\\( 120\). Following a change in Bob's working pattern, his mean weekly income for 40 randomly chosen weeks was \(\\) 581\(. You should assume that the standard deviation remains unchanged at \)\\( 120\).
  1. Test at the \(2.5 \%\) significance level whether Bob's mean weekly income has increased.
    Bob finds his mean weekly income for another random sample of 40 weeks and carries out a similar test at the \(2.5 \%\) significance level.
  2. Given that Bob's mean weekly income is now in fact \(\\) 595$, find the probability of a Type II error.
    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 S2 2020 November Q3
6 marks Standard +0.8
3 The masses, in kilograms, of female and male animals of a certain species have the distributions \(\mathrm { N } \left( 102,27 ^ { 2 } \right)\) and \(\mathrm { N } \left( 170,55 ^ { 2 } \right)\) respectively. Find the probability that a randomly chosen female has a mass that is less than half the mass of a randomly chosen male. \includegraphics[max width=\textwidth, alt={}, center]{fb305858-2d96-4a5d-b1a9-a965c248fb8d-06_76_1659_484_244}
CAIE S2 2020 November Q4
5 marks Moderate -0.5
4 \includegraphics[max width=\textwidth, alt={}, center]{fb305858-2d96-4a5d-b1a9-a965c248fb8d-07_316_984_260_577} The diagram shows the probability density function, \(\mathrm { f } ( x )\), of a random variable \(X\). For \(0 \leqslant x \leqslant a\), \(\mathrm { f } ( x ) = k\); elsewhere \(\mathrm { f } ( x ) = 0\).
  1. Express \(k\) in terms of \(a\).
  2. Given that \(\operatorname { Var } ( X ) = 3\), find \(a\).
CAIE S2 2020 November Q3
7 marks Moderate -0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{4a5f9f7e-b045-4c6f-8bda-6c4067668da2-04_332_1100_260_520} A random variable \(X\) takes values between 0 and 3 only and has probability density function as shown in the diagram, where \(c\) is a constant.
  1. Show that \(c = \frac { 2 } { 3 }\).
  2. Find \(\mathrm { P } ( X > 2 )\).
  3. Calculate \(\mathrm { E } ( X )\).
CAIE S2 2020 November Q3
6 marks Standard +0.8
3 The masses, in kilograms, of female and male animals of a certain species have the distributions \(\mathrm { N } \left( 102,27 ^ { 2 } \right)\) and \(\mathrm { N } \left( 170,55 ^ { 2 } \right)\) respectively. Find the probability that a randomly chosen female has a mass that is less than half the mass of a randomly chosen male. \includegraphics[max width=\textwidth, alt={}, center]{937c15d2-fb12-4af8-96d3-c54c81d771ba-06_76_1659_484_244}
CAIE S2 2020 November Q4
5 marks Moderate -0.5
4 \includegraphics[max width=\textwidth, alt={}, center]{937c15d2-fb12-4af8-96d3-c54c81d771ba-07_316_984_260_577} The diagram shows the probability density function, \(\mathrm { f } ( x )\), of a random variable \(X\). For \(0 \leqslant x \leqslant a\), \(\mathrm { f } ( x ) = k\); elsewhere \(\mathrm { f } ( x ) = 0\).
  1. Express \(k\) in terms of \(a\).
  2. Given that \(\operatorname { Var } ( X ) = 3\), find \(a\).
CAIE S2 Specimen Q1
4 marks Standard +0.3
1 Failures of two computers occur at random and independently. On average the first computer fails 1.2 times per year and the second computer fails 2.3 times per year. Find the probability that the total number of failures by the two computers in a 6-month period is more than 1 and less than 4 .
CAIE S2 Specimen Q2
6 marks Moderate -0.5
2 The mean and standard deviation of the time spent by people in a certain library are 29 minutes and 6 minutes respectively.
  1. Find the probability that the mean time spent in the library by a random sample of 120 people is more than 30 minutes.
  2. Explain whether it was necessary to assume that the time spent by people in the library is normally distributed in the solution to part (i).
CAIE S2 Specimen Q3
6 marks Moderate -0.3
3 Jagdeesh measured the lengths, \(x\) minutes, of 60 randomly chosen lectures. His results are summarised below.
  1. Calculate unbiased estimates of the population mean and variance.
  2. Calculate a \(98 \%\) confidence interval for the population mean.
CAIE S2 Specimen Q4
7 marks Moderate -0.8
4 A random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} k ( 3 - x ) & 1 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 2 } { 3 }\).
  2. Find the median of \(X\).
CAIE S2 Specimen Q5
7 marks Standard +0.3
5 On average, 1 in 2500 adults has a certain medical condition.
  1. Use a suitable approximation to find the probability that, in a random sample of 4000 people, more than 3 have this condition.
  2. In a random sample of \(n\) people, where \(n\) is large, the probability that none has the condition is less than 0.05 . Find the smallest possible value of \(n\).
CAIE S2 Specimen Q6
9 marks Standard +0.3
6 The weights, in kilograms, of men and women have the distributions \(\mathrm { N } \left( 78,7 ^ { 2 } \right)\) and \(\mathrm { N } \left( 66,5 ^ { 2 } \right)\) respectively.
  1. The maximum load that a certain cable car can carry safely is 1200 kg . If 9 randomly chosen men and 7 randomly chosen women enter the cable car, find the probability that the cable car can operate safely.
  2. Find the probability that a randomly chosen woman weighs more than a randomly chosen man. [4]
CAIE S2 Specimen Q7
11 marks Moderate -0.8
7 At a certain hospital it was found that the probability that a patient did not arrive for an appointment was 0.2 . The hospital carries out some publicity in the hope that this probability will be reduced. They wish to test whether the publicity has worked.
  1. It is suggested that the first 30 appointments on a Monday should be used for the test. Give a reason why this is not an appropriate sample.
    A suitable sample of 30 appointments is selected and the number of patients that do not arrive is noted. This figure is used to carry out a test at the 5\% significance level.
  2. Explain why the test is one-tail and state suitable null and alternative hypotheses.
  3. State what is meant by a Type I error in this context.
  4. Use the binomial distribution to find the critical region, and find the probability of a Type I error.
  5. In fact 3 patients out of the 30 do not arrive. State the conclusion of the test, explaining your answer.
CAIE Further Paper 4 2020 June Q1
6 marks Standard +0.3
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
5 marks Standard +0.8
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
9 marks Standard +0.3
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
8 marks Standard +0.8
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
11 marks Standard +0.8
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
11 marks Moderate -0.3
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
7 marks Standard +0.3
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.