Questions — CAIE (7646 questions)

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CAIE S2 2002 November Q4
7 marks Standard +0.3
The number of accidents per month at a certain road junction has a Poisson distribution with mean 4.8. A new road sign is introduced warning drivers of the danger ahead, and in a subsequent month 2 accidents occurred.
  1. A hypothesis test at the 10% level is used to determine whether there were fewer accidents after the new road sign was introduced. Find the critical region for this test and carry out the test. [5]
  2. Find the probability of a Type I error. [2]
CAIE S2 2002 November Q5
8 marks Standard +0.3
\(X\) and \(Y\) are independent random variables each having a Poisson distribution. \(X\) has mean 2.5 and \(Y\) has mean 3.1.
  1. Find P\((X + Y > 3)\). [4]
  2. A random sample of 80 values of \(X\) is taken. Find the probability that the sample mean is less than 2.4. [4]
CAIE S2 2002 November Q6
10 marks Moderate -0.3
The average speed of a bus, \(x\) km h\(^{-1}\), on a certain journey is a continuous random variable \(X\) with probability density function given by $$\text{f}(x) = \begin{cases} \frac{k}{x^2} & 20 \leq x \leq 28, \\ 0 & \text{otherwise}. \end{cases}$$
  1. Show that \(k = 70\). [3]
  2. Find E\((X)\). [3]
  3. Find P\((X < \text{E}(X))\). [2]
  4. Hence determine whether the mean is greater or less than the median. [2]
CAIE S2 2002 November Q7
10 marks Standard +0.3
Bottles of wine are stacked in racks of 12. The weights of these bottles are normally distributed with mean 1.3 kg and standard deviation 0.06 kg. The weights of the empty racks are normally distributed with mean 2 kg and standard deviation 0.3 kg.
  1. Find the probability that the total weight of a full rack of 12 bottles of wine is between 17 kg and 18 kg. [5]
  2. Two bottles of wine are chosen at random. Find the probability that they differ in weight by more than 0.05 kg. [5]
CAIE S2 2011 November Q1
4 marks Moderate -0.8
Test scores, \(X\), have mean 54 and variance 144. The scores are scaled using the formula \(Y = a + bX\), where \(a\) and \(b\) are constants and \(b > 0\). The scaled scores, \(Y\), have mean 50 and variance 100. Find the values of \(a\) and \(b\). [4]
CAIE S2 2011 November Q2
5 marks Standard +0.3
35% of a random sample of \(n\) students walk to college. This result is used to construct an approximate 98% confidence interval for the population proportion of students who walk to college. Given that the width of this confidence interval is 0.157, correct to 3 significant figures, find \(n\). [5]
CAIE S2 2011 November Q3
7 marks Easy -1.2
Jack has to choose a random sample of 8 people from the 750 members of a sports club.
  1. Explain fully how he can use random numbers to choose the sample. [3]
Jack asks each person in the sample how much they spent last week in the club café. The results, in dollars, were as follows. 15 \quad 25 \quad 30 \quad 8 \quad 12 \quad 18 \quad 27 \quad 25
  1. Find unbiased estimates of the population mean and variance. [3]
  2. Explain briefly what is meant by 'population' in this question. [1]
CAIE S2 2011 November Q4
7 marks Moderate -0.3
The random variable \(X\) has probability density function given by $$f(x) = \begin{cases} ke^{-x} & 0 \leqslant x \leqslant 1, \\ 0 & \text{otherwise}. \end{cases}$$
  1. Show that \(k = \frac{e}{e-1}\). [3]
  2. Find E(\(X\)) in terms of \(e\). [4]
CAIE S2 2011 November Q5
8 marks Standard +0.3
Records show that the distance driven by a bus driver in a week is normally distributed with mean 1150 km and standard deviation 105 km. New driving regulations are introduced and in the next 20 weeks he drives a total of 21 800 km.
  1. Stating any assumption(s), test, at the 1% significance level, whether his mean weekly driving distance has decreased. [6]
  2. A similar test at the 1% significance level was carried out using the data from another 20 weeks. State the probability of a Type I error and describe what is meant by a Type I error in this context. [2]
CAIE S2 2011 November Q6
8 marks Standard +0.3
Ranjit goes to mathematics lectures and physics lectures. The length, in minutes, of a mathematics lecture is modelled by the variable \(X\) with distribution N(36, 3.5²). The length, in minutes, of a physics lecture is modelled by the independent variable \(Y\) with distribution N(55, 5.2²).
  1. Find the probability that the total length of two mathematics lectures and one physics lecture is less than 140 minutes. [4]
  2. Ranjit calculates how long he will need to spend revising the content of each lecture as follows. Each minute of a mathematics lecture requires 1 minute of revision and each minute of a physics lecture requires 1½ minutes of revision. Find the probability that the total revision time required for one mathematics lecture and one physics lecture is more than 100 minutes. [4]
CAIE S2 2011 November Q7
11 marks Standard +0.8
The numbers of men and women who visit a clinic each hour are independent Poisson variables with means 2.4 and 2.8 respectively.
  1. Find the probability that, in a half-hour period,
    1. 2 or more men and 1 or more women will visit the clinic, [4]
    2. a total of 3 or more people will visit the clinic. [3]
  2. Find the probability that, in a 10-hour period, a total of more than 60 people will visit the clinic. [4]
CAIE S2 2020 Specimen Q1
4 marks Moderate -0.8
Leaves from a certain type of tree have lengths that are distributed with standard deviation 3 cm. A random sample of 6 of these leaves is taken and the mean length of this sample is found to be 8 cm.
  1. Calculate a 95\% confidence interval for the population mean length. [3]
  2. Write down the probability that the whole 95\% confidence interval will lie below the population mean. [1]
CAIE S2 2020 Specimen Q2
3 marks Easy -1.8
Describe briefly how to use a random number generator to obtain a sample of 10 students from a group of 50 students. [3]
CAIE S2 2020 Specimen Q3
5 marks Moderate -0.3
The number of calls received at a small call centre has a Poisson distribution with mean 2 calls per 5 minute period.
  1. Find the probability exactly 4 calls in a 10 minute period. [2]
  2. Find the probability at least 3 calls in a 3 minute period. [3]
CAIE S2 2020 Specimen Q3
4 marks Standard +0.3
The number of calls received at a large call centre has a Poisson distribution with mean 4 calls per 5 minute period.
  1. [(c)] Use an approximation to find the probability that the number of calls received in a 5 minute period is between 4 and 9 inclusive. [4]
CAIE S2 2020 Specimen Q4
10 marks Standard +0.3
The lifetimes, in hours, of light bulbs have an exponential distribution with parameter \(\frac{1}{500}\). Each bulb is tested and rejected if the lifetime is less than 500 hours.
  1. Find the probability that a bulb of this type has a lifetime of more than 500 hours. [4]
  2. Find the probability that the lifetime is at least three times the expected lifetime. [6]
CAIE S2 2020 Specimen Q5
7 marks Standard +0.3
The diagram shows the graph of the probability density function of a random variable \(X\), where $$f(x) = \begin{cases} \frac{1}{6}(3x - x^2) & 0 \leq x \leq 3, \\ 0 & \text{otherwise}. \end{cases}$$ \includegraphics{figure_1}
  1. State the values of E(\(X\)) and Var(\(X\)). [4]
  2. State the values of P(\(0.5 < X < 1\)). [1]
  3. Given that P(\(1 < X < 2\)) = \(\frac{13}{27}\), find P(\(X > 2\)). [2]
CAIE S2 2020 Specimen Q6
9 marks Standard +0.3
At a certain hospital it was found that the probability that a patient did not arrive for an appointment was 0.1. The hospital's business model assumed that this probability will be reduced. They wish to test whether this probability is now less than 0.1. A random sample of 50 appointments is selected and the number of patients that did not arrive is noted. This figure is used as a test statistic at the 5\% significance level.
  1. Explain why this test is a one-tailed test and state suitable null and alternative hypotheses. [2]
  2. Use a binomial distribution to find the critical region and find the probability of a Type I error. [5]
  3. In fact 3 patients out of the 50 did not arrive. State the conclusion of the test, explaining your answer. [2]
CAIE S2 2020 Specimen Q7
7 marks Standard +0.3
The mean weight of bags of carrots is \(\mu\) kilograms. An inspector wishes to test whether \(\mu = 20\). He weighs a random sample of 6 bags and the results are summarised as follows: $$\Sigma x = 430 \quad \Sigma x^2 = 40$$ Carry out the test at the 5\% significance level. [7]
CAIE Further Paper 4 2021 June Q1
6 marks Standard +0.3
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, \(x\) kg, of apples produced in a year are summarised as follows. $$\sum x = 152.0 \qquad \sum x^2 = 2313.0$$ Test, at the 5% significance level, whether Farmer B's claim is justified. [6]
CAIE Further Paper 4 2021 June Q2
7 marks Standard +0.3
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.
ChocolateABCDEFGHI
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. [7]
CAIE Further Paper 4 2021 June Q3
8 marks Standard +0.8
The heights, \(x\) m, of a random sample of 50 adult males from country A were recorded. The heights, \(y\) m, of a random sample of 40 adult males from country B were also recorded. The results are summarised as follows. $$\sum x = 89.0 \qquad \sum x^2 = 159.4 \qquad \sum y = 67.2 \qquad \sum y^2 = 113.1$$ Find a 95% confidence interval for the difference between the mean heights of adult males from country A and adult males from country B. [8]
CAIE Further Paper 4 2021 June Q4
5 marks Standard +0.8
\(X\) is a discrete random variable which takes the values 0, 2, 4, ... . The probability generating function of \(X\) is given by $$G_X(t) = \frac{1}{3 - 2t^2}.$$
  1. Find E\((X)\) and Var\((X)\). [5]
CAIE Further Paper 4 2021 June Q4
3 marks Standard +0.8
  1. Find P\((X = 4)\). [3]
CAIE Further Paper 4 2021 June Q5
10 marks Standard +0.3
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. [2]
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.
  1. Use the data from this larger sample to carry out a goodness of fit test, at the 10% significance level, to test the claim. [8]