Questions — CAIE S2 (737 questions)

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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]