Questions S2 (1690 questions)

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CAIE S2 2018 June Q7
12 marks Standard +0.3
7 A ten-sided spinner has edges numbered \(1,2,3,4,5,6,7,8,9,10\). Sanjeev claims that the spinner is biased so that it lands on the 10 more often than it would if it were unbiased. In an experiment, the spinner landed on the 10 in 3 out of 9 spins.
  1. Test at the \(1 \%\) significance level whether Sanjeev's claim is justified.
  2. Explain why a Type I error cannot have been made.
    In fact the spinner is biased so that the probability that it will land on the 10 on any spin is 0.5 .
  3. Another test at the \(1 \%\) significance level, also based on 9 spins, is carried out. Calculate 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 2019 June Q1
5 marks Easy -1.2
1 At an internet café, the charge for using a computer is 5 cents per minute. The number of minutes for which people use a computer has mean 23 and standard deviation 8.
  1. Find, in cents, the mean and standard deviation of the amount people pay when using a computer.
  2. Each day, 15 people use computers independently. Find, in cents, the mean and standard deviation of the total amount paid by 15 people.
CAIE S2 2019 June Q2
7 marks Moderate -0.3
2 The time, in minutes, that John takes to travel to work has a normal distribution. Last year the mean and standard deviation were 26.5 and 4.8 respectively. This year John uses a different route and he finds that the mean time for his first 150 journeys is 27.5 minutes.
  1. Stating a necessary assumption, test at the \(1 \%\) significance level whether the mean time for his journey to work has increased.
  2. State, with a reason, whether it was necessary to use the Central Limit theorem in your answer to part (i).
CAIE S2 2019 June Q3
7 marks Moderate -0.3
3 Sumitra has a six-sided die. She suspects that it is biased so that it shows a six less often than it would if it were fair. She decides to test the die by throwing it 30 times and noting the number of throws on which it shows a six.
  1. It shows a six on exactly 2 throws. Use a binomial distribution to carry out the test at the \(5 \%\) significance level.
  2. Later, Sumitra repeats the test at the \(5 \%\) significance level by throwing the die 30 times again. Find the probability of a Type I error in this second test.
CAIE S2 2019 June Q4
9 marks Standard +0.3
4

  1. [diagram]
    The diagram shows the graph of the probability density function, f , of a random variable \(X\), where \(a\) is a constant greater than 0.5 . The graph between \(x = 0\) and \(x = a\) is a straight line parallel to the \(x\)-axis.
    1. Find \(\mathrm { P } ( X < 0.5 )\) in terms of \(a\).
    2. Find \(\mathrm { E } ( X )\) in terms of \(a\).
    3. Show that \(\operatorname { Var } ( X ) = \frac { 1 } { 12 } a ^ { 2 }\).
  2. A random variable \(T\) has probability density function given by $$\operatorname { g } ( t ) = \begin{cases} \frac { 3 } { 2 ( t - 1 ) ^ { 2 } } & 2 \leqslant t \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$ Find the value of \(b\) such that \(\mathrm { P } ( T \leqslant b ) = \frac { 3 } { 4 }\).
CAIE S2 2019 June Q5
12 marks Standard +0.3
5
  1. The random variable \(X\) has the distribution \(\operatorname { Po } ( 2.3 )\).
    1. Find \(\mathrm { P } ( 2 \leqslant X \leqslant 4 )\).
    2. Find the probability that the sum of two independent values of \(X\) is greater than 2 .
    3. The random variable \(S\) is the sum of 50 independent values of \(X\). Use a suitable approximating distribution to find \(\mathrm { P } ( S \leqslant 110 )\).
  2. The random variable \(Y\) has the distribution \(\mathrm { Po } ( \lambda )\). Given that \(\mathrm { P } ( Y = 3 ) = \mathrm { P } ( Y = 5 )\), find \(\lambda\).
CAIE S2 2019 June Q6
10 marks Moderate -0.8
6 Ramesh plans to carry out a survey in order to find out what adults in his town think about local sports facilities. He chooses a random sample from the adult members of a tennis club and gives each of them a questionnaire.
  1. Give a reason why this will not result in Ramesh having a random sample of adults who live in the town.
  2. Describe briefly a valid method that Ramesh could use to choose a random sample of adults in the town.
    Ramesh now uses a valid method to choose a random sample of 350 adults from the town. He finds that 47 adults think that the local sports facilities are good.
  3. Calculate an approximate \(90 \%\) confidence interval for the proportion of all adults in the town who think that the local sports facilities are good.
  4. Ramesh calculates a confidence interval whose width is 1.25 times the width of this \(90 \%\) confidence interval. Ramesh's new interval is an \(x \%\) confidence interval. Find the value of \(x\).
    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 2019 June Q1
3 marks Moderate -0.3
1 The random variable \(X\) has the distribution \(\operatorname { Po } ( 5 )\).
  1. Find \(\mathrm { P } ( X = 2 )\).
    It is given that \(\mathrm { P } ( X = n ) = \mathrm { P } ( X = n + 1 )\).
  2. Write down an equation in \(n\).
  3. Hence or otherwise find the value of \(n\).
CAIE S2 2019 June Q2
4 marks Easy -1.2
2 The random variable \(X\) has mean 372 and standard deviation 54 .
  1. Describe fully the distribution of the mean of a random sample of 36 values of \(X\).
  2. The distribution in part (i) might be either exact or approximate. State a condition under which the distribution is exact.
CAIE S2 2019 June Q3
8 marks Standard +0.3
3 It is claimed that, on average, a particular train journey takes less than 1.9 hours. The times, \(t\) hours, taken for this journey on a random sample of 50 days were recorded. The results are summarised below. $$n = 50 \quad \Sigma t = 92.5 \quad \Sigma t ^ { 2 } = 175.25$$
  1. Calculate unbiased estimates of the population mean and variance.
  2. Test the claim at the \(5 \%\) significance level.
CAIE S2 2019 June Q4
7 marks Challenging +1.8
4 The heights of a certain variety of plant are normally distributed with mean 110 cm and variance \(1050 \mathrm {~cm} ^ { 2 }\). Two plants of this variety are chosen at random. Find the probability that the height of one of these plants is at least 1.5 times the height of the other.
CAIE S2 2019 June Q5
8 marks Standard +0.3
5 The manufacturer of a certain type of biscuit claims that \(10 \%\) of packets include a free offer printed on the packet. Jyothi suspects that the true proportion is less than \(10 \%\). He plans to test the claim by looking at 40 randomly selected packets and, if the number which include the offer is less than 2 , he will reject the manufacturer's claim.
  1. State suitable hypotheses for the test.
  2. Find the probability of a Type I error.
    On another occasion Jyothi looks at 80 randomly selected packets and finds that exactly 6 include the free offer.
  3. Calculate an approximate \(90 \%\) confidence interval for the proportion of packets that include the offer.
  4. Use your confidence interval to comment on the manufacturer's claim. \(6 X\) is a random variable with probability density function given by $$f ( x ) = \begin{cases} \frac { a } { x ^ { 2 } } & 1 \leqslant x \leqslant b \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are constants.
CAIE S2 2019 June Q7
11 marks Standard +0.3
7 All the seats on a certain daily flight are always sold. The number of passengers who have bought seats but fail to arrive for this flight on a particular day is modelled by the distribution \(\mathbf { B } ( 320,0.005 )\).
  1. Explain what the number 320 represents in this context.
  2. The total number of passengers who have bought seats but fail to arrive for this flight on 2 randomly chosen days is denoted by \(X\). Use a suitable approximating distribution to find \(\mathrm { P } ( 2 < X < 6 )\).
  3. Justify the use of your approximating distribution.
    After some changes, the airline wishes to test whether the mean number of passengers per day who fail to arrive for this flight has decreased.
  4. During 5 randomly chosen days, a total of 2 passengers failed to arrive. Carry out the test at the 2.5\% significance level.
    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 2019 June Q1
3 marks Moderate -0.5
1 A coin is thrown 100 times and it shows heads 60 times. Calculate an approximate \(98 \%\) confidence interval for the probability, \(p\), that the coin shows heads on any throw.
CAIE S2 2019 June Q2
3 marks Moderate -0.8
2 The length of worms is denoted by \(X \mathrm {~cm}\). The lengths of a random sample of 50 worms were measured. Some of the results were lost, but the following results are available.
  • \(\Sigma x ^ { 2 } = 4361\)
  • An unbiased estimate of the population variance of \(X\) is 9.62.
Calculate the mean length of the 50 worms.
CAIE S2 2019 June Q3
4 marks Easy -1.2
3 Luis has to choose one person at random from four people, \(A , B , C\) and \(D\). He throws a fair six-sided die. If the score is 1 , he will choose \(A\). If the score is 2 he will choose \(B\). If the score is 3 , he will choose \(C\). If the score is 4 or more he will choose \(D\).
  1. Explain why the choice made by this method is not random.
  2. Describe how Luis could use a single throw of the die to make a random choice.
    On another day, Luis has to choose two people at random from the same four people, \(A , B , C\) and \(D\).
  3. List the possible choices of two people and hence describe how Luis could use a single throw of the die to make this random choice.
CAIE S2 2019 June Q4
5 marks Standard +0.3
4 A factory supplies boxes of children's bricks. Each box contains 10 randomly chosen large bricks and 20 randomly chosen small bricks. The masses, in grams, of large and small bricks have the distributions \(\mathrm { N } ( 60,1.2 )\) and \(\mathrm { N } ( 30,0.7 )\) respectively. The mass of an empty box is 8 g . Find the probability that the total weight of a box and its contents is less than 1200 g .
CAIE S2 2019 June Q5
6 marks Challenging +1.2
5 The amount of money, in dollars, spent by a customer on one visit to a certain shop is modelled by the distribution \(\mathrm { N } ( \mu , 1.94 )\). In the past, the value of \(\mu\) has been found to be 20.00 , but following a rearrangement in the shop, the manager suspects that the value of \(\mu\) has changed. He takes a random sample of 6 customers and notes how much they each spend, in dollars. The results are as follows.
15.50
17.60
17.30
22.00
23.50
31.00 The manager carries out a hypothesis test using a significance level of \(\alpha \%\). The test does not support his suspicion. Find the largest possible value of \(\alpha\).
CAIE S2 2019 June Q6
9 marks Standard +0.3
6 A function f is defined by $$f ( x ) = \begin{cases} \frac { 3 x ^ { 2 } } { a ^ { 3 } } & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.
  1. Show that f is a probability density function for all positive values of \(a\).
    The random variable \(X\) has probability density function f and the median of \(X\) is 2 .
  2. Show that \(a = 2.52\), correct to 3 significant figures.
  3. Find \(\mathrm { E } ( X )\).
CAIE S2 2019 June Q7
10 marks Standard +0.3
7 Each day at a certain doctor's surgery there are 70 appointments available in the morning and 60 in the afternoon. All the appointments are filled every day. The probability that any patient misses a particular morning appointment is 0.04 , and the probability that any patient misses a particular afternoon appointment is 0.05 . All missed appointments are independent of each other. Use suitable approximating distributions to answer the following.
  1. Find the probability that on a randomly chosen morning there are at least 3 missed appointments.
  2. Find the probability that on a randomly chosen day there are a total of exactly 6 missed appointments.
  3. Find the probability that in a randomly chosen 10-day period there are more than 50 missed appointments.
CAIE S2 2019 June Q8
10 marks Standard +0.3
8 The four sides of a spinner are \(A , B , C , D\). The spinner is supposed to be fair, but Sonam suspects that the spinner is biased so that the probability, \(p\), that it will land on side \(A\) is greater than \(\frac { 1 } { 4 }\). He spins the spinner 10 times and finds that it lands on side \(A 6\) times.
  1. Test Sonam's suspicion using a \(1 \%\) significance level.
    Later Sonam carries out a similar test at the \(1 \%\) significance level, using another 10 spins of the spinner.
  2. Calculate the probability of a Type I error.
  3. Assuming that the value of \(p\) is actually \(\frac { 3 } { 5 }\), calculate 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 June Q1
6 marks Moderate -0.5
1 The lengths, \(X\) centimetres, of a random sample of 7 leaves from a certain variety of tree are as follows.
3.9
4.8
4.8
4.4
CAIE S2 2021 June Q6
6 marks Standard +0.3
6 The probability density function, f, of a random variable \(X\) is given by $$f ( x ) = \begin{cases} k \left( 6 x - x ^ { 2 } \right) & 0 \leqslant x \leqslant 6 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
State the value of \(\mathrm { E } ( X )\) and show that \(\operatorname { Var } ( X ) = \frac { 9 } { 5 }\).
CAIE S2 2021 June Q3
7 marks Moderate -0.8
3 \includegraphics[max width=\textwidth, alt={}, center]{189bcf7b-279f-457b-8232-ace7f0c9797f-05_456_668_260_735} The random variable \(X\) takes values in the range \(1 \leqslant x \leqslant p\), where \(p\) is a constant. The graph of the probability density function of \(X\) is shown in the diagram.
  1. Show that \(p = 2\).
  2. Find \(\mathrm { E } ( X )\).
CAIE S2 2016 March Q1
5 marks Moderate -0.8
1 A fair six-sided die is thrown 20 times and the number of sixes, \(X\), is recorded. Another fair six-sided die is thrown 20 times and the number of odd-numbered scores, \(Y\), is recorded. Find the mean and standard deviation of \(X + Y\).