Questions — CAIE S2 (737 questions)

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CAIE S2 2021 June Q4
6 marks Moderate -0.3
100 randomly chosen adults each throw a ball once. The length, \(l\) metres, of each throw is recorded. The results are summarised below. $$n = 100 \qquad \sum l = 3820 \qquad \sum l^2 = 182200$$ Calculate a 94% confidence interval for the population mean length of throws by adults. [6]
CAIE S2 2021 June Q5
7 marks Standard +0.3
On average, 1 in 75000 adults has a certain genetic disorder.
  1. Use a suitable approximating distribution to find the probability that, in a random sample of 10000 people, at least 1 has the genetic disorder. [3]
  2. In a random sample of \(n\) people, where \(n\) is large, the probability that no-one has the genetic disorder is more than 0.9. Find the largest possible value of \(n\). [4]
CAIE S2 2021 June Q6
6 marks Standard +0.3
The probability density function, f, of a random variable \(X\) is given by $$\text{f}(x) = \begin{cases} k(6x - x^2) & 0 \leq x \leq 6, \\ 0 & \text{otherwise,} \end{cases}$$ where \(k\) is a constant. State the value of \(\text{E}(X)\) and show that \(\text{Var}(X) = \frac{9}{5}\). [6]
CAIE S2 2021 June Q7
10 marks Standard +0.3
The masses, in kilograms, of large and small sacks of flour have the distributions \(\text{N}(55, 3^2)\) and \(\text{N}(27, 2.5^2)\) respectively.
  1. Some sacks are loaded onto a boat. The maximum load of flour that the boat can carry safely is 340 kg. Find the probability that the boat can carry safely 3 randomly chosen large sacks of flour and 6 randomly chosen small sacks of flour. [5]
  2. Find the probability that the mass of a randomly chosen large sack of flour is greater than the total mass of two randomly chosen small sacks of flour. [5]
CAIE S2 2021 June Q8
11 marks Standard +0.3
At a certain large school it was found that the proportion of students not wearing correct uniform was 0.15. The school sent a letter to parents asking them to ensure that their children wear the correct uniform. The school now wishes to test whether the proportion not wearing correct uniform has been reduced.
  1. It is suggested that a random sample of the students in Grade 12 should be used for the test. Give a reason why this would not be an appropriate sample. [1]
  2. State suitable null and alternative hypotheses. [1]
  3. Use a binomial distribution to find the probability of a Type I error. You must justify your answer fully. [5]
  4. In fact 4 students out of the 50 are not wearing correct uniform. State the conclusion of the test, explaining your answer. [2]
  5. State, with a reason, which of the errors, Type I or Type II, may have been made. [2]
A suitable sample of 50 students is selected and the number not wearing correct uniform is noted. This figure is used to carry out a test at the 5% significance level.
CAIE S2 2022 November Q1
7 marks Standard +0.3
Each of a random sample of 80 adults gave an estimate, \(h\) metres, of the height of a particular building. The results were summarised as follows. $$n = 80 \quad \sum h = 2048 \quad \sum h^2 = 52760$$
  1. Calculate unbiased estimates of the population mean and variance. [3]
  2. Using this sample, the upper boundary of an \(\alpha\%\) confidence interval for the population mean is 26.0. Find the value of \(\alpha\). [4]
CAIE S2 2022 November Q2
5 marks Moderate -0.3
In the past, the mean length of a particular variety of worm has been 10.3 cm, with standard deviation 2.6 cm. Following a change in the climate, it is thought that the mean length of this variety of worm has decreased. The lengths of a random sample of 100 worms of this variety are found and the mean of this sample is found to be 9.8 cm. Assuming that the standard deviation remains at 2.6 cm, carry out a test at the 2% significance level of whether the mean length has decreased. [5]
CAIE S2 2022 November Q3
6 marks Moderate -0.8
1.6% of adults in a certain town ride a bicycle. A random sample of 200 adults from this town is selected.
  1. Use a suitable approximating distribution to find the probability that more than 3 of these adults ride a bicycle. [4]
  2. Justify your approximating distribution. [2]
CAIE S2 2022 November Q4
8 marks Standard +0.3
The number of faults in cloth made on a certain machine has a Poisson distribution with mean 2.4 per 10 m\(^2\). An adjustment is made to the machine. It is required to test at the 5% significance level whether the mean number of faults has decreased. A randomly selected 30 m\(^2\) of cloth is checked and the number of faults is found.
  1. State suitable null and alternative hypotheses for the test. [1]
  2. Find the probability of a Type I error. [3]
Exactly 3 faults are found in the randomly selected 30 m\(^2\) of cloth.
  1. Carry out the test at the 5% significance level. [2]
Later a similar test was carried out at the 5% significance level, using another randomly selected 30 m\(^2\) of cloth.
  1. Given that the number of faults actually has a Poisson distribution with mean 0.5 per 10 m\(^2\), find the probability of a Type II error. [2]
CAIE S2 2022 November Q5
6 marks Moderate -0.8
\(X\) is a random variable with distribution B(10, 0.2). A random sample of 160 values of \(X\) is taken.
  1. Find the approximate distribution of the sample mean, including the values of the parameters. [3]
  2. Hence find the probability that the sample mean is less than 1.8. [3]
CAIE S2 2022 November Q6
10 marks Standard +0.3
The masses, in grams, of small and large bags of flour have the distributions N(510, 100) and N(1015, 324) respectively. André selects 4 small bags of flour and 2 large bags of flour at random.
  1. Find the probability that the total mass of these 6 bags of flour is less than 4130 g. [5]
  2. Find the probability that the total mass of the 4 small bags is more than the total mass of the 2 large bags. [5]
CAIE S2 2022 November Q7
8 marks Standard +0.3
\includegraphics{figure_7} The diagram shows the graph of the probability density function, f, of a random variable \(X\) which takes values between \(-3\) and 2 only.
  1. Given that the graph is symmetrical about the line \(x = -0.5\) and that P(\(X < 0\)) = \(p\), find P(\(-1 < X < 0\)) in terms of \(p\). [2]
  2. It is now given that the probability density function shown in the diagram is given by $$\text{f}(x) = \begin{cases} a - b(x^2 + x) & -3 \leq x \leq 2, \\ 0 & \text{otherwise,} \end{cases}$$ where \(a\) and \(b\) are positive constants.
    1. Show that \(30a - 55b = 6\). [3]
    2. By substituting a suitable value of \(x\) into f(\(x\)), find another equation relating \(a\) and \(b\) and hence determine the values of \(a\) and \(b\). [3]
CAIE S2 2023 November Q1
3 marks Moderate -0.5
A random variable \(X\) has the distribution N(410, 400). Find the probability that the mean of a random sample of 36 values of \(X\) is less than 405. [3]
CAIE S2 2023 November Q2
4 marks Standard +0.8
In a survey of 300 randomly chosen adults in Rickton, 134 said that they exercised regularly. This information was used to calculate an \(\alpha\)% confidence interval for the proportion of adults in Rickton who exercise regularly. The upper bound of the confidence interval was found to be 0.487, correct to 3 significant figures. Find the value of \(\alpha\) correct to the nearest integer. [4]
CAIE S2 2023 November Q3
10 marks Standard +0.3
A website owner finds that, on average, his website receives 0.3 hits per minute. He believes that the number of hits per minute follows a Poisson distribution.
  1. Assume that the owner is correct.
    1. Find the probability that there will be at least 4 hits during a 10-minute period. [3]
    2. Use a suitable approximating distribution to find the probability that there will be fewer than 40 hits during a 3-hour period. [4]
A friend agrees that the website receives, on average, 0.3 hits per minute. However, she notices that the number of hits during the day-time (9.00am to 9.00pm) is usually about twice the number of hits during the night-time (9.00pm to 9.00am).
    1. Explain why this fact contradicts the owner's belief that the number of hits per minute follows a Poisson distribution. [1]
    2. Specify separate Poisson distributions that might be suitable models for the number of hits during the day-time and during the night-time. [2]
CAIE S2 2023 November Q4
8 marks Standard +0.3
The masses, in kilograms, of chemicals \(A\) and \(B\) produced per day by a factory are modelled by the independent random variables \(X\) and \(Y\) respectively, where \(X \sim\) N(10.3, 5.76) and \(Y \sim\) N(11.4, 9.61). The income generated by the chemicals is \\(2.50 per kilogram for \)A\( and \\)3.25 per kilogram for \(B\).
  1. Find the mean and variance of the daily income generated by chemical \(A\). [2]
  2. Find the probability that, on a randomly chosen day, the income generated by chemical \(A\) is greater than the income generated by chemical \(B\). [6]
CAIE S2 2023 November Q5
5 marks Standard +0.3
In the past the number of enquiries per minute at a customer service desk has been modelled by a random variable with distribution Po(0.31). Following a change in the position of the desk, it is expected that the mean number of enquiries per minute will increase. In order to test whether this is the case, the total number of enquiries during a randomly chosen 5-minute period is noted. You should assume that a Poisson model is still appropriate. Given that the total number of enquiries is 5, carry out the test at the 2.5% significance level. [5]
CAIE S2 2023 November Q6
8 marks Standard +0.8
A continuous random variable \(X\) takes values from 0 to 6 only and has a probability distribution that is symmetrical. Two values, \(a\) and \(b\), of \(X\) are such that P\((a < X < b) = p\) and P\((b < X < 3) = \frac{13}{10}p\), where \(p\) is a positive constant.
  1. Show that \(p \leq \frac{5}{23}\). [1]
  2. Find P\((b < X < 6 - a)\) in terms of \(p\). [2]
It is now given that the probability density function of \(X\) is \(f\), where $$f(x) = \begin{cases} \frac{1}{36}(6x - x^2) & 0 \leq x \leq 6, \\ 0 & \text{otherwise}. \end{cases}$$
  1. Given that \(b = 2\) and \(p = \frac{5}{81}\), find the value of \(a\). [5]
CAIE S2 2023 November Q7
12 marks Standard +0.3
A biologist wishes to test whether the mean concentration \(\mu\), in suitable units, of a certain pollutant in a river is below the permitted level of 0.5. She measures the concentration, \(x\), of the pollutant at 50 randomly chosen locations in the river. The results are summarised below. \(n = 50 \quad \Sigma x = 23.0 \quad \Sigma x^2 = 13.02\)
  1. Carry out a test at the 5% significance level of the null hypothesis \(\mu = 0.5\) against the alternative hypothesis \(\mu < 0.5\). [7]
Later, a similar test is carried out at the 5% significance level using another sample of size 50 and the same hypotheses as before. You should assume that the standard deviation is unchanged.
  1. Given that, in fact, the value of \(\mu\) is 0.4, find the probability of a Type II error. [5]
CAIE S2 2024 November Q1
4 marks Moderate -0.8
The heights of a certain species of deer are known to have standard deviation \(0.35\) m. A zoologist takes a random sample of \(150\) of these deer and finds that the mean height of the deer in the sample is \(1.42\) m.
  1. Calculate a \(96\%\) confidence interval for the population mean height. [3]
  2. Bubay says that \(96\%\) of deer of this species are likely to have heights that are within this confidence interval. Explain briefly whether Bubay is correct. [1]
CAIE S2 2024 November Q2
5 marks Standard +0.3
The masses, in kilograms, of small and large bags of wheat have the independent distributions \(\text{N}(16.0, 0.4)\) and \(\text{N}(51.0, 0.9)\) respectively. Find the probability that the total mass of \(3\) randomly chosen small bags is greater than the mass of one randomly chosen large bag. [5]
CAIE S2 2024 November Q3
6 marks Standard +0.3
The times, \(T\) minutes, taken by a random sample of \(75\) students to complete a test were noted. The results were summarised by \(\sum t = 230\) and \(\sum t^2 = 930\).
  1. Calculate unbiased estimates of the population mean and variance of \(T\). [3]
You should now assume that your estimates from part (a) are the true values of the population mean and variance of \(T\).
  1. The times taken by another random sample of \(75\) students were noted, and the sample mean, \(\overline{T}\), was found. Find the value of \(a\) such that \(P(\overline{T} > a) = 0.234\). [3]
CAIE S2 2024 November Q4
6 marks Moderate -0.3
A random variable \(X\) has probability density function \(f\) defined by $$f(x) = \begin{cases} \frac{a}{x^2} - \frac{18}{x^3} & 2 \leqslant x < 3, \\ 0 & \text{otherwise}, \end{cases}$$ where \(a\) is a constant.
  1. Show that \(a = \frac{27}{2}\). [3]
  2. Show that \(\text{E}(X) = \frac{27}{2} \ln \frac{3}{2} - 3\). [3]
CAIE S2 2024 November Q5
6 marks Moderate -0.3
The lengths, in centimetres, of worms of a certain kind are normally distributed with mean \(\mu\) and standard deviation \(2.3\). An article in a magazine states that the value of \(\mu\) is \(12.7\). A scientist wishes to test whether this value is correct. He measures the lengths, \(x\) cm, of a random sample of \(50\) worms of this kind and finds that \(\sum x = 597.1\). He plans to carry out a test, at the \(1\%\) significance level, of whether the true value of \(\mu\) is different from \(12.7\).
  1. State, with a reason, whether he should use a one-tailed or a two-tailed test. [1]
  2. Carry out the test. [5]
CAIE S2 2024 November Q6
9 marks Standard +0.3
The numbers of customers arriving at service desks \(A\) and \(B\) during a \(10\)-minute period have the independent distributions \(\text{Po}(1.8)\) and \(\text{Po}(2.1)\) respectively.
  1. Find the probability that during a randomly chosen \(15\)-minute period more than \(2\) customers will arrive at desk \(A\). [2]
  2. Find the probability that during a randomly chosen \(5\)-minute period the total number of customers arriving at both desks is less than \(4\). [3]
  3. An inspector waits at desk \(B\). She wants to wait long enough to be \(90\%\) certain of seeing at least one customer arrive at the desk. Find the minimum time for which she should wait, giving your answer correct to the nearest minute. [4]