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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]
CAIE S2 2024 November Q7
14 marks Standard +0.8
The number of accidents per year on a certain road has the distribution \(\text{Po}(\lambda)\). In the past the value of \(\lambda\) was \(3.3\). Recently, a new speed limit was imposed and the council wishes to test whether the value of \(\lambda\) has decreased. The council notes the total number, \(X\), of accidents during two randomly chosen years after the speed limit was introduced and it carries out a test at the \(5\%\) significance level.
  1. Calculate the probability of a Type I error. [4]
  2. Given that \(X = 2\), carry out the test. [3]
  3. The council decides to carry out another similar test at the \(5\%\) significance level using the same hypotheses and two different randomly chosen years. Given that the true value of \(\lambda\) is \(0.6\), calculate the probability of a Type II error. [3]
  4. Using \(\lambda = 0.6\) and a suitable approximating distribution, find the probability that there will be more than \(10\) accidents in \(30\) years. [4]
CAIE S2 2011 June Q1
4 marks Moderate -0.5
The weights of bags of fuel have mean 3.2 kg and standard deviation 0.04 kg. The total weight of a random sample of three bags is denoted by \(T\) kg. Find the mean and standard deviation of \(T\). [4]
CAIE S2 2011 June Q2
5 marks Standard +0.3
\(X\) is a random variable having the distribution \(\text{B}(12, \frac{1}{4})\). A random sample of 60 values of \(X\) is taken. Find the probability that the sample mean is less than 2.8. [5]
CAIE S2 2011 June Q3
7 marks Moderate -0.3
The number of goals scored per match by Everly Rovers is represented by the random variable \(X\) which has mean 1.8.
  1. State two conditions for \(X\) to be modelled by a Poisson distribution. [2]
Assume now that \(X \sim \text{Po}(1.8)\).
  1. Find \(\text{P}(2 < X < 6)\). [2]
  2. The manager promises the team a bonus if they score at least 1 goal in each of the next 10 matches. Find the probability that they win the bonus. [3]
CAIE S2 2011 June Q4
8 marks Moderate -0.3
A doctor wishes to investigate the mean fat content in low-fat burgers. He takes a random sample of 15 burgers and sends them to a laboratory where the mass, in grams, of fat in each burger is determined. The results are as follows. \(9 \quad 7 \quad 8 \quad 9 \quad 6 \quad 11 \quad 7 \quad 9 \quad 8 \quad 9 \quad 8 \quad 10 \quad 7 \quad 9 \quad 9\) Assume that the mass, in grams, of fat in low-fat burgers is normally distributed with mean \(\mu\) and that the population standard deviation is 1.3.
  1. Calculate a 99\% confidence interval for \(\mu\). [4]
  2. Explain whether it was necessary to use the Central Limit theorem in the calculation in part (i). [2]
  3. The manufacturer claims that the mean mass of fat in burgers of this type is 8 g. Use your answer to part (i) to comment on this claim. [2]
CAIE S2 2011 June Q5
9 marks Standard +0.3
The number of adult customers arriving in a shop during a 5-minute period is modelled by a random variable with distribution \(\text{Po}(6)\). The number of child customers arriving in the same shop during a 10-minute period is modelled by an independent random variable with distribution \(\text{Po}(4.5)\).
  1. Find the probability that during a randomly chosen 2-minute period, the total number of adult and child customers who arrive in the shop is less than 3. [3]
  2. During a sale, the manager claims that more adult customers are arriving than usual. In a randomly selected 30-minute period during the sale, 49 adult customers arrive. Test the manager's claim at the 2.5\% significance level. [6]
CAIE S2 2011 June Q6
8 marks Moderate -0.3
Jeevan thinks that a six-sided die is biased in favour of six. In order to test this, Jeevan throws the die 10 times. If the die shows a six on at least 4 throws out of 10, she will conclude that she is correct.
  1. State appropriate null and alternative hypotheses. [1]
  2. Calculate the probability of a Type I error. [3]
  3. Explain what is meant by a Type II error in this situation. [1]
  4. If the die is actually biased so that the probability of throwing a six is \(\frac{1}{3}\), calculate the probability of a Type II error. [3]
CAIE S2 2011 June Q7
9 marks Moderate -0.3
A random variable \(X\) has probability density function given by $$f(x) = \begin{cases} k(1-x) & -1 \leq x \leq 1, \\ 0 & \text{otherwise}, \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac{1}{2}\). [2]
  2. Find \(\text{P}(X > \frac{1}{2})\). [1]
  3. Find the mean of \(X\). [3]
  4. Find \(a\) such that \(\text{P}(X < a) = \frac{1}{3}\). [3]
CAIE S2 2016 June Q1
3 marks Moderate -0.8
The time taken for a particular type of paint to dry was measured for a sample of 150 randomly chosen points on a wall. The sample mean was 192.4 minutes and an unbiased estimate of the population variance was 43.6 minutes\(^2\). Find a 98\% confidence interval for the mean drying time. [3]
CAIE S2 2016 June Q2
5 marks Moderate -0.3
In the past, the mean annual crop yield from a particular field has been 8.2 tonnes. During the last 16 years, a new fertiliser has been used on the field. The mean yield for these 16 years is 8.7 tonnes. Assume that yields are normally distributed with standard deviation 1.2 tonnes. Carry out a test at the 5\% significance level of whether the mean yield has increased. [5]
CAIE S2 2016 June Q3
6 marks Moderate -0.3
1\% of adults in a certain country own a yellow car.
  1. Use a suitable approximating distribution to find the probability that a random sample of 240 adults includes more than 2 who own a yellow car. [4]
  2. Justify your approximation. [2]
CAIE S2 2016 June Q4
7 marks Standard +0.3
The number of sightings of a golden eagle at a certain location has a Poisson distribution with mean 2.5 per week. Drilling for oil is started nearby. A naturalist wishes to test at the 5\% significance level whether there are fewer sightings since the drilling began. He notes that during the following 3 weeks there are 2 sightings.
  1. Find the critical region for the test and carry out the test. [5]
  2. State the probability of a Type I error. [1]
  3. State why the naturalist could not have made a Type II error. [1]
CAIE S2 2016 June Q5
10 marks Standard +0.3
The time, \(T\) minutes, taken by people to complete a test has probability density function given by $$\mathrm{f}(t) = \begin{cases} k(10t - t^2) & 5 \leq t \leq 10, \\ 0 & \text{otherwise}, \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac{3}{250}\). [3]
  2. Find \(\mathrm{E}(T)\). [3]
  3. Find the probability that a randomly chosen value of \(T\) lies between \(\mathrm{E}(T)\) and the median of \(T\). [3]
  4. State the greatest possible length of time taken to complete the test. [1]
CAIE S2 2016 June Q6
9 marks Standard +0.3
\(X\) and \(Y\) are independent random variables with distributions \(\mathrm{Po}(1.6)\) and \(\mathrm{Po}(2.3)\) respectively.
  1. Find \(\mathrm{P}(X + Y = 4)\). [3]
A random sample of 75 values of \(X\) is taken.
  1. State the approximate distribution of the sample mean, \(\overline{X}\), including the values of the parameters. [2]
  2. Hence find the probability that the sample mean is more than 1.7. [3]
  3. Explain whether the Central Limit theorem was needed to answer part (ii). [1]
CAIE S2 2016 June Q7
10 marks Standard +0.3
Bags of sugar are packed in boxes, each box containing 20 bags. The masses of the boxes, when empty, are normally distributed with mean 0.4 kg and standard deviation 0.01 kg. The masses of the bags are normally distributed with mean 1.02 kg and standard deviation 0.03 kg.
  1. Find the probability that the total mass of a full box of 20 bags is less than 20.6 kg. [5]
  2. Two full boxes are chosen at random. Find the probability that they differ in mass by less than 0.02 kg. [5]
CAIE S2 2002 November Q1
3 marks Moderate -0.8
The time taken, \(T\) minutes, for a special anti-rust paint to dry was measured for a random sample of 120 painted pieces of metal. The sample mean was 51.2 minutes and an unbiased estimate of the population variance was 37.4 minutes\(^2\). Determine a 99% confidence interval for the mean drying time. [3]
CAIE S2 2002 November Q2
5 marks Moderate -0.8
1.5% of the population of the UK can be classified as 'very tall'.
  1. The random variable \(X\) denotes the number of people in a sample of \(n\) people who are classified as very tall. Given that E\((X) = 2.55\), find \(n\). [2]
  2. By using the Poisson distribution as an approximation to a binomial distribution, calculate an approximate value for the probability that a sample of size 210 will contain fewer than 3 people who are classified as very tall. [3]
CAIE S2 2002 November Q3
7 marks Standard +0.3
From previous years' observations, the lengths of salmon in a river were found to be normally distributed with mean 65 cm. A researcher suspects that pollution in water is restricting growth. To test this theory, she measures the length \(x\) cm of a random sample of \(n\) salmon and calculates that \(\bar{x} = 64.3\) and \(s = 4.9\), where \(s^2\) is the unbiased estimate of the population variance. She then carries out an appropriate hypothesis test.
  1. Her test statistic \(z\) has a value of \(-1.807\) correct to 3 decimal places. Calculate the value of \(n\). [3]
  2. Using this test statistic, carry out the hypothesis test at the 5% level of significance and state what her conclusion should be. [4]