Questions — OCR S2 (169 questions)

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OCR S2 2010 January Q9
16 marks Standard +0.3
Buttercups in a meadow are distributed independently of one another and at a constant average incidence of 3 buttercups per square metre.
  1. Find the probability that in 1 square metre there are more than 7 buttercups. [2]
  2. Find the probability that in 4 square metres there are either 13 or 14 buttercups. [3]
  3. Use a suitable approximation to find the probability that there are no more than 69 buttercups in 20 square metres. [5]
    1. Without using an approximation, find an expression for the probability that in \(m\) square metres there are at least 2 buttercups. [2]
    2. It is given that the probability that there are at least 2 buttercups in \(m\) square metres is 0.9. Using your answer to part (a), show numerically that \(m\) lies between 1.29 and 1.3. [4]
OCR S2 2012 January Q1
4 marks Easy -1.2
A random sample of 50 observations of the random variable \(X\) is summarised by $$n = 50, \Sigma x = 182.5, \Sigma x^2 = 739.625.$$ Calculate unbiased estimates of the expectation and variance of \(X\). [4]
OCR S2 2012 January Q2
5 marks Standard +0.3
The random variable \(Y\) has the distribution B(140, 0.03). Use a suitable approximation to find P(\(Y = 5\)). Justify your approximation. [5]
OCR S2 2012 January Q3
6 marks Standard +0.8
The random variable \(G\) has a normal distribution. It is known that $$\text{P}(G < 56.2) = \text{P}(G > 63.8) = 0.1.$$ Find P(\(G > 65\)). [6]
OCR S2 2012 January Q4
5 marks Standard +0.3
The discrete random variable \(H\) takes values 1, 2, 3 and 4. It is given that E(\(H\)) = 2.5 and Var(\(H\)) = 1.25. The mean of a random sample of 50 observations of \(H\) is denoted by \(\bar{H}\). Use a suitable approximation to find P(\(\bar{H} < 2.6\)). [5]
OCR S2 2012 January Q5
10 marks Standard +0.3
  1. Six prizes are allocated, using random numbers, to a group of 12 girls and 8 boys. Calculate the probability that exactly 4 of the prizes are allocated to girls if
    1. the same child may win more than one prize, [2]
    2. no child may win more than one prize. [2]
  2. Sixty prizes are allocated, using random numbers, to a group of 1200 girls and 800 boys. Use a suitable approximation to calculate the probability that at least 30 of the prizes are allocated to girls. Does it affect your calculation whether or not the same child may win more than one prize? Justify your answer. [6]
OCR S2 2012 January Q6
8 marks Standard +0.3
The number of fruit pips in 1 cubic centimetre of raspberry jam has the distribution Po(\(\lambda\)). Under a traditional jam-making process it is known that \(\lambda = 6.3\). A new process is introduced and a random sample of 1 cubic centimetre of jam produced by the new process is found to contain 2 pips. Test, at the 5% significance level, whether this is evidence that under the new process the average number of pips has been reduced. [8]
OCR S2 2012 January Q7
9 marks Standard +0.3
  1. The continuous random variable \(X\) has the probability density function $$f(x) = \begin{cases} \frac{1}{2\sqrt{x}} & 1 < x < 4, \\ 0 & \text{otherwise}. \end{cases}$$ Find
    1. E(\(X\)), [3]
    2. the median of \(X\). [3]
  2. The continuous random variable \(Y\) has the probability density function $$g(y) = \begin{cases} \frac{1.5}{y^{2.5}} & y > 1, \\ 0 & \text{otherwise}. \end{cases}$$ Given that E(\(Y\)) = 3, show that Var(\(Y\)) is not finite. [3]
OCR S2 2012 January Q8
14 marks Standard +0.3
In a certain fluid, bacteria are distributed randomly and occur at a constant average rate of 2.5 in every 10 ml of the fluid.
  1. State a further condition needed for the number of bacteria in a fixed volume of the fluid to be well modelled by a Poisson distribution, explaining what your answer means. [2]
Assume now that a Poisson model is appropriate.
  1. Find the probability that in 10 ml there are at least 5 bacteria. [2]
  2. Find the probability that in 3.7 ml there are exactly 2 bacteria. [3]
  3. Use a suitable approximation to find the probability that in 1000 ml there are fewer than 240 bacteria, justifying your approximation. [7]
OCR S2 2012 January Q9
11 marks Standard +0.3
It is desired to test whether the average amount of sleep obtained by school pupils in Year 11 is 8 hours, based on a random sample of size 64. The population standard deviation is 0.87 hours and the sample mean is denoted by \(\bar{H}\). The critical values for the test are \(\bar{H} = 7.72\) and \(\bar{H} = 8.28\).
  1. State appropriate hypotheses for the test, explaining the meaning of any symbol you use. [3]
  2. Calculate the significance level of the test. [4]
  3. Explain what is meant by a Type I error in this context. [1]
  4. Given that in fact the average amount of sleep obtained by all pupils in Year 11 is 7.9 hours, find the probability that the test results in a Type II error. [3]
OCR S2 2016 June Q1
4 marks Easy -1.2
The results of 14 observations of a random variable \(V\) are summarised by $$n = 14, \quad \sum v = 3752, \quad \sum v^2 = 1007448.$$ Calculate unbiased estimates of E\((V)\) and Var\((V)\). [4]
OCR S2 2016 June Q2
6 marks Moderate -0.3
The mass, in kilograms, of a packet of flour is a normally distributed random variable with mean 1.03 and variance \(\sigma^2\). Given that 5% of packets have mass less than 1.00 kg, find the percentage of packets with mass greater than 1.05 kg. [6]
OCR S2 2016 June Q3
7 marks Standard +0.3
The random variable \(F\) has the distribution B\((40, 0.65)\). Use a suitable approximation to find P\((F \leq 30)\), justifying your approximation. [7]
OCR S2 2016 June Q4
5 marks Moderate -0.8
It is given that \(Y \sim\) Po\((\lambda)\), where \(\lambda \neq 0\), and that P\((Y = 4) =\) P\((Y = 5)\). Write down an equation for \(\lambda\). Hence find the value of \(\lambda\) and the corresponding value of P\((Y = 5)\). [5]
OCR S2 2016 June Q5
8 marks Standard +0.3
55% of the pupils in a large school are girls. A member of the student council claims that the probability that a girl rather than a boy becomes Head Student is greater than 0.55. As evidence for his claim he says that 6 of the last 8 Head Students have been girls.
  1. Use an exact binomial distribution to test the claim at the 10% significance level. [7]
  2. A statistics teacher says that considering only the last 8 Head Students may not be satisfactory. Explain what needs to be assumed about the data for the test to be valid. [1]
OCR S2 2016 June Q6
12 marks Moderate -0.3
The number of cars passing a point on a single-track one-way road during a one-minute period is denoted by \(X\). Cars pass the point at random intervals and the expected value of \(X\) is denoted by \(\lambda\).
  1. State, in the context of the question, two conditions needed for \(X\) to be well modelled by a Poisson distribution. [2]
  2. At a quiet time of the day, \(\lambda = 6.50\). Assuming that a Poisson distribution is valid, calculate P\((4 \leq X < 8)\). [3]
  3. At a busy time of the day, \(\lambda = 30\).
    1. Assuming that a Poisson distribution is valid, use a suitable approximation to find P\((X > 35)\). Justify your approximation. [6]
    2. Give a reason why a Poisson distribution might not be valid in this context when \(\lambda = 30\). [1]
OCR S2 2016 June Q7
11 marks Standard +0.3
A continuous random variable \(X\) has probability density function $$\text{f}(x) = \begin{cases} ax^{-3} + bx^{-4} & x \geq 1, \\ 0 & \text{otherwise,} \end{cases}$$ where \(a\) and \(b\) are constants.
  1. Explain what the letter \(x\) represents. [1]
It is given that P\((X > 2) = \frac{3}{16}\).
  1. Show that \(a = 1\), and find the value of \(b\). [7]
  2. Find E\((X)\). [3]
OCR S2 2016 June Q8
13 marks Standard +0.3
It is known that the lifetime of a certain species of animal in the wild has mean 13.3 years. A zoologist reads a study of 50 randomly chosen animals of this species that have been kept in zoos. According to the study, for these 50 animals the sample mean lifetime is 12.48 years and the population variance is 12.25 years\(^2\).
  1. Test at the 5% significance level whether these results provide evidence that animals of this species that have been kept in zoos have a shorter expected lifetime than those in the wild. [7]
  2. Subsequently the zoologist discovered that there had been a mistake in the study. The quoted variance of 12.25 years\(^2\) was in fact the sample variance. Determine whether this makes a difference to the conclusion of the test. [5]
  3. Explain whether the Central Limit Theorem is needed in these tests. [1]
OCR S2 2016 June Q9
6 marks Challenging +1.3
The random variable \(R\) has the distribution Po\((\lambda)\). A significance test is carried out at the 1% level of the null hypothesis H\(_0\): \(\lambda = 11\) against H\(_1\): \(\lambda > 11\), based on a single observation of \(R\). Given that in fact the value of \(\lambda\) is 14, find the probability that the result of the test is incorrect, and give the technical name for such an incorrect outcome. You should show the values of any relevant probabilities. [6]