Questions — OCR (4907 questions)

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OCR S2 2010 January Q1
4 marks Easy -1.2
The values of 5 independent observations from a population can be summarised by $$\sum x = 75.8, \quad \sum x^2 = 1154.58.$$ Find unbiased estimates of the population mean and variance. [4]
OCR S2 2010 January Q2
3 marks Easy -1.2
A college has 400 students. A journalist wants to carry out a survey about food preferences and she obtains a sample of 30 pupils from the college by the following method. • Obtain a list of all the students. • Number the students, with numbers running sequentially from 0 to 399. • Select 30 random integers in the range 000 to 999 inclusive. If a random integer is in the range 0 to 399, then the student with that number is selected. If the number is greater than 399, then 400 is subtracted from the number (if necessary more than once) until an answer in the range 0 to 399 is selected, and the student with that number is selected.
  1. Explain why this method is unsatisfactory. [2]
  2. Explain how it could be improved. [1]
OCR S2 2010 January Q3
6 marks Moderate -0.8
In a large town, 35% of the inhabitants have access to television channel \(C\). A random sample of 60 inhabitants is obtained. Use a suitable approximation to find the probability that 18 or fewer inhabitants in the sample have access to channel \(C\). [6]
OCR S2 2010 January Q4
7 marks Moderate -0.3
80 randomly chosen people are asked to estimate a time interval of 60 seconds without using a watch or clock. The mean of the 80 estimates is 58.9 seconds. Previous evidence shows that the population standard deviation of such estimates is 5.0 seconds. Test, at the 5% significance level, whether there is evidence that people tend to underestimate the time interval. [7]
OCR S2 2010 January Q5
8 marks Standard +0.3
The number of customers arriving at a store between 8.50 am and 9 am on Saturday mornings is a random variable which can be modelled by the distribution Po(11.0). Following a series of price cuts, on one particular Saturday morning 19 customers arrive between 8.50 am and 9 am. The store's management claims, first, that the mean number of customers has increased, and second, that this is due to the price cuts.
  1. Test the first part of the claim, at the 5% significance level. [7]
  2. Comment on the second part of the claim. [1]
OCR S2 2010 January Q6
7 marks Moderate -0.8
The continuous random variable \(X\) has the distribution N(\(\mu\), \(\sigma^2\)).
  1. Each of the three following sets of probabilities is impossible. Give a reason in each case why the probabilities cannot both be correct. (You should not attempt to find \(\mu\) or \(\sigma\).)
    1. P(\(X > 50\)) = 0.7 and P(\(X < 50\)) = 0.2 [1]
    2. P(\(X > 50\)) = 0.7 and P(\(X > 70\)) = 0.8 [1]
    3. P(\(X > 50\)) = 0.3 and P(\(X < 70\)) = 0.3 [1]
  2. Given that P(\(X > 50\)) = 0.7 and P(\(X < 70\)) = 0.7, find the values of \(\mu\) and \(\sigma\). [4]
OCR S2 2010 January Q7
13 marks Moderate -0.3
The continuous random variable \(T\) is equally likely to take any value from 5.0 to 11.0 inclusive.
  1. Sketch the graph of the probability density function of \(T\). [2]
  2. Write down the value of E(\(T\)) and find by integration the value of Var(\(T\)). [5]
  3. A random sample of 48 observations of \(T\) is obtained. Find the approximate probability that the mean of the sample is greater than 8.3, and explain why the answer is an approximation. [6]
OCR S2 2010 January Q8
8 marks Standard +0.3
The random variable \(R\) has the distribution B(10, \(p\)). The null hypothesis H\(_0\): \(p = 0.7\) is to be tested against the alternative hypothesis H\(_1\): \(p < 0.7\), at a significance level of 5%.
  1. Find the critical region for the test and the probability of making a Type I error. [3]
  2. Given that \(p = 0.4\), find the probability that the test results in a Type II error. [3]
  3. Given that \(p\) is equally likely to take the values 0.4 and 0.7, find the probability that the test results in a Type II error. [2]
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]