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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]
OCR MEI S2 2007 January Q1
18 marks Moderate -0.8
In a science investigation into energy conservation in the home, a student is collecting data on the time taken for an electric kettle to boil as the volume of water in the kettle is varied. The student's data are shown in the table below, where \(v\) litres is the volume of water in the kettle and \(t\) seconds is the time taken for the kettle to boil (starting with the water at room temperature in each case). Also shown are summary statistics and a scatter diagram on which the regression line of \(t\) on \(v\) is drawn.
\(v\)0.20.40.60.81.0
\(t\)4478114156172
\(n = 5\), \(\Sigma v = 3.0\), \(\Sigma t = 564\), \(\Sigma v^2 = 2.20\), \(\Sigma vt = 405.2\). \includegraphics{figure_1}
  1. Calculate the equation of the regression line of \(t\) on \(v\), giving your answer in the form \(t = a + bv\). [5]
  2. Use this equation to predict the time taken for the kettle to boil when the amount of water which it contains is
    1. 0.5 litres,
    2. 1.5 litres.
    Comment on the reliability of each of these predictions. [4]
  3. In the equation of the regression line found in part (i), explain the role of the coefficient of \(v\) in the relationship between time taken and volume of water. [2]
  4. Calculate the values of the residuals for \(v = 0.8\) and \(v = 1.0\). [4]
  5. Explain how, on a scatter diagram with the regression line drawn accurately on it, a residual could be measured and its sign determined. [3]
OCR MEI S2 2007 January Q2
18 marks Moderate -0.3
  1. A farmer grows Brussels sprouts. The diameter of sprouts in a particular batch, measured in mm, is Normally distributed with mean 28 and variance 16. Sprouts that are between 24 mm and 33 mm in diameter are sold to a supermarket.
    1. Find the probability that the diameter of a randomly selected sprout will be within this range. [4]
    2. The farmer sells the sprouts in this range to the supermarket for 10 pence per kilogram. The farmer sells sprouts under 24 mm in diameter to a frozen food factory for 5 pence per kilogram. Sprouts over 33 mm in diameter are thrown away. Estimate the total income received by the farmer for the batch, which weighs 25 500 kg. [3]
    3. By harvesting sprouts earlier, the mean diameter for another batch can be reduced to \(k\) mm. Find the value of \(k\) for which only 5\% of the sprouts will be above 33 mm in diameter. You may assume that the variance is still 16. [3]
  2. The farmer also grows onions. The weight in kilograms of the onions is Normally distributed with mean 0.155 and variance 0.005. He is trying out a new variety, which he hopes will yield a higher mean weight. In order to test this, he takes a random sample of 25 onions of the new variety and finds that their total weight is 4.77 kg. You should assume that the weight in kilograms of the new variety is Normally distributed with variance 0.005.
    1. Write down suitable null and alternative hypotheses for the test in terms of \(\mu\). State the meaning of \(\mu\) in this case. [2]
    2. Carry out the test at the 1\% level. [6]
OCR MEI S2 2007 January Q3
18 marks Standard +0.3
An electrical retailer gives customers extended guarantees on washing machines. Under this guarantee all repairs in the first 3 years are free. The retailer records the numbers of free repairs made to 80 machines.
Number of repairs0123\(>3\)
Frequency5320610
  1. Show that the sample mean is 0.4375. [1]
  2. The sample standard deviation \(s\) is 0.6907. Explain why this supports a suggestion that a Poisson distribution may be a suitable model for the distribution of the number of free repairs required by a randomly chosen washing machine. [2]
The random variable \(X\) denotes the number of free repairs required by a randomly chosen washing machine. For the remainder of this question you should assume that \(X\) may be modelled by a Poisson distribution with mean 0.4375.
  1. Find P\((X = 1)\). Comment on your answer in relation to the data in the table. [4]
  2. The manager decides to monitor 8 washing machines sold on one day. Find the probability that there are at least 12 free repairs in total on these 8 machines. You may assume that the 8 machines form an independent random sample. [3]
  3. A launderette with 8 washing machines has needed 12 free repairs. Why does your answer to part (iv) suggest that the Poisson model with mean 0.4375 is unlikely to be a suitable model for free repairs on the machines in the launderette? Give a reason why the model may not be appropriate for the launderette. [3]
The retailer also sells tumble driers with the same guarantee. The number of free repairs on a tumble drier in three years can be modelled by a Poisson distribution with mean 0.15. A customer buys a tumble drier and a washing machine.
  1. Assuming that free repairs are required independently, find the probability that
    1. the two appliances need a total of 3 free repairs between them,
    2. each appliance needs exactly one free repair.
    [5]
OCR MEI S2 2007 January Q4
18 marks Standard +0.3
Two educational researchers are investigating the relationship between personal ambitions and home location of students. The researchers classify students into those whose main personal ambition is good academic results and those who have some other ambition. A random sample of 480 students is selected.
  1. One researcher summarises the data as follows.
    \multirow{2}{*}{Observed}Home location
    \cline{2-3}CityNon-city
    \multirow{2}{*}{Ambition}Good results102147
    \cline{2-3}Other75156
    Carry out a test at the 5\% significance level to examine whether there is any association between home location and ambition. State carefully your null and alternative hypotheses. Your working should include a table showing the contributions of each cell to the test statistic. [9]
  2. The other researcher summarises the same data in a different way as follows.
    \multirow{2}{*}{Observed}Home location
    \cline{2-4}CityTownCountry
    \multirow{2}{*}{Ambition}Good results1028364
    \cline{2-4}Other756492
    1. Calculate the expected frequencies for both 'Country' cells. [2]
    2. The test statistic for these data is 10.94. Carry out a test at the 5\% level based on this table, using the same hypotheses as in part (i). [3]
    3. The table below gives the contribution of each cell to the test statistic. Discuss briefly how personal ambitions are related to home location. [2]
      \multirow{2}{*}{
      Contribution to the
      test statistic
      }      		Home location
      \cline{2-4}CityTownCountry
      \multirow{2}{*}{Ambition}Good results1.1290.5963.540
      \cline{2-4}Other1.2170.6433.816
  3. Comment briefly on whether the analysis in part (ii) means that the conclusion in part (i) is invalid. [2]
Edexcel S2 Q1
5 marks Moderate -0.8
A golfer believes that the distance, in metres, that she hits a ball with a 5 iron, follows a continuous uniform distribution over the interval \([100, 150]\).
  1. Find the median and interquartile range of the distance she hits a ball, that would be predicted by this model. [3 marks]
  2. Explain why the continuous uniform distribution may not be a suitable model. [2 marks]
Edexcel S2 Q2
8 marks Moderate -0.3
The continuous random variable \(X\) has the following cumulative distribution function: $$F(x) = \begin{cases} 0, & x < 0, \\ \frac{1}{64}(16x - x^2), & 0 \leq x \leq 8, \\ 1, & x > 8. \end{cases}$$
  1. Find \(P(X > 5)\). [2 marks]
  2. Find and specify fully the probability density function \(f(x)\) of \(X\). [3 marks]
  3. Sketch \(f(x)\) for all values of \(x\). [3 marks]
Edexcel S2 Q3
10 marks Moderate -0.3
An electrician records the number of repairs of different types of appliances that he makes each day. His records show that over 40 working days he repaired a total of 180 CD players.
  1. Explain why a Poisson distribution may be suitable for modelling the number of CD players he repairs each day and find the parameter for this distribution. [4 marks]
  2. Find the probability that on one particular day he repairs
    1. no CD players,
    2. more than 6 CD players. [3 marks]
  3. Find the probability that over 10 working days he will repair more than 6 CD players on exactly 3 of the days. [3 marks]