Questions S2 (1597 questions)

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OCR S2 2008 January Q3
8 marks Standard +0.8
3 The random variable \(G\) has the distribution \(\operatorname { Po } ( \lambda )\). A test is carried out of the null hypothesis \(\mathrm { H } _ { 0 } : \lambda = 4.5\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : \lambda \neq 4.5\), based on a single observation of \(G\). The critical region for the test is \(G \leqslant 1\) and \(G \geqslant 9\).
  1. Find the significance level of the test.
  2. Given that \(\lambda = 5.5\), calculate the probability that the test results in a Type II error.
OCR S2 2008 January Q4
7 marks Moderate -0.8
4 The random variable \(Y\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). The results of 40 independent observations of \(Y\) are summarised by $$\Sigma y = 3296.0 , \quad \Sigma y ^ { 2 } = 286800.40$$
  1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
  2. Use your answers to part (i) to estimate the probability that a single random observation of \(Y\) will be less than 60.0.
  3. Explain whether it is necessary to know that \(Y\) is normally distributed in answering part (i) of this question.
OCR S2 2008 January Q5
9 marks Standard +0.3
5 Over a long period the number of visitors per week to a stately home was known to have the distribution \(\mathrm { N } \left( 500,100 ^ { 2 } \right)\). After higher car parking charges were introduced, a sample of four randomly chosen weeks gave a mean number of visitors per week of 435 . You should assume that the number of visitors per week is still normally distributed with variance \(100 ^ { 2 }\).
  1. Test, at the \(10 \%\) significance level, whether there is evidence that the mean number of visitors per week has fallen.
  2. Explain why it is necessary to assume that the distribution of the number of visitors per week (after the introduction of higher charges) is normal in order to carry out the test.
OCR S2 2008 January Q6
11 marks Standard +0.3
6 The number of house sales per week handled by an estate agent is modelled by the distribution \(\operatorname { Po } ( 3 )\).
  1. Find the probability that, in one randomly chosen week, the number of sales handled is
    (a) greater than 4 ,
    (b) exactly 4 .
  2. Use a suitable approximation to the Poisson distribution to find the probability that, in a year consisting of 50 working weeks, the estate agent handles more than 165 house sales.
  3. One of the conditions needed for the use of a Poisson model to be valid is that house sales are independent of one another.
    (a) Explain, in non-technical language, what you understand by this condition.
    (b) State another condition that is needed.
OCR S2 2008 January Q7
13 marks Moderate -0.3
7 A continuous random variable \(X _ { 1 }\) has probability density function given by $$f ( x ) = \begin{cases} k x & 0 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 1 } { 2 }\).
  2. Sketch the graph of \(y = \mathrm { f } ( x )\).
  3. Find \(\mathrm { E } \left( X _ { 1 } \right)\) and \(\operatorname { Var } \left( X _ { 1 } \right)\).
  4. Sketch the graph of \(y = \mathrm { f } ( x - 1 )\).
  5. The continuous random variable \(X _ { 2 }\) has probability density function \(\mathrm { f } ( x - 1 )\) for all \(x\). Write down the values of \(\mathrm { E } \left( X _ { 2 } \right)\) and \(\operatorname { Var } \left( X _ { 2 } \right)\).
OCR S2 2008 January Q8
13 marks Standard +0.3
8 Consultations are taking place as to whether a site currently in use as a car park should be developed as a shopping mall. An agency acting on behalf of a firm of developers claims that at least \(65 \%\) of the local population are in favour of the development. In a survey of a random sample of 12 members of the local population, 6 are in favour of the development.
  1. Carry out a test, at the \(10 \%\) significance level, to determine whether the result of the survey is consistent with the claim of the agency.
  2. A local residents' group claims that no more than \(35 \%\) of the local population are in favour of the development. Without further calculations, state with a reason what can be said about the claim of the local residents' group.
  3. A test is carried out, at the \(15 \%\) significance level, of the agency's claim. The test is based on a random sample of size \(2 n\), and exactly \(n\) of the sample are in favour of the development. Find the smallest possible value of \(n\) for which the outcome of the test is to reject the agency's claim.
    [0pt] [4] 4
OCR S2 2012 January Q1
4 marks Easy -1.2
1 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 \(X\).
OCR S2 2012 January Q2
5 marks Moderate -0.3
2 The random variable \(Y\) has the distribution \(\mathrm { B } ( 140,0.03 )\). Use a suitable approximation to find \(\mathrm { P } ( Y = 5 )\). Justify your approximation.
OCR S2 2012 January Q3
6 marks Moderate -0.3
3 The random variable \(G\) has a normal distribution. It is known that $$\mathrm { P } ( G < 56.2 ) = \mathrm { P } ( G > 63.8 ) = 0.1 \text {. }$$ Find \(\mathrm { P } ( G > 65 )\).
OCR S2 2012 January Q4
5 marks Standard +0.3
4 The discrete random variable \(H\) takes values 1, 2, 3 and 4. It is given that \(\mathrm { E } ( H ) = 2.5\) and \(\operatorname { Var } ( H ) = 1.25\). The mean of a random sample of 50 observations of \(H\) is denoted by \(\bar { H }\).
Use a suitable approximation to find \(\mathrm { P } ( \bar { H } < 2.6 )\).
OCR S2 2012 January Q5
10 marks Standard +0.3
5
  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
    (a) the same child may win more than one prize,
    (b) no child may win more than one prize.
  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.
OCR S2 2012 January Q6
8 marks Standard +0.3
6 The number of fruit pips in 1 cubic centimetre of raspberry jam has the distribution \(\operatorname { Po } ( \lambda )\). 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. Find (a) \(\mathrm { E } ( X )\),
(ii) The continuous random variable \(Y\) has the probability density function $$g ( y ) = \left\{ \begin{array} { l r } \frac { 1.5 } { y ^ { 2.5 } } & y \geqslant 1 \\ 0 & \text { otherwise. } \end{array} \right.$$ Given that \(\mathrm { E } ( Y ) = 3\), show that \(\operatorname { Var } ( Y )\) is not finite.
OCR S2 2012 January Q8
14 marks Standard +0.3
8 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. Assume now that a Poisson model is appropriate.
  2. Find the probability that in 10 ml there are at least 5 bacteria.
  3. Find the probability that in 3.7 ml there are exactly 2 bacteria.
  4. Use a suitable approximation to find the probability that in 1000 ml there are fewer than 240 bacteria, justifying your approximation.
OCR S2 2012 January Q9
11 marks Standard +0.3
9 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.
  2. Calculate the significance level of the test.
  3. Explain what is meant by a Type I error in this context.
  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. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}
OCR S2 2005 June Q1
4 marks Easy -1.8
1 It is desired to obtain a random sample of 15 pupils from a large school. One pupil suggests listing all the pupils in the school in alphabetical order and choosing the first 15 names on the list.
  1. Explain why this method is unsatisfactory.
  2. Suggest a better method.
OCR S2 2005 June Q2
4 marks Moderate -0.3
2 A continuous random variable has a normal distribution with mean 25.0 and standard deviation \(\sigma\). The probability that any one observation of the random variable is greater than 20,0 is 0.75 . Find the value of \(\sigma\).
OCR S2 2005 June Q3
8 marks Standard +0.3
3
  1. The random variable \(X\) has a \(\mathrm { B } ( 60,0.02 )\) distribution. Use an appropriate approximation to find \(\mathrm { P } ( X \leqslant 2 )\).
  2. The random variable \(Y\) has a \(\operatorname { Po } ( 30 )\) distribution. Use an appropriate approximation to find \(\mathrm { P } ( Y \leqslant 38 )\).
OCR S2 2005 June Q4
9 marks Moderate -0.3
4 The height of sweet pea plants grown in a nursery is a random variable. A random sample of 50 plants is measured and is found to have a mean height 1.72 m and variance \(0.0967 \mathrm {~m} ^ { 2 }\).
  1. Calculate an unbiased estimate for the population variance of the heights of sweet pea plants.
  2. Hence test, at the \(10 \%\) significance level, whether the mean height of sweet pea plants grown by the nursery is 1.8 m , stating your hypotheses clearly.
OCR S2 2005 June Q5
11 marks Moderate -0.3
5 The random variable \(W\) has the distribution \(\mathbf { B } ( 30 , p )\).
  1. Use the exact binomial distribution to calculate \(\mathbf { P } ( W = 10 )\) when \(p = 0.4\).
  2. Find the range of values of \(p\) for which you would expect that a normal distribution could be used as an approximation to the distribution of \(W\).
  3. Use a normal approximation to calculate \(\mathrm { P } ( W = 10 )\) when \(p = 0.4\).
OCR S2 2005 June Q6
11 marks Standard +0.3
6 A factory makes chocolates of different types. The proportion of milk chocolates made on any day is denoted by \(p\). It is desired to test the null hypothesis \(\mathrm { H } _ { 0 } : p = 0.8\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : p < 0.8\). The test consists of choosing a random sample of 25 chocolates. \(\mathrm { H } _ { 0 }\) is rejected if the number of milk chocolates is \(k\) or fewer. The test is carried out at a significance level as close to \(5 \%\) as possible.
  1. Use tables to find the value of \(k\), giving the values of any relevant probabilities.
  2. The test is carried out 20 times, and each time the value of \(p\) is 0.8 . Each of the tests is independent of all the others. State the expected number of times that the test will result in rejection of the null hypothesis.
  3. The test is carried out once. If in fact the value of \(p\) is 0.6 , find the probability of rejecting \(\mathrm { H } _ { 0 }\).
  4. The test is carried out twice. Each time the value of \(p\) is equally likely to be 0.8 or 0.6 . Find the probability that exactly one of the two tests results in rejection of the null hypothesis.
OCR S2 2005 June Q7
13 marks Standard +0.3
7 The continuous random variable \(X\) has the probability density function shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{b69b1fe8-790d-4727-a892-8ab2ade08962-3_364_766_1229_699}
  1. Find the value of the constant \(k\).
  2. Write down the mean of \(X\), and use integration to find the variance of \(X\).
  3. Three observations of \(X\) are made. Find the probability that \(X < 9\) for all three observations.
  4. The mean of 32 observations of \(X\) is denoted by \(\bar { X }\). State the approximate distribution of \(\bar { X }\), giving its mean and variance. \section*{[Question 8 is printed overleaf.]}
OCR S2 2005 June Q8
12 marks Standard +0.3
8 In excavating an archaeological site, Roman coins are found scattered throughout the site.
  1. State two assumptions needed to model the number of coins found per square metre of the site by a Poisson distribution. Assume now that the number of coins found per square metre of the site can be modelled by a Poisson distribution with mean \(\lambda\).
  2. Given that \(\lambda = 0.75\), calculate the probability that exactly 3 coins are found in a region of the site of area \(7.20 \mathrm {~m} ^ { 2 }\). A test is carried out, at the \(5 \%\) significance level, of the null hypothesis \(\lambda = 0.75\), against the alternative hypothesis \(\lambda > 0.75\), in Region LVI which has area \(4 \mathrm {~m} ^ { 2 }\).
  3. Determine the smallest number of coins that, if found in Region LVI, would lead to rejection of the null hypothesis, stating also the values of any relevant probabilities.
  4. Given that, in fact, \(\lambda = 1.2\) in Region LVI, find the probability that the test results in a Type II error.
OCR S2 2006 June Q1
6 marks Moderate -0.5
1 Calculate the variance of the continuous random variable with probability density function given by $$f ( x ) = \begin{cases} \frac { 3 } { 37 } x ^ { 2 } & 3 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$
OCR S2 2006 June Q2
7 marks Standard +0.3
2
  1. The random variable \(R\) has the distribution \(\mathrm { B } ( 6 , p )\). A random observation of \(R\) is found to be 6. Carry out a \(5 \%\) significance test of the null hypothesis \(\mathrm { H } _ { 0 } : p = 0.45\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : p \neq 0.45\), showing all necessary details of your calculation.
  2. The random variable \(S\) has the distribution \(\mathrm { B } ( n , p ) . \mathrm { H } _ { 0 }\) and \(\mathrm { H } _ { 1 }\) are as in part (i). A random observation of \(S\) is found to be 1 . Use tables to find the largest value of \(n\) for which \(\mathrm { H } _ { 0 }\) is not rejected. Show the values of any relevant probabilities.
OCR S2 2006 June Q3
8 marks Standard +0.3
3 The continuous random variable \(T\) has mean \(\mu\) and standard deviation \(\sigma\). It is known that \(\mathrm { P } ( T < 140 ) = 0.01\) and \(\mathrm { P } ( T < 300 ) = 0.8\).
  1. Assuming that \(T\) is normally distributed, calculate the values of \(\mu\) and \(\sigma\). In fact, \(T\) represents the time, in minutes, taken by a randomly chosen runner in a public marathon, in which about \(10 \%\) of runners took longer than 400 minutes.
  2. State with a reason whether the mean of \(T\) would be higher than, equal to, or lower than the value calculated in part (i).