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OCR S2 2013 January Q1
4 marks Standard +0.8
1 A random variable has the distribution \(\mathrm { B } ( n , p )\). It is required to test \(\mathrm { H } _ { 0 } : p = \frac { 2 } { 3 }\) against \(\mathrm { H } _ { 1 } : p < \frac { 2 } { 3 }\) at a significance level as close to \(1 \%\) as possible, using a sample of size \(n = 8,9\) or 10 . Use tables to find which value of \(n\) gives such a test, stating the critical region for the test and the corresponding significance level.
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OCR S2 2013 January Q2
6 marks Moderate -0.3
2 A random variable \(C\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). A random sample of 10 observations of \(C\) is obtained, and the results are summarised as $$n = 10 , \Sigma c = 380 , \Sigma c ^ { 2 } = 14602 .$$
  1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
  2. Hence calculate an estimate of the probability that \(C > 40\).
OCR S2 2013 January Q3
8 marks Moderate -0.8
3 A factory produces 9000 music DVDs each day. A random sample of 100 such DVDs is obtained.
  1. Explain how to obtain this sample using random numbers.
  2. Given that \(24 \%\) of the DVDs produced by the factory are classical, use a suitable approximation to find the probability that, in the sample of 100 DVDs, fewer than 20 are classical.
OCR S2 2013 January Q4
9 marks Moderate -0.5
4 A continuous random variable \(X\) has probability density function $$\mathrm { f } ( x ) = \left\{ \begin{array} { c l } k x & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{array} \right.$$ where \(k\) and \(a\) are constants.
  1. State what the letter \(x\) represents.
  2. Find \(k\) in terms of \(a\).
  3. Find \(\operatorname { Var } ( X )\) in terms of \(a\).
OCR S2 2013 January Q5
8 marks Standard +0.3
5 In a mine, a deposit of the substance pitchblende emits radioactive particles. The number of particles emitted has a Poisson distribution with mean 70 particles per second. The warning level is reached if the total number of particles emitted in one minute is more than 4350.
  1. A one-minute period is chosen at random. Use a suitable approximation to show that the probability that the warning level is reached during this period is 0.01 , correct to 2 decimal places. You should calculate the answer correct to 4 decimal places.
  2. Use a suitable approximation to find the probability that in 30 one-minute periods the warning level is reached on at least 4 occasions. (You should use the given rounded value of 0.01 from part (i) in your calculation.)
OCR S2 2013 January Q6
10 marks Standard +0.3
6 Gordon is a cricketer. Over a long period he knows that his population mean score, in number of runs per innings, is 28 , and the population standard deviation is 12 . In a new season he adopts a different batting style and he finds that in 30 innings using this style his mean score is 28.98 .
  1. Stating a necessary assumption, test at the \(5 \%\) significance level whether his population mean score has increased.
  2. Explain whether it was necessary to use the Central Limit Theorem in part (i).
OCR S2 2013 January Q7
9 marks Standard +0.8
7 The continuous random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). The mean of a random sample of \(n\) observations of \(X\) is denoted by \(\bar { X }\). It is given that \(\mathrm { P } ( \bar { X } < 35.0 ) = 0.9772\) and \(\mathrm { P } ( \bar { X } < 20.0 ) = 0.1587\).
  1. Obtain a formula for \(\sigma\) in terms of \(n\). Two students are discussing this question. Aidan says "If you were told another probability, for instance \(\mathrm { P } ( \bar { X } > 32 ) = 0.1\), you could work out the value of \(\sigma\)." Binya says, "No, the value of \(\mathrm { P } ( \bar { X } > 32 )\) is fixed by the information you know already."
  2. State which of Aidan and Binya is right. If you think that Aidan is right, calculate the value of \(\sigma\) given that \(\mathrm { P } ( \bar { X } > 32 ) = 0.1\). If you think that Binya is right, calculate the value of \(\mathrm { P } ( \bar { X } > 32 )\).
OCR S2 2013 January Q8
10 marks Standard +0.3
8 In a large city the number of traffic lights that fail in one day of 24 hours is denoted by \(Y\). It may be assumed that failures occur randomly.
  1. Explain what the statement "failures occur randomly" means.
  2. State, in context, two different conditions that must be satisfied if \(Y\) is to be modelled by a Poisson distribution, and for each condition explain whether you think it is likely to be met in this context.
  3. For this part you may assume that \(Y\) is well modelled by the distribution \(\operatorname { Po } ( \lambda )\). It is given that \(\mathrm { P } ( Y = 7 ) = \mathrm { P } ( Y = 8 )\). Use an algebraic method to calculate the value of \(\lambda\) and hence calculate the corresponding value of \(\mathrm { P } ( Y = 7 )\).
OCR S2 2013 January Q9
8 marks Standard +0.8
9 The random variable \(A\) has the distribution \(\mathrm { B } ( 30 , p )\). A test is carried out of the hypotheses \(\mathrm { H } _ { 0 } : p = 0.6\) against \(\mathrm { H } _ { 1 } : p < 0.6\). The critical region is \(A \leqslant 13\).
  1. State the probability that \(\mathrm { H } _ { 0 }\) is rejected when \(p = 0.6\).
  2. Find the probability that a Type II error occurs when \(p = 0.5\).
  3. It is known that on average \(p = 0.5\) on one day in five, and on other days the value of \(p\) is 0.6 . On each day two tests are carried out. If the result of the first test is that \(\mathrm { H } _ { 0 }\) is rejected, the value of \(p\) is adjusted if necessary, to ensure that \(p = 0.6\) for the rest of the day. Otherwise the value of \(p\) remains the same as for the first test. Calculate the probability that the result of the second test is to reject \(\mathrm { H } _ { 0 }\). \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}
OCR S2 2015 June Q1
6 marks Standard +0.3
1 The random variable \(Y\) is normally distributed with mean \(\mu\) and variance \(\sigma ^ { 2 }\). It is found that \(\mathrm { P } ( Y > 150.0 ) = 0.0228\) and \(\mathrm { P } ( Y > 143.0 ) = 0.9332\). Find the values of \(\mu\) and \(\sigma\).
OCR S2 2015 June Q2
6 marks Moderate -0.8
2 A class investigated the number of dead rabbits found along a particular stretch of road.
  1. The class agrees that dead rabbits occur randomly along the road. Explain what this statement means.
  2. State, in this context, an assumption needed for the number of dead rabbits in a fixed length of road to be modelled by a Poisson distribution, and explain what your statement means. Assume now that the number of dead rabbits in a fixed length of road can be well modelled by a Poisson distribution with mean 1 per 600 m of road.
  3. Use an appropriate formula, showing your working, to find the probability that in a road of length 1650 m there are exactly 3 dead rabbits.
OCR S2 2015 June Q3
10 marks Standard +0.3
3 A continuous random variable \(X\) has probability density function $$f ( x ) = \left\{ \begin{array} { c l } \frac { 3 } { 2 a ^ { 3 } } x ^ { 2 } & - a \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{array} \right.$$ where \(a\) is a constant.
  1. It is given that \(\mathrm { P } ( - 3 \leqslant X \leqslant 3 ) = 0.125\). Find the value of \(a\) in this case.
  2. It is given instead that \(\operatorname { Var } ( X ) = 1.35\). Find the value of \(a\) in this case.
  3. Explain the relationship between \(x\) and \(X\) in this question.
OCR S2 2015 June Q4
10 marks Standard +0.3
4 A continuous random variable is normally distributed with mean \(\mu\). A significance test for \(\mu\) is carried out, at the \(5 \%\) significance level, on 90 independent occasions.
  1. Given that the null hypothesis is correct on all 90 occasions, use a suitable approximation to find the probability that on 6 or fewer occasions the test results in a Type I error. Justify your approximation.
  2. Given instead that on all 90 occasions the probability of a Type II error is 0.35 , use a suitable approximation to find the probability that on fewer than 29 occasions the test results in a Type II error.
OCR S2 2015 June Q5
8 marks Standard +0.3
5
  1. State an advantage of using random numbers in selecting samples.
  2. It is known that in analysing the digits in large sets of financial records, the probability that the leading digit is 1 is 0.25 . A random sample of 18 leading digits from a certain large set of financial records is obtained and it is found that 8 of the leading digits are 1 s . Test, at the \(5 \%\) significance level, whether the probability that the leading digit is 1 in this set of records is greater than 0.25 .
OCR S2 2015 June Q6
12 marks Standard +0.3
6 Records for a doctors' surgery over a long period suggest that the time taken for a consultation, \(T\) minutes, has a mean of 11.0. Following the introduction of new regulations, a doctor believes that the average time has changed. She finds that, with new regulations, the consultation times for a random sample of 120 patients can be summarised as $$n = 120 , \Sigma t = 1411.20 , \Sigma t ^ { 2 } = 18737.712 .$$
  1. Test, at the \(10 \%\) significance level, whether the doctor's belief is correct.
  2. Explain whether, in answering part (i), it was necessary to assume that the consultation times were normally distributed.
OCR S2 2015 June Q7
13 marks Standard +0.3
7 A large railway network suffers points failures at an average rate of 1 every 3 days. Assume that the number of points failures can be modelled by a Poisson distribution. The network employs a new firm of engineers. After the new engineers have become established, it is found that in a randomly chosen period of 15 days there are 2 instances of points failures.
  1. Test, at the \(5 \%\) significance level, whether there is evidence that the mean number of points failures has been reduced.
  2. A new test is carried out over a period of 150 days. Use a suitable approximation to find the greatest number of points failures there could be in 150 days that would lead to a \(5 \%\) significance test concluding that the average number of points failures had been reduced.
OCR S2 2015 June Q8
7 marks Standard +0.8
8 The random variable \(S\) has the distribution \(\mathrm { B } ( 14 , p )\). A significance test is carried out of the null hypothesis \(\mathrm { H } _ { 0 } : p = 0.3\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : p > 0.3\). The critical region for the test is \(S \geqslant 8\).
  1. Find the significance level of the test, correct to 3 significant figures.
  2. It is given that, on each occasion that the test is carried out, the true value of \(p\) is equally likely to be \(0.3,0.5\) or 0.7 , independently of any other test. Four independent tests are carried out. Find the probability that at least one of the tests results in a Type II error.
OCR S2 2016 June Q1
4 marks Easy -1.2
1 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 \(\mathrm { E } ( V )\) and \(\operatorname { Var } ( V )\).
OCR S2 2016 June Q2
6 marks Standard +0.3
2 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 .
OCR S2 2016 June Q3
7 marks Moderate -0.8
3 The random variable \(F\) has the distribution \(\mathrm { B } ( 40,0.65 )\). Use a suitable approximation to find \(\mathrm { P } ( F \leqslant 30 )\), justifying your approximation.
OCR S2 2016 June Q4
5 marks Standard +0.3
4 It is given that \(Y \sim \operatorname { Po } ( \lambda )\), where \(\lambda \neq 0\), and that \(\mathrm { P } ( Y = 4 ) = \mathrm { P } ( Y = 5 )\). Write down an equation for \(\lambda\). Hence find the value of \(\lambda\) and the corresponding value of \(\mathrm { P } ( Y = 5 )\).
\(555 \%\) 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.
  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.
OCR S2 2016 June Q6
12 marks Moderate -0.8
6 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. At a quiet time of the day, \(\lambda = 6.50\). Assuming that a Poisson distribution is valid, calculate \(\mathrm { P } ( 4 \leqslant X < 8 )\).
  3. At a busy time of the day, \(\lambda = 30\).
    (a) Assuming that a Poisson distribution is valid, use a suitable approximation to find \(\mathrm { P } ( X > 35 )\). Justify your approximation.
    (b) Give a reason why a Poisson distribution might not be valid in this context when \(\lambda = 30\).
OCR S2 2016 June Q7
11 marks Standard +0.3
7 A continuous random variable \(X\) has probability density function $$f ( x ) = \left\{ \begin{array} { c c } a x ^ { - 3 } + b x ^ { - 4 } & x \geqslant 1 \\ 0 & \text { otherwise } \end{array} \right.$$ where \(a\) and \(b\) are constants.
  1. Explain what the letter \(x\) represents. It is given that \(\mathrm { P } ( X > 2 ) = \frac { 3 } { 16 }\).
  2. Show that \(a = 1\), and find the value of \(b\).
  3. Find \(\mathrm { E } ( X )\).
OCR S2 2016 June Q8
13 marks Standard +0.3
8 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.
  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.
  3. Explain whether the Central Limit Theorem is needed in these tests.
OCR S2 2016 June Q9
6 marks Challenging +1.2
9 The random variable \(R\) has the distribution \(\operatorname { Po } ( \lambda )\). A significance test is carried out at the \(1 \%\) level of the null hypothesis \(\mathrm { H } _ { 0 } : \lambda = 11\) against \(\mathrm { H } _ { 1 } : \lambda > 11\), 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. \section*{END OF QUESTION PAPER}