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OCR MEI S3 2013 June Q4
18 marks Standard +0.3
4 A company that makes meat pies includes a "small" size in its product range. These pies consist of a pastry case and meat filling, the weights of which are independent of each other. The weight of the pastry case, \(C\), is Normally distributed with mean 96 g and variance \(21 \mathrm {~g} ^ { 2 }\). The weight of the meat filling, \(M\), is Normally distributed with mean 57 g and variance \(14 \mathrm {~g} ^ { 2 }\).
  1. Find the probability that, in a randomly chosen pie, the weight of the pastry case is between 90 and 100 g .
  2. The wrappers on the pies state that the weight is 145 g . Find the proportion of pies that are underweight.
  3. The pies are sold in packs of 4 . Find the value of \(w\) such that, in \(95 \%\) of packs, the total weight of the 4 pies in a randomly chosen pack exceeds \(w \mathrm {~g}\).
  4. It is required that the weight of the meat filling in a pie should be at least \(35 \%\) of the total weight. Show that this means that \(0.65 M - 0.35 C \geqslant 0\). Hence find the probability that, in a randomly chosen pie, this requirement is met.
OCR MEI S3 2014 June Q1
17 marks Standard +0.3
1
  1. Let \(X\) be a random variable with variance \(\sigma ^ { 2 }\). The independent random variables \(X _ { 1 }\) and \(X _ { 2 }\) are both distributed as \(X\). Write down the variances of \(X _ { 1 } + X _ { 2 }\) and \(2 X\); explain why they are different. A large company has produced an aptitude test which consists of three parts. The parts are called mathematical ability, spatial awareness and communication. The scores obtained by candidates in the three parts are continuous random variables \(X , Y\) and \(W\) which have been found to have independent Normal distributions with means and standard deviations as shown in the table.
    MeanStandard deviation
    Mathematical ability, \(X\)30.15.1
    Spatial awareness, \(Y\)25.44.2
    Communication, \(W\)28.23.9
  2. Find the probability that a randomly selected candidate obtains a score of less than 22 in the mathematical ability part of the test.
  3. Find the probability that a randomly selected candidate obtains a total score of at least 100 in the whole test.
  4. For a particular role in the company, the score \(2 X + Y\) is calculated. Find the score that is exceeded by only \(2 \%\) of candidates.
  5. For a different role, a candidate must achieve a score in communication which is at least \(60 \%\) of the score obtained in mathematical ability. What proportion of candidates do not achieve this?
OCR MEI S3 2014 June Q2
19 marks Standard +0.3
2
  1. Explain what is meant by a simple random sample. A manufacturer produces tins of paint which nominally contain 1 litre. The quantity of paint delivered by the machine that fills the tins can be assumed to be a Normally distributed random variable. The machine is designed to deliver an average of 1.05 litres to each tin. However, over time paint builds up in the delivery nozzle of the machine, reducing the quantity of paint delivered. Random samples of 10 tins are taken regularly from the production process. If a significance test, carried out at the \(5 \%\) level, suggests that the average quantity of paint delivered is less than 1.02 litres, the machine is cleaned.
  2. By carrying out an appropriate test, determine whether or not the sample below leads to the machine being cleaned. $$\begin{array} { l l l l l l l l l l } 0.994 & 1.010 & 1.021 & 1.015 & 1.016 & 1.022 & 1.009 & 1.007 & 1.011 & 1.026 \end{array}$$ Each time the machine has been cleaned, a random sample of 10 tins is taken to determine whether or not the average quantity of paint delivered has returned to 1.05 litres.
  3. On one occasion after the machine has been cleaned, the quality control manager thinks that the distribution of the quantity of paint is symmetrical but not necessarily Normal. The sample on this occasion is as follows.
    1.0551.0641.0631.0431.0621.0701.0591.0441.054
    1.053
    By carrying out an appropriate test at the \(5 \%\) level of significance, determine whether or not this sample supports the conclusion that the average quantity of paint delivered is 1.05 litres.
OCR MEI S3 2014 June Q3
19 marks Standard +0.3
3
  1. A personal trainer believes that drinking a glass of beetroot juice an hour before exercising enables endurance tests to be completed more quickly. To test his belief he takes a random sample of 12 of his trainees and, on two occasions, asks them to carry out 100 repetitions of a particular exercise as quickly as possible. Each trainee drinks a glass of water on one occasion and a glass of beetroot juice on the other occasion. The times in seconds taken by the trainees are given in the table.
    TraineeWaterBeetroot juice
    A75.172.9
    B86.279.9
    C77.371.6
    D89.190.2
    E67.968.2
    F101.595.2
    G82.576.5
    H83.380.2
    I102.599.1
    J91.382.2
    K92.590.1
    L77.277.9
    The trainer wishes to test his belief using a paired \(t\) test at the \(1 \%\) level of significance. Assuming any necessary assumptions are valid, carry out a test of the hypotheses \(\mathrm { H } _ { 0 } : \mu _ { D } = 0 , \mathrm { H } _ { 1 } : \mu _ { D } < 0\), where \(\mu _ { D }\) is the population mean difference in times (time with beetroot juice minus time with water).
  2. An ornithologist believes that the number of birds landing on the bird feeding station in her garden in a given interval of time during the morning should follow a Poisson distribution. In order to test her belief, she makes the following observations in 60 randomly chosen minutes one morning.
    Number of birds0123456\(\geqslant 7\)
    Frequency25101714741
    Given that the data in the table have a mean value of 3.3, use a goodness of fit test, with a significance level of \(5 \%\), to investigate whether the ornithologist is justified in her belief.
OCR MEI S3 2014 June Q4
17 marks Challenging +1.2
4 The probability density function of a random variable \(X\) is given by $$\mathrm { f } ( x ) = \begin{cases} k x & 0 \leqslant x \leqslant a \\ k ( 2 a - x ) & a < x \leqslant 2 a \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) and \(k\) are positive constants.
  1. Sketch \(\mathrm { f } ( x )\). Hence explain why \(\mathrm { E } ( X ) = a\).
  2. Show that \(k = \frac { 1 } { a ^ { 2 } }\).
  3. Find \(\operatorname { Var } ( X )\) in terms of \(a\). In order to estimate the value of \(a\), a random sample of size 50 is taken from the distribution. It is found that the sample mean and standard deviation are \(\bar { x } = 1.92\) and \(s = 0.8352\).
  4. Construct a symmetrical \(95 \%\) confidence interval for \(a\). Give one reason why the answer is only approximate.
  5. A non-statistician states that the probability that \(a\) lies in the interval found in part (iv) is 0.95 . Comment on this statement. \section*{END OF QUESTION PAPER} \section*{OCR \(^ { \text {® } }\)}
OCR MEI S3 2016 June Q1
18 marks Standard +0.8
1 A game consists of 20 rounds. Each round is denoted as either a starter, middle or final round. The times taken for each round are independently and Normally distributed with the following parameters (given in seconds).
Type of roundMeanStandard deviation
Starter20015
Middle22025
Final25020
The game consists of 4 starter, 12 middle and 4 final rounds. Find the probability that
  1. the mean time per round for the 4 final rounds will exceed 260 seconds,
  2. all 20 rounds will be completed in a total time of 75 minutes or less,
  3. the 12 middle rounds will take at least 3.5 times as long in total as the 4 starter rounds,
  4. the mean time per round for the 12 middle rounds will be at least 25 seconds less than the mean time per round for the 4 final rounds.
OCR MEI S3 2016 June Q2
18 marks Standard +0.3
2
  1. A genetic model involving body colour and eye colour of fruit flies predicts that offspring will consist of four phenotypes in the ratio \(9 : 3 : 3 : 1\). A random sample of 200 such offspring is taken. Their phenotypes are found to be as follows.
    PhenotypeBrown body Red eyeBrown body Brown eyeBlack body Red eyeBlack body Brown eye
    Frequency12537326
    Relative proportion from model9331
    Carry out a test, using a \(2.5 \%\) level of significance, of the goodness of fit of the genetic model to these data.
  2. The median length of European fruit flies is 2.5 mm . South American fruit flies are believed to be larger than European fruit flies. A random sample of 12 South American fruit flies is taken. The flies are found to have the following lengths (in mm). \(1.7 \quad 1.4\) \(3.1 \quad 3.5\) 3.8
    4.2
    2.2
    2.9
    4.4
    2.6 \(3.9 \quad 3.2\) Carry out a Wilcoxon signed rank test, using a \(5 \%\) level of significance, to test this belief.
OCR MEI S3 2016 June Q3
18 marks Standard +0.3
3 The random variable \(X\) has the following probability density function: $$\mathrm { f } ( x ) = \begin{cases} k \left( 1 - x ^ { 2 } \right) & - 1 \leqslant x \leqslant 1 \\ 0 & \text { elsewhere } \end{cases}$$ where \(k\) is a positive constant.
  1. Calculate the value of \(k\).
  2. Sketch the probability density function.
  3. Calculate \(\operatorname { Var } ( X )\).
  4. Find a cubic equation satisfied by the upper quartile \(q\), and hence verify that \(q = 0.35\) to 2 decimal places.
  5. A random sample of 40 values of \(X\) is taken. Using a suitable approximating distribution, calculate the probability that the mean of these values is greater than 0.125 . Justify your choice of distribution.
OCR MEI S3 2016 June Q4
18 marks Standard +0.3
4 An insurance company is investigating a new system designed to reduce the average time taken to process claim forms. The company has decided to use 10 experienced employees to process claims using the old system and the new system. Two procedures for comparing the systems are proposed.
Procedure \(A\) There are two sets of claim forms, set 1 and set 2. Each contains the same number of forms. Each employee processes set 1 on the old system and set 2 on the new system. The times taken are compared. Procedure \(B\) There is just one set of claim forms which each employee processes firstly on the old system and then on the new system. The times taken are compared.
  1. State one weakness of each of these procedures. In fact a third procedure which avoids these two weaknesses is adopted. In this procedure each employee is given a randomly selected set of claim forms. Each set contains the same number of forms. The employees each process their set of claim forms on both systems. The times taken, in minutes, are shown in the table.
    Employee12345678910
    Old system40.542.952.851.777.266.765.249.255.658.3
    New system39.240.750.650.771.470.571.147.752.155.5
  2. Carry out a paired \(t\) test at the \(5 \%\) level of significance to investigate whether the mean length of time taken to process a set of forms has reduced using the new system.
  3. State fully the usual conditions for a paired \(t\) test.
  4. Construct a \(99 \%\) confidence interval for the mean reduction in time taken to process a set of forms using the new system.
OCR S4 2009 June Q2
11 marks Standard +0.8
2 A company wishes to buy a new lathe for making chair legs. Two models of lathe, 'Allegro' and 'Vivace', were trialled. The company asked 12 randomly selected employees to make a particular type of chair leg on each machine. The times, in seconds, for each employee are shown in the table.
Employee123456789101112
Time on Allegro162111194159202210183168165150185160
Time on Vivace182130193181192205186184192180178189
The company wishes to test whether there is any difference in average times for the two machines.
  1. State the circumstances under which a non-parametric test should be used.
  2. Use two different non-parametric tests and show that they lead to different conclusions at the 5\% significance level.
  3. State, with a reason, which conclusion is to be preferred.
OCR S4 2009 June Q3
9 marks Standard +0.8
3 The continuous random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} \mathrm { e } ^ { 2 x } & x < 0 \\ \mathrm { e } ^ { - 2 x } & x \geqslant 0 \end{cases}$$
  1. Show that the moment generating function of \(X\) is \(\frac { 4 } { 4 - t ^ { 2 } }\), where \(| t | < 2\), and explain why the condition \(| t | < 2\) is necessary.
  2. Find \(\operatorname { Var } ( X )\).
OCR S4 2009 June Q4
10 marks Challenging +1.2
4 The probability generating function of the discrete random variable \(Y\) is given by $$\mathrm { G } _ { Y } ( t ) = \frac { a + b t ^ { 3 } } { t }$$ where \(a\) and \(b\) are constants.
  1. Given that \(\mathrm { E } ( Y ) = - 0.7\), find the values of \(a\) and \(b\).
  2. Find \(\operatorname { Var } ( Y )\).
  3. Find the probability that the sum of 10 random observations of \(Y\) is - 7 .
OCR S4 2009 June Q5
13 marks Standard +0.3
5 Alana and Ben work for an estate agent. The joint probability distribution of the number of houses they sell in a randomly chosen week, \(X _ { A }\) and \(X _ { B }\) respectively, is shown in the table. \includegraphics[max width=\textwidth, alt={}, center]{f1879b0f-17e3-41b4-af38-a843b67c5301-3_405_602_370_781}
  1. Find \(\mathrm { E } \left( X _ { A } \right)\) and \(\operatorname { Var } \left( X _ { A } \right)\).
  2. Determine whether \(X _ { A }\) and \(X _ { B }\) are independent.
  3. Given that \(\mathrm { E } \left( X _ { B } \right) = 1.15 , \operatorname { Var } \left( X _ { B } \right) = 0.8275\) and \(\mathrm { E } \left( X _ { A } X _ { B } \right) = 1.09\), find \(\operatorname { Cov } \left( X _ { A } , X _ { B } \right)\) and \(\operatorname { Var } \left( X _ { A } - X _ { B } \right)\).
  4. During a particular week only one house was sold by Alana and Ben. Find the probability that it was sold by Alana.
OCR S4 2009 June Q6
13 marks Challenging +1.8
6 The continuous random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} 0 & x < a , \\ \mathrm { e } ^ { - ( x - a ) } & x \geqslant a , \end{cases}$$ where \(a\) is a constant. \(X _ { 1 } , X _ { 2 } , \ldots , X _ { n }\) are \(n\) independent observations of \(X\), where \(n \geqslant 4\).
  1. Show that \(\mathrm { E } ( X ) = a + 1\). \(T _ { 1 }\) and \(T _ { 2 }\) are proposed estimators of \(a\), where $$T _ { 1 } = X _ { 1 } + 2 X _ { 2 } - X _ { 3 } - X _ { 4 } - 1 \quad \text { and } \quad T _ { 2 } = \frac { X _ { 1 } + X _ { 2 } } { 4 } + \frac { X _ { 3 } + X _ { 4 } + \ldots + X _ { n } } { 2 ( n - 2 ) } - 1 .$$
  2. Show that \(T _ { 1 }\) and \(T _ { 2 }\) are unbiased estimators of \(a\).
  3. Determine which is the more efficient estimator.
  4. Suggest another unbiased estimator of \(a\) using all of the \(n\) observations.
OCR S4 2009 June Q7
11 marks Standard +0.3
7 A particular disease occurs in a proportion \(p\) of the population of a town. A diagnostic test has been developed, in which a positive result indicates the presence of the disease. It has a probability 0.98 of giving a true positive result, i.e. of indicating the presence of the disease when it is actually present. The test will give a false positive result with probability 0.08 when the disease is not present. A randomly chosen person is given the test.
  1. Find, in terms of \(p\), the probability that
    1. the person has the disease when the result is positive,
    2. the test will lead to a wrong conclusion. It is decided that if the result of the test on someone is positive, that person is tested again. The result of the second test is independent of the result of the first test.
    3. Find the probability that the person has the disease when the result of the second test is positive.
    4. The town has 24000 children and plans to test all of them at a cost of \(\pounds 5\) per test. Assuming that \(p = 0.001\), calculate the expected total cost of carrying out these tests.
OCR S4 2010 June Q1
4 marks Standard +0.3
1 For the variables \(A\) and \(B\), it is given that \(\operatorname { Var } ( A ) = 9 , \operatorname { Var } ( B ) = 6\) and \(\operatorname { Var } ( 2 A - 3 B ) = 18\).
  1. Find \(\operatorname { Cov } ( A , B )\).
  2. State with a reason whether \(A\) and \(B\) are independent.
OCR S4 2010 June Q2
6 marks Standard +0.3
2 The probability generating function of the discrete random variable \(X\) is \(\frac { \mathrm { e } ^ { 4 t ^ { 2 } } } { \mathrm { e } ^ { 4 } }\). Find
  1. \(\mathrm { E } ( X )\),
  2. \(\mathrm { P } ( X = 2 )\). \(3 X _ { 1 }\) and \(X _ { 2 }\) are continuous random variables. Random samples of 5 observations of \(X _ { 1 }\) and 6 observations of \(X _ { 2 }\) are taken. No two observations are equal. The 11 observations are ranked, lowest first, and the sum of the ranks of the observations of \(X _ { 1 }\) is denoted by \(R\).
OCR S4 2010 June Q4
10 marks Standard +0.8
4 The moment generating function of a continuous random variable \(Y\), which has a \(\chi ^ { 2 }\) distribution with \(n\) degrees of freedom, is \(( 1 - 2 t ) ^ { - \frac { 1 } { 2 } n }\), where \(0 \leqslant t < \frac { 1 } { 2 }\).
  1. Find \(\mathrm { E } ( Y )\) and \(\operatorname { Var } ( Y )\). For the case \(n = 1\), the sum of 60 independent observations of \(Y\) is denoted by \(S\).
  2. Write down the moment generating function of \(S\) and hence identify the distribution of \(S\).
  3. Use a normal approximation to estimate \(\mathrm { P } ( S \geqslant 70 )\).
OCR S4 2010 June Q5
11 marks Standard +0.3
5 In order to test whether the median salary of employees in a certain industry who had worked for three years was \(\pounds 19500\), the salaries \(x\), in thousands of pounds, of 50 randomly chosen employees were obtained.
  1. The values \(| x - 19.5 |\) were calculated and ranked. No two values of \(x\) were identical and none was equal to 19.5 . The sum of the ranks corresponding to positive values of \(( x - 19.5 )\) was 867. Stating a required assumption, carry out a suitable test at the \(5 \%\) significance level.
  2. If the assumption you stated in part (i) does not hold, what test could have been used?
OCR S4 2010 June Q6
13 marks Standard +0.8
6 Nuts and raisins occur in randomly chosen squares of a particular brand of chocolate. The numbers of nuts and raisins are denoted by \(N\) and \(R\) respectively and the joint probability distribution of \(N\) and \(R\) is given by $$f ( n , r ) = \begin{cases} c ( n + 2 r ) & n = 0,1,2 \text { and } r = 0,1,2 \\ 0 & \text { otherwise } \end{cases}$$ where \(c\) is a constant.
  1. Find the value of \(c\).
  2. Find the probability that there is exactly one nut in a randomly chosen square.
  3. Find the probability that the total number of nuts and raisins in a randomly chosen square is more than 2 .
  4. For squares in which there are 2 raisins, find the mean number of nuts.
  5. Determine whether \(N\) and \(R\) are independent.
OCR S4 2010 June Q7
15 marks Challenging +1.2
7 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { x } { 2 \theta ^ { 2 } } & 0 \leqslant x \leqslant 2 \theta \\ 0 & \text { otherwise } \end{cases}$$ where \(\theta\) is an unknown positive constant.
  1. Find \(\mathrm { E } \left( X ^ { n } \right)\), where \(n \neq - 2\), and hence write down the value of \(\mathrm { E } ( X )\).
  2. Find
    1. \(\operatorname { Var } ( X )\),
    2. \(\operatorname { Var } \left( X ^ { 2 } \right)\).
    3. Find \(\mathrm { E } \left( X _ { 1 } + X _ { 2 } + X _ { 3 } \right)\) and \(\mathrm { E } \left( X _ { 1 } ^ { 2 } + X _ { 2 } ^ { 2 } + X _ { 3 } ^ { 2 } \right)\), where \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\) are independent observations of \(X\). Hence construct unbiased estimators, \(T _ { 1 }\) and \(T _ { 2 }\), of \(\theta\) and \(\operatorname { Var } ( X )\) respectively, which are based on \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\).
    4. Find \(\operatorname { Var } \left( T _ { 2 } \right)\).
OCR S4 2010 June Q8
6 marks Standard +0.3
8 For the events \(L\) and \(M , \mathrm { P } ( L \mid M ) = 0.2 , \mathrm { P } ( M \mid L ) = 0.4\) and \(\mathrm { P } ( M ) = 0.6\).
  1. Find \(\mathrm { P } ( L )\) and \(\mathrm { P } \left( L ^ { \prime } \cup M ^ { \prime } \right)\).
  2. Given that, for the event \(N , \mathrm { P } ( N \mid ( L \cap M ) ) = 0.3\), find \(\mathrm { P } \left( L ^ { \prime } \cup M ^ { \prime } \cup N ^ { \prime } \right)\).
OCR S4 2015 June Q1
5 marks Moderate -0.8
1 For the events \(A\) and \(B\) it is given that $$\mathrm { P } ( A ) = 0.6 , \mathrm { P } ( B ) = 0.3 \text { and } \mathrm { P } ( A \text { or } B \text { but not both } ) = 0.4 \text {. }$$
  1. Find \(\mathrm { P } ( A \cap B )\).
  2. Find \(\mathrm { P } \left( A ^ { \prime } \cap B \right)\).
  3. State, giving a reason, whether \(A\) and \(B\) are independent.
OCR S4 2015 June Q2
8 marks Standard +0.3
2 The manufacturer of a painkiller, designed to relieve headaches, claims that people taking the painkiller feel relief in at most 30 minutes, on average. A random sample of eight users of the painkiller recorded the times it took for them to feel relief from their headaches. These times, in minutes, were as follows: $$\begin{array} { l l l l l l l l } 33 & 39 & 29 & 35 & 40 & 32 & 26 & 37 \end{array}$$ Use a Wilcoxon single-sample signed-rank test at the \(5 \%\) significance level to test the manufacturer's claim, stating a necessary assumption.
OCR S4 2015 June Q3
6 marks Challenging +1.2
3 The manufacturer of electronic components uses the following process to test the proportion of defective items produced. A random sample of 20 is taken from a large batch of components.
  • If no defective item is found, the batch is accepted.
  • If two or more defective items are found, the batch is rejected.
  • If one defective item is found, a second random sample of 20 is taken. If two or more defective items are found in this second sample, the batch is rejected, otherwise the batch is accepted.
The proportion of defective items in the batch is denoted by \(p\), and \(q = 1 - p\).
  1. Show that the probability that a batch is accepted is \(q ^ { 20 } + 20 p q ^ { 38 } ( q + 20 p )\). For a particular component, \(p = 0.01\).
  2. Given that a batch is accepted, find the probability that it is accepted as a result of the first sample.