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OCR S3 2016 June Q5
11 marks Standard +0.8
5 The independent random variables \(X\) and \(Y\) have distributions \(\mathrm { N } \left( 30 , \sigma ^ { 2 } \right)\) and \(\mathrm { N } \left( 20 , \sigma ^ { 2 } \right)\) respectively. The random variable \(a X + b Y\), where \(a\) and \(b\) are constants, has the distribution \(\mathrm { N } \left( 410,130 \sigma ^ { 2 } \right)\).
  1. Given that \(a\) and \(b\) are integers, find the value of \(a\) and the value of \(b\).
  2. Given that \(\mathrm { P } ( X > Y ) = 0.966\), find \(\sigma ^ { 2 }\).
OCR S3 2016 June Q6
11 marks Challenging +1.2
6 The masses at birth, in kg, of random samples of babies were recorded for each of the years 1970 and 2010. The table shows the sample mean and an unbiased estimate of the population variance for each year.
YearNo. of babies
Sample
mean
Unbiased estimate of
population variance
19702853.3030.2043
20102603.3520.2323
  1. A researcher tests the null hypothesis that babies born in 2010 are 0.04 kg heavier, on average, than babies born in 1970, against the alternative hypothesis that they are more than 0.04 kg heavier on average. Show that, at the \(5 \%\) level of significance, the null hypothesis is not rejected.
  2. Another researcher chooses samples of equal size, \(n\), for the two years. Using the same hypothesis as in part (i), she finds that the null hypothesis is rejected at the \(5 \%\) level of significance. Assuming that the sample means and unbiased estimates of population variance for the two years are as given in the table, find the smallest possible value of \(n\).
OCR S3 2016 June Q7
12 marks Standard +0.8
7 A continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} a x ^ { 3 } & 0 \leqslant x \leqslant 1 \\ a x ^ { 2 } & 1 < x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.
  1. Show that \(a = \frac { 12 } { 31 }\).
  2. Find \(\mathrm { E } ( X )\). It is thought that the time taken by a student to complete a task can be well modelled by \(X\). The times taken by 992 randomly chosen students are summarised in the table, together with some of the expected frequencies.
    Time\(0 \leqslant x < 0.5\)\(0.5 \leqslant x < 1\)\(1 \leqslant x < 1.5\)\(1.5 \leqslant x \leqslant 2\)
    Observed frequency892279613
    Expected frequency690
  3. Find the other expected frequencies and test, at the \(5 \%\) level of significance, whether the data can be well modelled by \(X\).
OCR S3 2016 June Q8
10 marks Challenging +1.2
8 The radius, \(R\), of a sphere is a random variable with a continuous uniform distribution between 0 and 10 .
  1. Find the cumulative distribution function and probability density function of \(A\), the surface area of the sphere.
  2. Find \(\mathrm { P } ( \mathrm { A } \leqslant 200 \pi )\). \section*{END OF QUESTION PAPER}
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
    (a) the person has the disease when the result is positive,
    (b) 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.
  2. Find the probability that the person has the disease when the result of the second test is positive.
  3. 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
    (a) \(\operatorname { Var } ( X )\),
    (b) \(\operatorname { Var } \left( X ^ { 2 } \right)\).
  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
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.
OCR S4 2015 June Q4
9 marks Challenging +1.2
4 The discrete random variable \(Y\) has probability generating function $$\mathrm { G } _ { Y } ( t ) = 0.09 t ^ { 2 } + 0.24 t ^ { 3 } + 0.34 t ^ { 4 } + 0.24 t ^ { 5 } + 0.09 t ^ { 6 }$$
  1. Find the mean and variance of \(Y\). \(Y\) is the sum of two independent observations of a random variable \(X\).
  2. Find the probability generating function of \(X\), expressing your answer as a cubic polynomial in \(t\).
  3. Write down the value of \(\mathrm { P } ( X = 2 )\).
OCR S4 2015 June Q5
9 marks Standard +0.3
5 The random variable \(X\) has a Poisson distribution with mean \(\lambda\). It is given that the moment generating function of \(X\) is \(e ^ { \lambda \left( e ^ { t } - 1 \right) }\).
  1. Use the moment generating function to verify that the mean of \(X\) is \(\lambda\), and to show that the variance of \(X\) is also \(\lambda\).
  2. Five independent observations of \(X\) are added to produce a new variable \(Y\). Find the moment generating function of \(Y\), simplifying your answer.
OCR S4 2015 June Q6
9 marks Moderate -0.8
6 In a two-tail Wilcoxon rank-sum test, the sample sizes are 13 and 15. The sum of the ranks for the sample of size 13 is 135 . Carry out the test at the \(5 \%\) level of significance.
OCR S4 2015 June Q7
14 marks Challenging +1.2
7 The discrete random variable \(X\) can take the values 0,1 and 2 with equal probabilities.
The random variables \(X _ { 1 }\) and \(X _ { 2 }\) are independent observations of \(X\), and the random variables \(Y\) and \(Z\) are defined as follows: \(Y\) is the smaller of \(X _ { 1 }\) and \(X _ { 2 }\), or their common value if they are equal; \(Z = \left| X _ { 1 } - X _ { 2 } \right|\).
  1. Draw up a table giving the joint distribution of \(Y\) and \(Z\).
  2. Find \(P ( Y = 0 \mid Z = 0 )\).
  3. Find \(\operatorname { Cov } ( Y , Z )\).
OCR S4 2015 June Q8
12 marks Challenging +1.2
8 The independent random variables \(X _ { 1 }\) and \(X _ { 2 }\) have the distributions \(\mathrm { B } \left( n _ { 1 } , \theta \right)\) and \(\mathrm { B } \left( n _ { 2 } , \theta \right)\) respectively. Two possible estimators for \(\theta\) are $$T _ { 1 } = \frac { 1 } { 2 } \left( \frac { X _ { 1 } } { n _ { 1 } } + \frac { X _ { 2 } } { n _ { 2 } } \right) \text { and } T _ { 2 } = \frac { X _ { 1 } + X _ { 2 } } { n _ { 1 } + n _ { 2 } } .$$
  1. Show that \(T _ { 1 }\) and \(T _ { 2 }\) are both unbiased estimators, and calculate their variances.
  2. Find \(\frac { \operatorname { Var } \left( T _ { 1 } \right) } { \operatorname { Var } \left( T _ { 2 } \right) }\). Given that \(n _ { 1 } \neq n _ { 2 }\), use the inequality \(\left( n _ { 1 } - n _ { 2 } \right) ^ { 2 } > 0\) to find which of \(T _ { 1 }\) and \(T _ { 2 }\) is the more efficient estimator.