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OCR S2 2011 June Q7
14 marks Standard +0.8
7 The number of customer complaints received by a company per day is denoted by \(X\). Assume that \(X\) has the distribution \(\operatorname { Po } ( 2.2 )\).
  1. In a week of 5 working days, the probability there are at least \(n\) customer complaints is 0.146 correct to 3 significant figures. Use tables to find the value of \(n\).
  2. Use a suitable approximation to find the probability that in a period of 20 working days there are fewer than 38 customer complaints. A week of 5 working days in which at least \(n\) customer complaints are received, where \(n\) is the value found in part (i), is called a 'bad' week.
  3. Use a suitable approximation to find the probability that, in 40 randomly chosen weeks, more than 7 are bad.
OCR S2 2011 June Q8
13 marks Standard +0.3
8
  1. A group of students is discussing the conditions that are needed if a Poisson distribution is to be a good model for the number of telephone calls received by a fire brigade on a working day.
    1. Alice says "Events must be independent". Explain why this condition may not hold in this context.
    2. State a different condition that is needed. Explain whether it is likely to hold in this context.
  2. The random variables \(R , S\) and \(T\) have independent Poisson distributions with means \(\lambda , \mu\) and \(\lambda + \mu\) respectively.
    1. In the case \(\lambda = 2.74\), find \(\mathrm { P } ( R > 2 )\).
    2. In the case \(\lambda = 2\) and \(\mu = 3\), find \(\mathrm { P } ( R = 0\) and \(S = 1 ) + \mathrm { P } ( R = 1\) and \(S = 0 )\). Give your answer correct to 4 decimal places.
    3. In the general case, show algebraically that $$\mathrm { P } ( R = 0 \text { and } S = 1 ) + \mathrm { P } ( R = 1 \text { and } S = 0 ) = \mathrm { P } ( T = 1 ) .$$
OCR S2 2012 June Q1
2 marks Easy -1.8
1 In one day's production, a machine produces 1000 CDs . Explain how to take a random sample of 15 CDs chosen from one day's production.
OCR S2 2012 June Q2
6 marks Standard +0.3
2
  1. For the continuous random variable \(V\), it is known that \(\mathrm { E } ( V ) = 72.0\). The mean of a random sample of 40 observations of \(V\) is denoted by \(\bar { V }\). Given that \(\mathrm { P } ( \bar { V } < 71.2 ) = 0.35\), estimate the value of \(\operatorname { Var } ( V )\).
  2. Explain why you need to use the Central Limit Theorem in part (i), and why its use is justified.
OCR S2 2012 June Q3
7 marks Moderate -0.3
3 It is known that on average one person in three prefers the colour of a certain object to be blue. In a psychological test, 12 randomly chosen people were seated in a room with blue walls, and asked to state independently which colour they preferred for the object. Seven of the 12 people said that they preferred blue. Carry out a significance test, at the \(5 \%\) level, of whether the statement "on average one person in three prefers the colour of the object to be blue" is true for people who are seated in a room with blue walls.
OCR S2 2012 June Q4
11 marks Moderate -0.3
4 In a rock, small crystal formations occur at a constant average rate of 3.2 per cubic metre.
  1. State a further assumption needed to model the number of crystal formations in a fixed volume of rock by a Poisson distribution. In the remainder of the question, you should assume that a Poisson model is appropriate.
  2. Calculate the probability that in one cubic metre of rock there are exactly 5 crystal formations.
  3. Calculate the probability that in 0.74 cubic metres of rock there are at least 3 crystal formations.
  4. Use a suitable approximation to calculate the probability that in 10 cubic metres of rock there are at least 36 crystal formations.
OCR S2 2012 June Q5
11 marks Moderate -0.3
5 The acidity \(A\) (measured in pH ) of soil of a particular type has a normal distribution. The pH values of a random sample of 80 soil samples from a certain region can be summarised as $$\Sigma a = 496 , \quad \Sigma a ^ { 2 } = 3126 .$$ Test, at the \(10 \%\) significance level, whether in this region the mean pH of soil is 6.1 .
OCR S2 2012 June Q6
11 marks Moderate -0.3
6 At a tourist car park, a survey is made of the regions from which cars come.
  1. It is given that \(40 \%\) of cars come from the London region. Use a suitable approximation to find the probability that, in a random sample of 32 cars, more than 17 come from the London region. Justify your approximation.
  2. It is given that \(1 \%\) of cars come from France. Use a suitable approximation to find the probability that, in a random sample of 90 cars, exactly 3 come from France.
OCR S2 2012 June Q7
12 marks Standard +0.3
7 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} k x ^ { 2 } & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) and \(k\) are constants.
  1. Sketch the graph of \(y = \mathrm { f } ( x )\) and explain in non-technical language what this tells you about \(X\).
  2. Given that \(\mathrm { E } ( X ) = 4.5\), find
    (a) the value of \(a\),
    (b) \(\operatorname { Var } ( X )\).
OCR S2 2012 June Q8
12 marks Standard +0.8
8 The random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , 8 ^ { 2 } \right)\). A test is carried out, at the \(5 \%\) significance level, of \(\mathrm { H } _ { 0 } : \mu = 30\) against \(\mathrm { H } _ { 1 } : \mu > 30\), based on a random sample of size 18 .
  1. Find the critical region for the test.
  2. If \(\mu = 30\) and the outcome of the test is that \(\mathrm { H } _ { 0 }\) is rejected, state the type of error that is made. On a particular day this test is carried out independently a total of 20 times, and for 4 of these tests the outcome is that \(\mathrm { H } _ { 0 }\) is rejected. It is known that the value of \(\mu\) remains the same throughout these 20 tests.
  3. Find the probability that \(\mathrm { H } _ { 0 }\) is rejected at least 4 times if \(\mu = 30\). Hence state whether you think that \(\mu = 30\), giving a reason.
  4. Given that the probability of making an error of the type different from that stated in part (ii) is 0.4 , calculate the actual value of \(\mu\), giving your answer correct to 4 significant figures. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}
OCR S2 2013 June Q1
4 marks Moderate -0.8
1 It is required to select a random sample of 30 pupils from a school with 853 pupils. A student suggests the following method.
"Give each pupil sequentially a three-digit number from 001 to 853 . Use a calculator to generate random three-digit numbers from 0.000 to 0.999 inclusive, multiply the answer by 853 , add 1 and round off to the nearest whole number. Select the corresponding pupil, and repeat as necessary".
  1. Determine which pupil would be picked for each of the following calculator outputs: $$0.103 , \quad 0.104 , \quad 0.105 , \quad 0.106 , \quad 0.107$$
  2. Use your answers to part (i) to show that this method is biased, and suggest an improvement.
OCR S2 2013 June Q2
4 marks Standard +0.3
2 The number of neutrinos that pass through a certain region in one second is a random variable with the distribution \(\operatorname { Po } \left( 5 \times 10 ^ { 4 } \right)\). Use a suitable approximation to calculate the probability that the number of neutrinos passing through the region in 40 seconds is less than \(1.999 \times 10 ^ { 6 }\).
OCR S2 2013 June Q3
9 marks Challenging +1.2
3 The mean of a sample of 80 independent observations of a continuous random variable \(Y\) is denoted by \(\bar { Y }\). It is given that \(\mathrm { P } ( \bar { Y } \leqslant 157.18 ) = 0.1\) and \(\mathrm { P } ( \bar { Y } \geqslant 164.76 ) = 0.7\).
  1. Calculate \(\mathrm { E } ( Y )\) and the standard deviation of \(Y\).
  2. State
    (a) where in your calculations you have used the Central Limit Theorem,
    (b) why it was necessary to use the Central Limit Theorem,
    (c) why it was possible to use the Central Limit Theorem.
OCR S2 2013 June Q4
7 marks Standard +0.3
4 The number of floods in a certain river plain is known to have a Poisson distribution. It is known that up until 10 years ago the mean number of floods per year was 0.32 . During the last 10 years there were 6 floods. Test at the \(1 \%\) significance level whether there is evidence of an increase in the mean number of floods per year.
OCR S2 2013 June Q5
10 marks Moderate -0.3
5 Two random variables \(S\) and \(T\) have probability density functions given by $$\begin{aligned} & f _ { S } ( x ) = \begin{cases} \frac { 3 } { a ^ { 3 } } ( x - a ) ^ { 2 } & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases} \\ & f _ { T } ( x ) = \begin{cases} c & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases} \end{aligned}$$ where \(a\) and \(c\) are constants.
  1. On a single diagram sketch both probability density functions.
  2. Calculate the mean of \(S\), in terms of \(a\).
  3. Use your diagram to explain which of \(S\) or \(T\) has the bigger variance. (Answers obtained by calculation will score no marks.)
OCR S2 2013 June Q6
11 marks Standard +0.3
6 The random variable \(X\) denotes the yield, in kilograms per acre, of a certain crop. Under the standard treatment it is known that \(\mathrm { E } ( X ) = 38.4\). Under a new treatment, the yields of 50 randomly chosen regions can be summarised as $$n = 50 , \quad \sum x = 1834.0 , \quad \sum x ^ { 2 } = 70027.37 .$$ Test at the \(1 \%\) level whether there has been a change in the mean crop yield.
OCR S2 2013 June Q7
11 marks Standard +0.3
7 Past experience shows that \(35 \%\) of the senior pupils in a large school know the regulations about bringing cars to school. The head teacher addresses this subject in an assembly, and afterwards a random sample of 120 senior pupils is selected. In this sample it is found that 50 of these pupils know the regulations. Use a suitable approximation to test, at the \(10 \%\) significance level, whether there is evidence that the proportion of senior pupils who know the regulations has increased. Justify your approximation.
OCR S2 2013 June Q8
6 marks Challenging +1.2
8 The random variable \(R\) has the distribution \(\mathrm { B } ( 14 , p )\). A test is carried out at the \(\alpha \%\) significance level of the null hypothesis \(\mathrm { H } _ { 0 } : p = 0.25\), against \(\mathrm { H } _ { 1 } : p > 0.25\).
  1. Given that \(\alpha\) is as close to 5 as possible, find the probability of a Type II error when the true value of \(p\) is 0.4 .
  2. State what happens to the probability of a Type II error as
    (a) \(p\) increases from 0.4,
    (b) \(\alpha\) increases, giving a reason.
OCR S2 2013 June Q9
10 marks Standard +0.3
9 The managers of a car breakdown recovery service are discussing whether the number of breakdowns per day can be modelled by a Poisson distribution. They agree that breakdowns occur randomly. Manager \(A\) says, "it must be assumed that breakdowns occur at a constant rate throughout the day".
  1. Give an improved version of Manager \(A\) 's statement, and explain why the improvement is necessary.
  2. Explain whether you think your improved statement is likely to hold in this context. Assume now that the number \(B\) of breakdowns per day can be modelled by the distribution \(\operatorname { Po } ( \lambda )\).
  3. Given that \(\lambda = 9.0\) and \(\mathrm { P } \left( B > B _ { 0 } \right) < 0.1\), use tables to find the smallest possible value of \(B _ { 0 }\), and state the corresponding value of \(\mathrm { P } \left( B > B _ { 0 } \right)\).
  4. Given that \(\mathrm { P } ( B = 2 ) = 0.0072\), show that \(\lambda\) satisfies an equation of the form \(\lambda = 0.12 \mathrm { e } ^ { k \lambda }\), for a value of \(k\) to be stated. Evaluate the expression \(0.12 \mathrm { e } ^ { k \lambda }\) for \(\lambda = 8.5\) and \(\lambda = 8.6\), giving your answers correct to 4 decimal places. What can be deduced about a possible value of \(\lambda\) ?
OCR S3 2009 January Q1
4 marks Moderate -0.3
1 At a particular hospital, admissions of patients as a result of visits to the Accident and Emergency Department occur randomly at a uniform average rate of 0.75 per day. Independently, admissions that result from G.P. referrals occur randomly at a uniform average rate of 6.4 per week. The total number of admissions from these two causes over a randomly chosen period of four weeks is denoted by \(T\). State the distribution of \(T\) and obtain its expectation and variance.
OCR S3 2009 January Q2
5 marks Standard +0.3
2 The continuous random variable \(U\) has (cumulative) distribution function given by $$\mathrm { F } ( u ) = \begin{cases} \frac { 1 } { 5 } \mathrm { e } ^ { u } & u < 0 \\ 1 - \frac { 4 } { 5 } \mathrm { e } ^ { - \frac { 1 } { 4 } u } & u \geqslant 0 \end{cases}$$
  1. Find the upper quartile of \(U\).
  2. Find the probability density function of \(U\).
OCR S3 2009 January Q3
8 marks Moderate -0.3
3 In a random sample of credit card holders, it was found that \(28 \%\) of them used their card for internet purchases.
  1. Given that the sample size is 1200 , find a \(98 \%\) confidence interval for the percentage of all credit card holders who use their card for internet purchases.
  2. Estimate the smallest sample size for which a \(98 \%\) confidence interval would have a width of at most \(5 \%\), and state why the value found is only an estimate.
OCR S3 2009 January Q4
7 marks Standard +0.3
4 The weekly sales of petrol, \(X\) thousand litres, at a garage may be modelled by a continuous random variable with probability density function given by $$f ( x ) = \begin{cases} c & 25 \leqslant x \leqslant 45 \\ 0 & \text { otherwise } \end{cases}$$ where \(c\) is a constant. The weekly profit, in \(\pounds\), is given by \(( 400 \sqrt { X } - 240 )\).
  1. Obtain the value of \(c\).
  2. Find the expected weekly profit.
  3. Find the probability that the weekly profit exceeds \(\pounds 2000\).
OCR S3 2009 January Q5
10 marks Standard +0.8
5 The concentration level of mercury in a large lake is known to have a normal distribution with standard deviation 0.24 in suitable units. At the beginning of June 2008, the mercury level was measured at five randomly chosen places on the lake, and the sample mean is denoted by \(\bar { x } _ { 1 }\). Towards the end of June 2008 there was a spillage in the lake which may have caused the mercury level to rise. Because of this the level was then measured at six randomly chosen points of the lake, and the mean of this sample is denoted by \(\bar { x } _ { 2 }\).
  1. State hypotheses for a test based on the two samples for whether, on average, the level of mercury had increased. Define any parameters that you use.
  2. Find the set of values of \(\bar { x } _ { 2 } - \bar { x } _ { 1 }\) for which there would be evidence at the 5\% significance level that, on average, the level of mercury had increased.
  3. Given that the average level had actually increased by 0.3 units, find the probability of making a Type II error in your test, and comment on its value.
OCR S3 2009 January Q6
13 marks Standard +0.3
6 A mathematics examination is taken by 29 boys and 26 girls. Experience has shown that the probability that any boy forgets to bring a calculator to the examination is 0.3 , and that any girl forgets is 0.2 . Whether or not any student forgets to bring a calculator is independent of all other students. The numbers of boys and girls who forget to bring a calculator are denoted by \(B\) and \(G\) respectively, and \(F = B + G\).
  1. Find \(\mathrm { E } ( F )\) and \(\operatorname { Var } ( F )\).
  2. Using suitable approximations to the distributions of \(B\) and \(G\), which should be justified, find the smallest number of spare calculators that should be available in order to be at least \(99 \%\) certain that all 55 students will have a calculator.