Questions — OCR S2 (169 questions)

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OCR S2 2010 June Q1
7 marks Moderate -0.3
1
  1. The number of inhabitants of a village who are selected for jury service in the course of a 10-year period is a random variable with the distribution \(\operatorname { Po } ( 4.2 )\).
    1. Find the probability that in the course of a 10-year period, at least 7 inhabitants are selected for jury service.
    2. Find the probability that in 1 year, exactly 2 inhabitants are selected for jury service.
    3. Explain why the number of inhabitants of the village who contract influenza in 1 year can probably not be well modelled by a Poisson distribution.
OCR S2 2010 June Q2
5 marks Moderate -0.8
2 A university has a large number of students, of whom \(35 \%\) are studying science subjects. A sample of 10 students is obtained by listing all the students, giving each a serial number and selecting by using random numbers.
  1. Find the probability that fewer than 3 of the sample are studying science subjects.
  2. It is required that, in selecting the sample, the same student is not selected twice. Explain whether this requirement invalidates your calculation in part (i).
OCR S2 2010 June Q3
9 marks Standard +0.3
3 Tennis balls are dropped from a standard height, and the height of bounce, \(H \mathrm {~cm}\), is measured. \(H\) is a random variable with the distribution \(\mathrm { N } \left( 40 , \sigma ^ { 2 } \right)\). It is given that \(\mathrm { P } ( H < 32 ) = 0.2\).
  1. Find the value of \(\sigma\).
  2. 90 tennis balls are selected at random. Use an appropriate approximation to find the probability that more than 19 have \(H < 32\).
OCR S2 2010 June Q4
7 marks Moderate -0.3
4 The proportion of commuters in a town who travel to work by train is 0.4 . Following the opening of a new station car park, a random sample of 16 commuters is obtained, and 11 of these travel to work by train. Test at the \(1 \%\) significance level whether there is evidence of an increase in the proportion of commuters in this town who travel to work by train.
OCR S2 2010 June Q5
11 marks Challenging +1.2
5 The time \(T\) seconds needed for a computer to be ready to use, from the moment it is switched on, is a normally distributed random variable with standard deviation 5 seconds. The specification of the computer says that the population mean time should be not more than 30 seconds.
  1. A test is carried out, at the \(5 \%\) significance level, of whether the specification is being met, using the mean \(\bar { t }\) of a random sample of 10 times.
    1. Find the critical region for the test, in terms of \(\bar { t }\).
    2. Given that the population mean time is in fact 35 seconds, find the probability that the test results in a Type II error.
    3. Because of system degradation and memory load, the population mean time \(\mu\) seconds increases with the number of months of use, \(m\). A formula for \(\mu\) in terms of \(m\) is \(\mu = 20 + 0.6 m\). Use this formula to find the value of \(m\) for which the probability that the test results in rejection of the null hypothesis is 0.5 .
OCR S2 2010 June Q6
10 marks Standard +0.3
6
  1. The random variable \(D\) has the distribution \(\operatorname { Po } ( 24 )\). Use a suitable approximation to find \(P ( D > 30 )\).
  2. An experiment consists of 200 trials. For each trial, the probability that the result is a success is 0.98 , independent of all other trials. The total number of successes is denoted by \(E\).
    1. Explain why the distribution of \(E\) cannot be well approximated by a Poisson distribution.
    2. By considering the number of failures, use an appropriate Poisson approximation to find \(\mathrm { P } ( E \leqslant 194 )\).
OCR S2 2010 June Q7
11 marks Standard +0.3
7 A machine is designed to make paper with mean thickness 56.80 micrometres. The thicknesses, \(x\) micrometres, of a random sample of 300 sheets are summarised by $$n = 300 , \quad \Sigma x = 17085.0 , \quad \Sigma x ^ { 2 } = 973847.0 .$$ Test, at the \(10 \%\) significance level, whether the machine is producing paper of the designed thickness.
OCR S2 2010 June Q8
12 marks Standard +0.3
8 The continuous random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} k x ^ { - a } & x \geqslant 1 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) and \(a\) are constants and \(a\) is greater than 1 .
  1. Show that \(k = a - 1\).
  2. Find the variance of \(X\) in the case \(a = 4\).
  3. It is given that \(\mathrm { P } ( X < 2 ) = 0.9\). Find the value of \(a\), correct to 3 significant figures.
OCR S2 2011 June Q1
3 marks Easy -1.8
1 In Fisher Avenue there are 263 houses, numbered 1 to 263. Explain how to obtain a random sample of 20 of these houses.
OCR S2 2011 June Q2
7 marks Standard +0.3
2 The random variable \(Y\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). It is given that $$\mathrm { P } ( Y < 48.0 ) = \mathrm { P } ( Y > 57.0 ) = 0.0668 .$$ Find the value \(y _ { 0 }\) such that \(\mathrm { P } \left( Y > y _ { 0 } \right) = 0.05\).
OCR S2 2011 June Q3
7 marks Challenging +1.2
3 The random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , 5 ^ { 2 } \right)\). A hypothesis test is carried out of \(\mathrm { H } _ { 0 } : \mu = 20.0\) against \(\mathrm { H } _ { 1 } : \mu < 20.0\), at the \(1 \%\) level of significance, based on the mean of a sample of size 16. Given that in fact \(\mu = 15.0\), find the probability that the test results in a Type II error.
OCR S2 2011 June Q4
8 marks Standard +0.3
4 A continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 3 } { 16 } ( x - 2 ) ^ { 2 } & 0 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$
  1. Sketch the graph of \(y = \mathrm { f } ( x )\).
  2. Calculate the variance of \(X\).
  3. A student writes " \(X\) is more likely to occur when \(x\) takes values further away from 2 ". Explain whether you agree with this statement.
OCR S2 2011 June Q5
8 marks Moderate -0.3
5 A travel company finds from its records that \(40 \%\) of its customers book with travel agents. The company redesigns its website, and then carries out a survey of 10 randomly chosen customers. The result of the survey is that 1 of these customers booked with a travel agent.
  1. Test at the \(5 \%\) significance level whether the percentage of customers who book with travel agents has decreased.
  2. The managing director says that "Our redesigned website has resulted in a decrease in the percentage of our customers who book with travel agents." Comment on this statement.
OCR S2 2011 June Q6
12 marks Standard +0.3
6 Records show that before the year 1990 the maximum daily temperature \(T ^ { \circ } \mathrm { C }\) at a seaside resort in August can be modelled by a distribution with mean 24.3. The maximum temperatures of a random sample of 50 August days since 1990 can be summarised by $$n = 50 , \quad \Sigma t = 1314.0 , \quad \Sigma t ^ { 2 } = 36602.17 .$$
  1. Test, at the \(1 \%\) significance level, whether there is evidence of a change in the mean maximum daily temperature in August since 1990.
  2. Give a reason why it is possible to use the Central Limit Theorem in your test.
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
    1. the value of \(a\),
    2. \(\operatorname { Var } ( X )\).
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.