Questions — OCR S2 (167 questions)

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OCR S2 2011 January Q5
5 A temporary job is advertised annually. The number of applicants for the job is a random variable which is known from many years' experience to have a distribution \(\operatorname { Po } ( 12 )\). In 2010 there were 19 applicants for the job. Test, at the 10\% significance level, whether there is evidence of an increase in the mean number of applicants for the job.
OCR S2 2011 January Q6
6 The number of randomly occurring events in a given time interval is denoted by \(R\). In order that \(R\) is well modelled by a Poisson distribution, it is necessary that events occur independently.
  1. Let \(R\) represent the number of customers dining at a restaurant on a randomly chosen weekday lunchtime. Explain what the condition 'events occur independently' means in this context, and give a reason why it would probably not hold in this context. Let \(D\) represent the number of tables booked at the restaurant on a randomly chosen day. Assume that \(D\) can be well modelled by distribution \(\operatorname { Po } ( 7 )\).
  2. Find \(\mathrm { P } ( D < 5 )\).
  3. Use a suitable approximation to find the probability that, in five randomly chosen days, the total number of tables booked is greater than 40 .
OCR S2 2011 January Q7
7 Two continuous random variables \(S\) and \(T\) have probability density functions \(\mathrm { f } _ { S }\) and \(\mathrm { f } _ { T }\) given respectively by $$\begin{aligned} & f _ { S } ( x ) = \begin{cases} \frac { a } { x ^ { 2 } } & 1 \leqslant x \leqslant 3
0 & \text { otherwise } \end{cases}
& f _ { T } ( x ) = \begin{cases} b & 1 \leqslant x \leqslant 3
0 & \text { otherwise } \end{cases} \end{aligned}$$ where \(a\) and \(b\) are constants.
  1. Sketch on the same axes the graphs of \(y = \mathrm { f } _ { S } ( x )\) and \(y = \mathrm { f } _ { T } ( x )\).
  2. Find the value of \(a\).
  3. Find \(\mathrm { E } ( S )\).
  4. A student gave the following description of the distribution of \(T\) : "The probability that \(T\) occurs is constant". Give an improved description, in everyday terms.
OCR S2 2011 January Q8
8 A company has 3600 employees, of whom \(22.5 \%\) live more than 30 miles from their workplace. A random sample of 40 employees is obtained.
  1. Use a suitable approximation, which should be justified, to find the probability that more than 5 of the employees in the sample live more than 30 miles from their workplace.
  2. Describe how to use random numbers to select a sample of 40 from a population of 3600 employees.
OCR S2 2011 January Q9
9 A pharmaceutical company is developing a new drug to treat a certain disease. The company will continue to develop the drug if the proportion \(p\) of those who have the disease and show a substantial improvement after treatment is greater than 0.7 . The company carries out a test, at the \(5 \%\) significance level, on a random sample of 14 patients who suffer from the disease.
  1. Find the critical region for the test.
  2. Given that 12 of the 14 patients in the sample show a substantial improvement, carry out the test.
  3. Find the probability that the test results in a Type II error if in fact \(p = 0.8\). RECOGNISING ACHIEVEMENT
OCR S2 2011 January Q10
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    OCR S2 2011 January Q12
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    OCR S2 2011 January Q13
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  • \section*{PLEASE DO NOT WRITE ON THIS PAGE} RECOGNISING ACHIEVEMENT
    OCR S2 2009 June Q1
    1 The random variable \(H\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). It is given that \(\mathrm { P } ( H < 105.0 ) = 0.2420\) and \(\mathrm { P } ( H > 110.0 ) = 0.6915\). Find the values of \(\mu\) and \(\sigma\), giving your answers to a suitable degree of accuracy.
    OCR S2 2009 June Q2
    2 The random variable \(D\) has the distribution \(\operatorname { Po } ( 20 )\). Using an appropriate approximation, which should be justified, calculate \(\mathrm { P } ( D \geqslant 25 )\).
    OCR S2 2009 June Q3
    3 An electronics company is developing a new sound system. The company claims that \(60 \%\) of potential buyers think that the system would be good value for money. In a random sample of 12 potential buyers, 4 thought that it would be good value for money. Test, at the 5\% significance level, whether the proportion claimed by the company is too high.
    OCR S2 2009 June Q4
    4 A survey is to be carried out to draw conclusions about the proportion \(p\) of residents of a town who support the building of a new supermarket. It is proposed to carry out the survey by interviewing a large number of people in the high street of the town, which attracts a large number of tourists.
    1. Give two different reasons why this proposed method is inappropriate.
    2. Suggest a good method of carrying out the survey.
    3. State two statistical properties of your survey method that would enable reliable conclusions about \(p\) to be drawn.
    OCR S2 2009 June Q5
    5 In a large region of derelict land, bricks are found scattered in the earth.
    1. State two conditions needed for the number of bricks per cubic metre to be modelled by a Poisson distribution. Assume now that the number of bricks in 1 cubic metre of earth can be modelled by the distribution Po(3).
    2. Find the probability that the number of bricks in 4 cubic metres of earth is between 8 and 14 inclusive.
    3. Find the size of the largest volume of earth for which the probability that no bricks are found is at least 0.4.
    OCR S2 2009 June Q6
    6 The continuous random variable \(R\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). The results of 100 observations of \(R\) are summarised by $$\Sigma r = 3360.0 , \quad \Sigma r ^ { 2 } = 115782.84 .$$
    1. Calculate an unbiased estimate of \(\mu\) and an unbiased estimate of \(\sigma ^ { 2 }\).
    2. The mean of 9 observations of \(R\) is denoted by \(\bar { R }\). Calculate an estimate of \(\mathrm { P } ( \bar { R } > 32.0 )\).
    3. Explain whether you need to use the Central Limit Theorem in your answer to part (ii).
    OCR S2 2009 June Q7
    7 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 2 } { 9 } x ( 3 - x ) & 0 \leqslant x \leqslant 3 ,
    0 & \text { otherwise } . \end{cases}$$
    1. Find the variance of \(X\).
    2. Show that the probability that a single observation of \(X\) lies between 0.0 and 0.5 is \(\frac { 2 } { 27 }\).
    3. 108 observations of \(X\) are obtained. Using a suitable approximation, find the probability that at least 10 of the observations lie between 0.0 and 0.5 .
    4. The mean of 108 observations of \(X\) is denoted by \(\bar { X }\). Write down the approximate distribution of \(\bar { X }\), giving the value(s) of any parameter(s).
    OCR S2 2009 June Q8
    8 In a large company the time taken for an employee to carry out a certain task is a normally distributed random variable with mean 78.0 s and unknown variance. A new training scheme is introduced and after its introduction the times taken by a random sample of 120 employees are recorded. The mean time for the sample is 76.4 s and an unbiased estimate of the population variance is \(68.9 \mathrm {~s} ^ { 2 }\).
    1. Test, at the \(1 \%\) significance level, whether the mean time taken for the task has changed.
    2. It is required to redesign the test so that the probability of making a Type I error is less than 0.01 when the sample mean is 77.0 s . Calculate an estimate of the smallest sample size needed, and explain why your answer is only an estimate.
    OCR S2 2010 June Q1
    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 \(\operatorname { Po } ( 4.2 )\).
      (a) Find the probability that in the course of a 10-year period, at least 7 inhabitants are selected for jury service.
      (b) Find the probability that in 1 year, exactly 2 inhabitants are selected for jury service.
    2. 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
    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
    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
    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
    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.
      (a) Find the critical region for the test, in terms of \(\bar { t }\).
      (b) Given that the population mean time is in fact 35 seconds, find the probability that the test results in a Type II error.
    2. 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
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    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
    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
    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
    1 In Fisher Avenue there are 263 houses, numbered 1 to 263. Explain how to obtain a random sample of 20 of these houses.