Questions — OCR S2 (169 questions)

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OCR S2 2009 January Q4
10 marks Moderate -0.3
4 A television company believes that the proportion of adults who watched a certain programme is 0.14 . Out of a random sample of 22 adults, it is found that 2 watched the programme.
  1. Carry out a significance test, at the \(10 \%\) level, to determine, on the basis of this sample, whether the television company is overestimating the proportion of adults who watched the programme.
  2. The sample was selected randomly. State what properties of this method of sampling are needed to justify the use of the distribution used in your test.
OCR S2 2009 January Q5
9 marks Standard +0.3
5 The continuous random variables \(S\) and \(T\) have probability density functions as follows. $$\begin{array} { l l } S : & \mathrm { f } ( x ) = \begin{cases} \frac { 1 } { 4 } & - 2 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases} \\ T : & \mathrm { g } ( x ) = \begin{cases} \frac { 5 } { 64 } x ^ { 4 } & - 2 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases} \end{array}$$
  1. Sketch, on the same axes, the graphs of f and g .
  2. Describe in everyday terms the difference between the distributions of the random variables \(S\) and \(T\). (Answers that comment only on the shapes of the graphs will receive no credit.)
  3. Calculate the variance of \(T\).
OCR S2 2009 January Q6
11 marks Standard +0.3
6 The weight of a plastic box manufactured by a company is \(W\) grams, where \(W \sim \mathrm {~N} ( \mu , 20.25 )\). A significance test of the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 50.0\), against the alternative hypothesis \(\mathrm { H } _ { 1 } : \mu \neq 50.0\), is carried out at the \(5 \%\) significance level, based on a sample of size \(n\).
  1. Given that \(n = 81\),
    1. find the critical region for the test, in terms of the sample mean \(\bar { W }\),
    2. find the probability that the test results in a Type II error when \(\mu = 50.2\).
    3. State how the probability of this Type II error would change if \(n\) were greater than 81 .
OCR S2 2009 January Q7
12 marks Standard +0.3
7 A motorist records the time taken, \(T\) minutes, to drive a particular stretch of road on each of 64 occasions. Her results are summarised by $$\Sigma t = 876.8 , \quad \Sigma t ^ { 2 } = 12657.28$$
  1. Test, at the \(5 \%\) significance level, whether the mean time for the motorist to drive the stretch of road is greater than 13.1 minutes.
  2. Explain whether it is necessary to use the Central Limit Theorem in your test.
OCR S2 2009 January Q8
14 marks Moderate -0.3
8 A sales office employs 21 representatives. Each day, for each representative, the probability that he or she achieves a sale is 0.7 , independently of other representatives. The total number of representatives who achieve a sale on any one day is denoted by \(K\).
  1. Using a suitable approximation (which should be justified), find \(\mathrm { P } ( K \geqslant 16 )\).
  2. Using a suitable approximation (which should be justified), find the probability that the mean of 36 observations of \(K\) is less than or equal to 14.0 . 4
OCR S2 2011 January Q1
4 marks Easy -1.2
1 A random sample of nine observations of a random variable is obtained. The results are summarised as $$\Sigma x = 468 , \quad \Sigma x ^ { 2 } = 24820 .$$ Calculate unbiased estimates of the population mean and variance.
OCR S2 2011 January Q2
6 marks Standard +0.3
2 The random variable \(H\) has the distribution \(\mathrm { N } \left( \mu , 5 ^ { 2 } \right)\). The mean of a sample of \(n\) observations of \(H\) is denoted by \(\bar { H }\). It is given that \(\mathrm { P } ( \bar { H } > 53.28 ) = 0.0250\) and \(\mathrm { P } ( \bar { H } < 51.65 ) = 0.0968\), both correct to 4 decimal places. Find the values of \(\mu\) and \(n\).
OCR S2 2011 January Q3
6 marks Moderate -0.8
3 The probability that a randomly chosen PPhone has a faulty casing is 0.0228 . A random sample of 200 PPhones is obtained. Use a suitable approximation to find the probability that the number of PPhones in the sample with a faulty casing is 2 or fewer. Justify your approximation.
OCR S2 2011 January Q4
7 marks Standard +0.3
4 The continuous random variable \(X\) has mean \(\mu\) and standard deviation 45. A significance test is to be carried out of the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 230\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : \mu \neq 230\), at the \(1 \%\) significance level. A random sample of size 50 is obtained, and the sample mean is found to be 213.4.
  1. Carry out the test.
  2. Explain whether it is necessary to use the Central Limit Theorem in your test.
OCR S2 2011 January Q5
7 marks Standard +0.3
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
10 marks Standard +0.3
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
10 marks Moderate -0.8
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
11 marks Moderate -0.3
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
11 marks Standard +0.3
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
Moderate -0.5
10
7
7
  • 7
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    OCR S2 2011 January Q12
    Moderate -0.5
    12
    8
    {}
    OCR S2 2011 January Q13
    Moderate -0.5
    13
    8
    		(continued)
    8
  • 9
  • 9
  • {}
    9
  • 9
  • \section*{PLEASE DO NOT WRITE ON THIS PAGE} RECOGNISING ACHIEVEMENT
    OCR S2 2009 June Q1
    6 marks Standard +0.3
    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
    6 marks Moderate -0.5
    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
    7 marks Moderate -0.3
    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
    7 marks Moderate -0.8
    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
    9 marks Moderate -0.8
    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
    10 marks Moderate -0.3
    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
    16 marks Standard +0.3
    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
    11 marks Standard +0.3
    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.