Questions — OCR S2 (167 questions)

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OCR S2 2014 June Q4
7 marks
4 A zoologist investigates the number of snakes found in a given region of land. The zoologist intends to use a Poisson distribution to model the number of snakes.
[0pt]
  1. One condition for a Poisson distribution to be valid is that snakes must occur at constant average rate. State another condition needed for a Poisson distribution to be valid. [1] Assume now that the number of snakes found in 1 acre of a region can be modelled by the distribution Po(4).
    [0pt]
  2. Find the probability that, in 1 acre of the region, at least 6 snakes are found. [2]
    [0pt]
  3. Find the probability that, in 0.77 acres of the region, the number of snakes found is either 2 or 3. [4]
OCR S2 2014 June Q5
13 marks
5 A continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 1 } { 2 } \pi \sin ( \pi x ) & 0 \leqslant x \leqslant 1
0 & \text { otherwise } \end{cases}$$
  1. Show that this is a valid probability density function. [4]
  2. Sketch the curve \(\boldsymbol { y } = \mathbf { f } ( \boldsymbol { x } )\) and write down the value of \(\mathbf { E } \boldsymbol { ( } \boldsymbol { X } \boldsymbol { ) }\). [3]
  3. Find the value \(q\) such that \(\mathrm { P } ( X > q ) = 0.75\). [3]
  4. Write down an expression, including an integral, for \(\operatorname { Var } ( X )\). (Do not attempt to evaluate the integral.) [2]
  5. A student states that " \(X\) is more likely to occur when \(x\) is close to \(\mathrm { E } ( X )\)." Give an improved version of this statement. [1]
OCR S2 2014 June Q6
12 marks
6 In a city the proportion of inhabitants from ethnic group \(\mathbf { Z }\) is known to be \(\mathbf { 0 . 4 }\). A sample of \(\mathbf { 1 2 }\) employees of a large company in this city is obtained and it is found that 2 of them are from ethnic group \(Z\). A test is carried out, at the \(5 \%\) significance level, of whether the proportion of employees in this company from ethnic group \(Z\) is less than in the city as a whole.
[0pt]
  1. State an assumption that must be made about the sample for a significance test to be valid. [1]
    [0pt]
  2. Describe briefly an appropriate way of obtaining the sample. [2]
    [0pt]
  3. Carry out the test. [7]
  4. A manager believes that the company discriminates against ethnic group \(Z\). Explain whether carrying out the test at the 10\% significance level would be more supportive or less supportive of the manager's belief. [2]
OCR S2 2014 June Q7
15 marks
7 An examination board is developing a new syllabus and wants to know if the question papers are the right length. A random sample of 50 candidates was given a pre-test on a dummy paper. The times, \(t\) minutes, taken by these candidates to complete the paper can be summarised by
\(n = 50\),
\(\sum \boldsymbol { t } = \mathbf { 4 0 5 0 }\),
\(\sum \boldsymbol { t } ^ { \mathbf { 2 } } \boldsymbol { = } \mathbf { 3 2 9 8 0 0 }\).
Assume that times are normally distributed.
[0pt]
  1. Estimate the proportion of candidates that could not complete the paper within 90 minutes. [6]
  2. Test, at the \(10 \%\) significance level, whether the mean time for all candidates to complete this paper is \(\mathbf { 8 0 }\) minutes. Use a two-tail test. [7]
  3. Explain whether the assumption that times are normally distributed is necessary in answering
    (a) part (i),
    [0pt] (b) part (ii). [2]
OCR S2 2014 June Q8
6 marks
8 The random variable \(W\) has the distribution \(\operatorname { Po } ( \lambda )\). A significance test is carried out of the null hypothesis \(\mathrm { H } _ { 0 } : \lambda = 3.60\), against the alternative hypothesis \(\mathrm { H } _ { 1 } : \lambda < 3.60\). The test is based on a single observation of \(W\). The critical region is \(W = 0\).
[0pt]
  1. Find the significance level of the test. [2]
  2. It is known that, when \(\boldsymbol { \lambda } = \boldsymbol { \lambda } _ { \mathbf { 0 } }\), the probability that the test results in a Type II error is \(\mathbf { 0 . 8 }\). Find the value of \(\lambda _ { 0 }\). [4] \section*{END OF QUESTION PAPER}
OCR S2 Specimen Q1
1 The standard deviation of a random variable \(F\) is 12.0. The mean of \(n\) independent observations of \(F\) is denoted by \(\bar { F }\).
  1. Given that the standard deviation of \(\bar { F }\) is 1.50 , find the value of \(n\).
  2. For this value of \(n\), state, with justification, what can be said about the distribution of \(\bar { F }\).
OCR S2 Specimen Q2
2 A certain neighbourhood contains many small houses (with small gardens) and a few large houses (with large gardens). A sample survey of all houses is to be carried out in this neighbourhood. A student suggests that the sample could be selected by sticking a pin into a map of the neighbourhood the requisite number of times, while blindfolded.
  1. Give two reasons why this method does not produce a random sample.
  2. Describe a better method.
OCR S2 Specimen Q3
3 Sixty people each make two throws with a fair six-sided die.
  1. State the probability of one particular person obtaining two sixes.
  2. Using a suitable approximation, calculate the probability that at least four of the sixty obtain two sixes.
OCR S2 Specimen Q4
4 The random variable \(G\) has mean 20.0 and standard deviation \(\sigma\). It is given that \(\mathrm { P } ( G > 15.0 ) = 0.6\). Assume that \(G\) is normally distributed.
  1. (a) Find the value of \(\sigma\).
    (b) Given that \(\mathrm { P } ( G > g ) = 0.4\), find the value of \(\mathrm { P } ( G > 2 g )\).
  2. It is known that no values of \(G\) are ever negative. State with a reason what this tells you about the assumption that \(G\) is normally distributed.
OCR S2 Specimen Q5
5 The mean solubility rating of widgets inserted into beer cans is thought to be 84.0, in appropriate units. A random sample of 50 widgets is taken. The solubility ratings, \(x\), are summarised by $$n = 50 , \quad \Sigma x = 4070 , \quad \Sigma x ^ { 2 } = 336100$$ Test, at the \(5 \%\) significance level, whether the mean solubility rating is less than 84.0 .
OCR S2 Specimen Q6
6 On average a motorway police force records one car that has run out of petrol every two days.
  1. (a) Using a Poisson distribution, calculate the probability that, in one randomly chosen day, the police force records exactly two cars that have run out of petrol.
    (b) Using a Poisson distribution and a suitable approximation to the binomial distribution, calculate the probability that, in one year of 365 days, there are fewer than 205 days on which the police force records no cars that have run out of petrol.
  2. State an assumption needed for the Poisson distribution to be appropriate in part (i), and explain why this assumption is unlikely to be valid.
OCR S2 Specimen Q7
7 The time, in minutes, for which a customer is prepared to wait on a telephone complaints line is modelled by the random variable \(X\). The probability density function of \(X\) is given by $$\mathrm { f } ( x ) = \begin{cases} k x \left( 9 - x ^ { 2 } \right) & 0 \leqslant x \leqslant 3
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 4 } { 81 }\).
  2. Find \(\mathrm { E } ( X )\).
  3. (a) Show that the value \(y\) which satisfies \(\mathrm { P } ( X < y ) = \frac { 3 } { 5 }\) satisfies $$5 y ^ { 4 } - 90 y ^ { 2 } + 243 = 0 .$$ (b) Using the substitution \(w = y ^ { 2 }\), or otherwise, solve the equation in part (a) to find the value of \(y\).
OCR S2 Specimen Q8
8 The proportion of left-handed adults in a country is known to be \(15 \%\). It is suggested that for mathematicians the proportion is greater than \(15 \%\). A random sample of 12 members of a university mathematics department is taken, and it is found to include five who are left-handed.
  1. Stating your hypotheses, test whether the suggestion is justified, using a significance level as close to \(5 \%\) as possible.
  2. In fact the significance test cannot be carried out at a significance level of exactly \(5 \%\). State the probability of making a Type I error in the test.
  3. Find the probability of making a Type II error in the test for the case when the proportion of mathematicians who are left-handed is actually \(20 \%\).
  4. Determine, as accurately as the tables of cumulative binomial probabilities allow, the actual proportion of mathematicians who are left-handed for which the probability of making a Type II error in the test is 0.01 .
OCR S2 2013 January Q1
4 marks
1 A random variable has the distribution \(\mathrm { B } ( n , p )\). It is required to test \(\mathrm { H } _ { 0 } : p = \frac { 2 } { 3 }\) against \(\mathrm { H } _ { 1 } : p < \frac { 2 } { 3 }\) at a significance level as close to \(1 \%\) as possible, using a sample of size \(n = 8,9\) or 10 . Use tables to find which value of \(n\) gives such a test, stating the critical region for the test and the corresponding significance level.
[0pt] [4]
OCR S2 2013 January Q2
2 A random variable \(C\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). A random sample of 10 observations of \(C\) is obtained, and the results are summarised as $$n = 10 , \Sigma c = 380 , \Sigma c ^ { 2 } = 14602 .$$
  1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
  2. Hence calculate an estimate of the probability that \(C > 40\).
OCR S2 2013 January Q3
3 A factory produces 9000 music DVDs each day. A random sample of 100 such DVDs is obtained.
  1. Explain how to obtain this sample using random numbers.
  2. Given that \(24 \%\) of the DVDs produced by the factory are classical, use a suitable approximation to find the probability that, in the sample of 100 DVDs, fewer than 20 are classical.
OCR S2 2013 January Q4
4 A continuous random variable \(X\) has probability density function $$\mathrm { f } ( x ) = \left\{ \begin{array} { c l } k x & 0 \leqslant x \leqslant a
0 & \text { otherwise } \end{array} \right.$$ where \(k\) and \(a\) are constants.
  1. State what the letter \(x\) represents.
  2. Find \(k\) in terms of \(a\).
  3. Find \(\operatorname { Var } ( X )\) in terms of \(a\).
OCR S2 2013 January Q5
5 In a mine, a deposit of the substance pitchblende emits radioactive particles. The number of particles emitted has a Poisson distribution with mean 70 particles per second. The warning level is reached if the total number of particles emitted in one minute is more than 4350.
  1. A one-minute period is chosen at random. Use a suitable approximation to show that the probability that the warning level is reached during this period is 0.01 , correct to 2 decimal places. You should calculate the answer correct to 4 decimal places.
  2. Use a suitable approximation to find the probability that in 30 one-minute periods the warning level is reached on at least 4 occasions. (You should use the given rounded value of 0.01 from part (i) in your calculation.)
OCR S2 2013 January Q6
6 Gordon is a cricketer. Over a long period he knows that his population mean score, in number of runs per innings, is 28 , and the population standard deviation is 12 . In a new season he adopts a different batting style and he finds that in 30 innings using this style his mean score is 28.98 .
  1. Stating a necessary assumption, test at the \(5 \%\) significance level whether his population mean score has increased.
  2. Explain whether it was necessary to use the Central Limit Theorem in part (i).
OCR S2 2013 January Q7
7 The continuous random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). The mean of a random sample of \(n\) observations of \(X\) is denoted by \(\bar { X }\). It is given that \(\mathrm { P } ( \bar { X } < 35.0 ) = 0.9772\) and \(\mathrm { P } ( \bar { X } < 20.0 ) = 0.1587\).
  1. Obtain a formula for \(\sigma\) in terms of \(n\). Two students are discussing this question. Aidan says "If you were told another probability, for instance \(\mathrm { P } ( \bar { X } > 32 ) = 0.1\), you could work out the value of \(\sigma\)." Binya says, "No, the value of \(\mathrm { P } ( \bar { X } > 32 )\) is fixed by the information you know already."
  2. State which of Aidan and Binya is right. If you think that Aidan is right, calculate the value of \(\sigma\) given that \(\mathrm { P } ( \bar { X } > 32 ) = 0.1\). If you think that Binya is right, calculate the value of \(\mathrm { P } ( \bar { X } > 32 )\).
OCR S2 2013 January Q8
8 In a large city the number of traffic lights that fail in one day of 24 hours is denoted by \(Y\). It may be assumed that failures occur randomly.
  1. Explain what the statement "failures occur randomly" means.
  2. State, in context, two different conditions that must be satisfied if \(Y\) is to be modelled by a Poisson distribution, and for each condition explain whether you think it is likely to be met in this context.
  3. For this part you may assume that \(Y\) is well modelled by the distribution \(\operatorname { Po } ( \lambda )\). It is given that \(\mathrm { P } ( Y = 7 ) = \mathrm { P } ( Y = 8 )\). Use an algebraic method to calculate the value of \(\lambda\) and hence calculate the corresponding value of \(\mathrm { P } ( Y = 7 )\).
OCR S2 2013 January Q9
9 The random variable \(A\) has the distribution \(\mathrm { B } ( 30 , p )\). A test is carried out of the hypotheses \(\mathrm { H } _ { 0 } : p = 0.6\) against \(\mathrm { H } _ { 1 } : p < 0.6\). The critical region is \(A \leqslant 13\).
  1. State the probability that \(\mathrm { H } _ { 0 }\) is rejected when \(p = 0.6\).
  2. Find the probability that a Type II error occurs when \(p = 0.5\).
  3. It is known that on average \(p = 0.5\) on one day in five, and on other days the value of \(p\) is 0.6 . On each day two tests are carried out. If the result of the first test is that \(\mathrm { H } _ { 0 }\) is rejected, the value of \(p\) is adjusted if necessary, to ensure that \(p = 0.6\) for the rest of the day. Otherwise the value of \(p\) remains the same as for the first test. Calculate the probability that the result of the second test is to reject \(\mathrm { H } _ { 0 }\). \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}
OCR S2 2015 June Q1
1 The random variable \(Y\) is normally distributed with mean \(\mu\) and variance \(\sigma ^ { 2 }\). It is found that \(\mathrm { P } ( Y > 150.0 ) = 0.0228\) and \(\mathrm { P } ( Y > 143.0 ) = 0.9332\). Find the values of \(\mu\) and \(\sigma\).
OCR S2 2015 June Q2
2 A class investigated the number of dead rabbits found along a particular stretch of road.
  1. The class agrees that dead rabbits occur randomly along the road. Explain what this statement means.
  2. State, in this context, an assumption needed for the number of dead rabbits in a fixed length of road to be modelled by a Poisson distribution, and explain what your statement means. Assume now that the number of dead rabbits in a fixed length of road can be well modelled by a Poisson distribution with mean 1 per 600 m of road.
  3. Use an appropriate formula, showing your working, to find the probability that in a road of length 1650 m there are exactly 3 dead rabbits.
OCR S2 2015 June Q3
3 A continuous random variable \(X\) has probability density function $$f ( x ) = \left\{ \begin{array} { c l } \frac { 3 } { 2 a ^ { 3 } } x ^ { 2 } & - a \leqslant x \leqslant a
0 & \text { otherwise } \end{array} \right.$$ where \(a\) is a constant.
  1. It is given that \(\mathrm { P } ( - 3 \leqslant X \leqslant 3 ) = 0.125\). Find the value of \(a\) in this case.
  2. It is given instead that \(\operatorname { Var } ( X ) = 1.35\). Find the value of \(a\) in this case.
  3. Explain the relationship between \(x\) and \(X\) in this question.