Questions S2 (1597 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
OCR S2 2006 June Q4
10 marks Standard +0.3
4
  1. Explain briefly what is meant by a random sample. Random numbers are used to select, with replacement, a sample of size \(n\) from a population numbered 000, 001, 002, ..., 799.
  2. If \(n = 6\), find the probability that exactly 4 of the selected sample have numbers less than 500 .
  3. If \(n = 60\), use a suitable approximation to calculate the probability that at least 40 of the selected sample have numbers less than 500 .
OCR S2 2006 June Q5
9 marks Standard +0.3
5 An airline has 300 seats available on a flight to Australia. It is known from experience that on average only \(99 \%\) of those who have booked seats actually arrive to take the flight, the remaining \(1 \%\) being called 'no-shows'. The airline therefore sells more than 300 seats. If more than 300 passengers then arrive, the flight is over-booked. Assume that the number of no-show passengers can be modelled by a binomial distribution.
  1. If the airline sells 303 seats, state a suitable distribution for the number of no-show passengers, and state a suitable approximation to this distribution, giving the values of any parameters. Using the distribution and approximation in part (i),
  2. show that the probability that the flight is over-booked is 0.4165 , correct to 4 decimal places,
  3. find the largest number of seats that can be sold for the probability that the flight is over-booked to be less than 0.2.
OCR S2 2006 June Q6
14 marks Moderate -0.3
6 Customers arrive at a post office at a constant average rate of 0.4 per minute.
  1. State an assumption needed to model the number of customers arriving in a given time interval by a Poisson distribution. Assuming that the use of a Poisson distribution is justified,
  2. find the probability that more than 2 customers arrive in a randomly chosen 1 -minute interval,
  3. use a suitable approximation to calculate the probability that more than 55 customers arrive in a given two-hour interval,
  4. calculate the smallest time for which the probability that no customers arrive in that time is less than 0.02 , giving your answer to the nearest second.
OCR S2 2006 June Q7
18 marks Standard +0.3
7 Three independent researchers, \(A , B\) and \(C\), carry out significance tests on the power consumption of a manufacturer's domestic heaters. The power consumption, \(X\) watts, is a normally distributed random variable with mean \(\mu\) and standard deviation 60. Each researcher tests the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 4000\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : \mu > 4000\). Researcher \(A\) uses a sample of size 50 and a significance level of \(5 \%\).
  1. Find the critical region for this test, giving your answer correct to 4 significant figures. In fact the value of \(\mu\) is 4020 .
  2. Calculate the probability that Researcher \(A\) makes a Type II error.
  3. Researcher \(B\) uses a sample bigger than 50 and a significance level of \(5 \%\). Explain whether the probability that Researcher \(B\) makes a Type II error is less than, equal to, or greater than your answer to part (ii).
  4. Researcher \(C\) uses a sample of size 50 and a significance level bigger than \(5 \%\). Explain whether the probability that Researcher \(C\) makes a Type II error is less than, equal to, or greater than your answer to part (ii).
  5. State with a reason whether it is necessary to use the Central Limit Theorem at any point in this question.
OCR S2 2007 June Q1
6 marks Moderate -0.8
1 A random sample of observations of a random variable \(X\) is summarised by $$n = 100 , \quad \Sigma x = 4830.0 , \quad \Sigma x ^ { 2 } = 249 \text { 509.16. }$$
  1. Obtain unbiased estimates of the mean and variance of \(X\).
  2. The sample mean of 100 observations of \(X\) is denoted by \(\bar { X }\). Explain whether you would need any further information about the distribution of \(X\) in order to estimate \(\mathrm { P } ( \bar { X } > 60 )\). [You should not attempt to carry out the calculation.]
OCR S2 2007 June Q2
5 marks Moderate -0.3
2 It is given that on average one car in forty is yellow. Using a suitable approximation, find the probability that, in a random sample of 130 cars, exactly 4 are yellow.
OCR S2 2007 June Q3
3 marks Easy -1.8
3 The proportion of adults in a large village who support a proposal to build a bypass is denoted by \(p\). A random sample of size 20 is selected from the adults in the village, and the members of the sample are asked whether or not they support the proposal.
  1. Name the probability distribution that would be used in a hypothesis test for the value of \(p\).
  2. State the properties of a random sample that explain why the distribution in part (i) is likely to be a good model.
    \(4 X\) is a continuous random variable.
OCR S2 2007 June Q5
7 marks Standard +0.3
5 The number of system failures per month in a large network is a random variable with the distribution \(\operatorname { Po } ( \lambda )\). A significance test of the null hypothesis \(\mathrm { H } _ { 0 } : \lambda = 2.5\) is carried out by counting \(R\), the number of system failures in a period of 6 months. The result of the test is that \(\mathrm { H } _ { 0 }\) is rejected if \(R > 23\) but is not rejected if \(R \leqslant 23\).
  1. State the alternative hypothesis.
  2. Find the significance level of the test.
  3. Given that \(\mathrm { P } ( R > 23 ) < 0.1\), use tables to find the largest possible actual value of \(\lambda\). You should show the values of any relevant probabilities.
OCR S2 2007 June Q6
9 marks Standard +0.3
6 In a rearrangement code, the letters of a message are rearranged so that the frequency with which any particular letter appears is the same as in the original message. In ordinary German the letter \(e\) appears \(19 \%\) of the time. A certain encoded message of 20 letters contains one letter \(e\).
  1. Using an exact binomial distribution, test at the \(10 \%\) significance level whether there is evidence that the proportion of the letter \(e\) in the language from which this message is a sample is less than in German, i.e., less than \(19 \%\).
  2. Give a reason why a binomial distribution might not be an appropriate model in this context.
OCR S2 2007 June Q7
10 marks Moderate -0.3
7 Two continuous random variables \(S\) and \(T\) have probability density functions as follows. $$\begin{array} { l l } S : & f ( x ) = \begin{cases} \frac { 1 } { 2 } & - 1 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases} \\ T : & g ( x ) = \begin{cases} \frac { 3 } { 2 } x ^ { 2 } & - 1 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases} \end{array}$$
  1. Sketch on the same axes the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\). [You should not use graph paper or attempt to plot points exactly.]
  2. Explain in everyday terms the difference between the two random variables.
  3. Find the value of \(t\) such that \(\mathrm { P } ( T > t ) = 0.2\).
OCR S2 2007 June Q8
13 marks Standard +0.3
8 A random variable \(Y\) is normally distributed with mean \(\mu\) and variance 12.25. Two statisticians carry out significance tests of the hypotheses \(\mathrm { H } _ { 0 } : \mu = 63.0 , \mathrm { H } _ { 1 } : \mu > 63.0\).
  1. Statistician \(A\) uses the mean \(\bar { Y }\) of a sample of size 23, and the critical region for his test is \(\bar { Y } > 64.20\). Find the significance level for \(A\) 's test.
  2. Statistician \(B\) uses the mean of a sample of size 50 and a significance level of \(5 \%\).
    (a) Find the critical region for \(B\) 's test.
    (b) Given that \(\mu = 65.0\), find the probability that \(B\) 's test results in a Type II error.
  3. Given that, when \(\mu = 65.0\), the probability that \(A\) 's test results in a Type II error is 0.1365 , state with a reason which test is better.
OCR S2 2007 June Q9
13 marks Standard +0.3
9
  1. The random variable \(G\) has the distribution \(\mathrm { B } ( n , 0.75 )\). Find the set of values of \(n\) for which the distribution of \(G\) can be well approximated by a normal distribution.
  2. The random variable \(H\) has the distribution \(\mathrm { B } ( n , p )\). It is given that, using a normal approximation, \(\mathrm { P } ( H \geqslant 71 ) = 0.0401\) and \(\mathrm { P } ( H \leqslant 46 ) = 0.0122\).
    1. Find the mean and standard deviation of the approximating normal distribution.
    2. Hence find the values of \(n\) and \(p\). 4
OCR S2 2014 June Q1
5 marks Moderate -0.8
1 The random variable \(F\) has the distribution \(B ( 50,0.7 )\). Use a suitable approximation to find \(\mathbf { P } \boldsymbol { ( } \mathbf { F > } \mathbf { 4 0 } \boldsymbol { ) }\). [5]
OCR S2 2014 June Q2
7 marks Standard +0.3
2 The events organiser of a school sends out invitations to \(\mathbf { 1 5 0 }\) people to attend its prize day. From past experience the organiser knows that the number of those who will come to the prize day can be modelled by the distribution \(\mathbf { B } ( \mathbf { 1 5 0 } , \mathbf { 0 . 9 8 } )\).
[0pt]
  1. Explain why this distribution cannot be well approximated by either a normal or a Poisson distribution. [3]
    [0pt]
  2. By considering the number of those who do not attend, use a suitable approximation to find the probability that fewer than 146 people attend. [4]
OCR S2 2014 June Q3
7 marks Standard +0.3
3 The random variable \(G\) has the distribution \(\mathbf { N } \left( \mu , \boldsymbol { \sigma } ^ { 2 } \right)\). One hundred observations of \(G\) are taken. The results are summarised in the following table.
Interval\(G < 40.0\)\(40.0 \leqslant G < 60.0\)\(G \geqslant 60.0\)
Frequency175825
  1. By considering \(\mathrm { P } ( G < 40.0 )\), write down an equation involving \(\mu\) and \(\sigma\). [2]
  2. Find a second equation involving \(\mu\) and \(\sigma\). Hence calculate values for \(\mu\) and \(\sigma\). [4]
    [0pt]
  3. Explain why your answers are only estimates. [1]
OCR S2 2014 June Q4
7 marks Easy -1.2
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 Moderate -0.3
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 Standard +0.3
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 Standard +0.3
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 Challenging +1.2
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
5 marks Moderate -0.8
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
5 marks Easy -1.2
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
6 marks Moderate -0.3
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
9 marks Standard +0.3
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
10 marks Standard +0.3
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 .