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OCR S2 2007 January Q2
5 marks Moderate -0.8
2 A school has 900 pupils. For a survey, Jan obtains a list of all the pupils, numbered 1 to 900 in alphabetical order. She then selects a sample by the following method. Two fair dice, one red and one green, are thrown, and the number in the list of the first pupil in the sample is determined by the following table.
\cline { 3 - 8 } \multicolumn{2}{c|}{}Score on green dice
\cline { 3 - 8 } \multicolumn{2}{c|}{}123456
Score on
red dice
1,2 or 3123456
For example, if the scores on the red and green dice are 5 and 2 respectively, then the first member of the sample is the pupil numbered 8 in the list. Starting with this first number, every 12th number on the list is then used, so that if the first pupil selected is numbered 8 , the others will be numbered \(20,32,44 , \ldots\).
  1. State the size of the sample.
  2. Explain briefly whether the following statements are true.
    1. Each pupil in the school has an equal probability of being in the sample.
    2. The pupils in the sample are selected independently of one another.
    3. Give a reason why the number of the first pupil in the sample should not be obtained simply by adding together the scores on the two dice. Justify your answer.
OCR S2 2007 January Q3
6 marks Moderate -0.5
3 A fair dice is thrown 90 times. Use an appropriate approximation to find the probability that the number 1 is obtained 14 or more times.
OCR S2 2007 January Q4
7 marks Moderate -0.8
4 A set of observations of a random variable \(W\) can be summarised as follows: $$n = 14 , \quad \Sigma w = 100.8 , \quad \Sigma w ^ { 2 } = 938.70 .$$
  1. Calculate an unbiased estimate of the variance of \(W\).
  2. The mean of 70 observations of \(W\) is denoted by \(\bar { W }\). State the approximate distribution of \(\bar { W }\), including unbiased estimate(s) of any parameter(s).
OCR S2 2007 January Q5
12 marks Standard +0.3
5 On a particular night, the number of shooting stars seen per minute can be modelled by the distribution \(\operatorname { Po(0.2). }\)
  1. Find the probability that, in a given 6 -minute period, fewer than 2 shooting stars are seen.
  2. Find the probability that, in 20 periods of 6 minutes each, the number of periods in which fewer than 2 shooting stars are seen is exactly 13 .
  3. Use a suitable approximation to find the probability that, in a given 2-hour period, fewer than 30 shooting stars are seen.
OCR S2 2007 January Q6
13 marks Standard +0.3
6 The continuous random variable \(X\) has the following probability density function: $$f ( x ) = \begin{cases} a + b x & 0 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are constants.
  1. Show that \(2 a + 2 b = 1\).
  2. It is given that \(\mathrm { E } ( X ) = \frac { 11 } { 9 }\). Use this information to find a second equation connecting \(a\) and \(b\), and hence find the values of \(a\) and \(b\).
  3. Determine whether the median of \(X\) is greater than, less than, or equal to \(\mathrm { E } ( X )\).
OCR S2 2007 January Q7
11 marks Standard +0.3
7 A television company believes that the proportion of households that can receive Channel C is 0.35 .
  1. In a random sample of 14 households it is found that 2 can receive Channel C. Test, at the \(2.5 \%\) significance level, whether there is evidence that the proportion of households that can receive Channel C is less than 0.35.
  2. On another occasion the test is carried out again, with the same hypotheses and significance level as in part (i), but using a new sample, of size \(n\). It is found that no members of the sample can receive Channel C. Find the largest value of \(n\) for which the null hypothesis is not rejected. Show all relevant working.
OCR S2 2007 January Q8
14 marks Challenging +1.8
8 The quantity, \(X\) milligrams per litre, of silicon dioxide in a certain brand of mineral water is a random variable with distribution \(\mathrm { N } \left( \mu , 5.6 ^ { 2 } \right)\).
  1. A random sample of 80 observations of \(X\) has sample mean 100.7. Test, at the \(1 \%\) significance level, the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 102\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : \mu \neq 102\).
  2. The test is redesigned so as to meet the following conditions.
    • The hypotheses are \(\mathrm { H } _ { 0 } : \mu = 102\) and \(\mathrm { H } _ { 1 } : \mu < 102\).
    • The significance level is \(1 \%\).
    • The probability of making a Type II error when \(\mu = 100\) is to be (approximately) 0.05 .
    The sample size is \(n\), and the critical region is \(\bar { X } < c\), where \(\bar { X }\) denotes the sample mean.
    1. Show that \(n\) and \(c\) satisfy (approximately) the equation \(102 - c = \frac { 13.0256 } { \sqrt { n } }\).
    2. Find another equation satisfied by \(n\) and \(c\).
    3. Hence find the values of \(n\) and \(c\).
OCR C1 Q1
3 marks Moderate -0.8
  1. Find the set of values of the constant \(k\) such that the equation
$$x ^ { 2 } - 6 x + k = 0$$ has real and distinct roots.
OCR C1 Q2
4 marks Moderate -0.8
2. The points \(A , B\) and \(C\) have coordinates \(( - 3,0 ) , ( 5 , - 2 )\) and \(( 4,1 )\) respectively. Find an equation for the straight line which passes through \(C\) and is parallel to \(A B\). Give your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.
OCR C1 Q3
4 marks Easy -1.2
3. (i) Express \(\frac { 18 } { \sqrt { 3 } }\) in the form \(k \sqrt { 3 }\).
(ii) Express \(( 1 - \sqrt { 3 } ) ( 4 - 2 \sqrt { 3 } )\) in the form \(a + b \sqrt { 3 }\) where \(a\) and \(b\) are integers.
OCR C1 Q4
4 marks Moderate -0.5
4. Solve the inequality $$2 x ^ { 2 } - 9 x + 4 < 0 .$$
OCR C1 Q5
7 marks Moderate -0.5
  1. Given that
$$\left( x ^ { 2 } + 2 x - 3 \right) \left( 2 x ^ { 2 } + k x + 7 \right) \equiv 2 x ^ { 4 } + A x ^ { 3 } + A x ^ { 2 } + B x - 21 ,$$ find the values of the constants \(k , A\) and \(B\).
OCR C1 Q6
8 marks Moderate -0.8
6. \includegraphics[max width=\textwidth, alt={}, center]{00364339-8108-4031-8e67-6100810e8297-2_549_885_251_370} The diagram shows the graph of \(y = \mathrm { f } ( x )\).
  1. Write down the number of solutions that exist for the equation
    1. \(\mathrm { f } ( x ) = 1\),
    2. \(\mathrm { f } ( x ) = - x\).
  2. Labelling the axes in a similar way, sketch on separate diagrams the graphs of
    1. \(\quad y = \mathrm { f } ( x - 2 )\),
    2. \(y = \mathrm { f } ( 2 x )\).
OCR C1 Q7
9 marks Moderate -0.8
7. $$f ( x ) = x ^ { 3 } - 9 x ^ { 2 }$$
  1. Find \(\mathrm { f } ^ { \prime } ( x )\).
  2. Find \(\mathrm { f } ^ { \prime \prime } ( x )\).
  3. Find the coordinates of the stationary points of the curve \(y = \mathrm { f } ( x )\).
  4. Determine whether each stationary point is a maximum or a minimum point.
OCR C1 Q8
10 marks Moderate -0.8
8. $$f ( x ) = 9 + 6 x - x ^ { 2 } .$$
  1. Find the values of \(A\) and \(B\) such that $$\mathrm { f } ( x ) = A - ( x + B ) ^ { 2 }$$
  2. State the maximum value of \(\mathrm { f } ( x )\).
  3. Solve the equation \(\mathrm { f } ( x ) = 0\), giving your answers in the form \(a + b \sqrt { 2 }\) where \(a\) and \(b\) are integers.
  4. Sketch the curve \(y = \mathrm { f } ( x )\).
OCR C1 Q9
10 marks Moderate -0.3
9. The circle \(C\) has centre \(( - 3,2 )\) and passes through the point \(( 2,1 )\).
  1. Find an equation for \(C\).
  2. Show that the point with coordinates \(( - 4,7 )\) lies on \(C\).
  3. Find an equation for the tangent to \(C\) at the point ( - 4 , 7). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
OCR C1 Q10
13 marks Standard +0.3
10. A curve has the equation \(y = ( \sqrt { x } - 3 ) ^ { 2 } , x \geq 0\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1 - \frac { 3 } { \sqrt { x } }\). The point \(P\) on the curve has \(x\)-coordinate 4 .
  2. Find an equation for the normal to the curve at \(P\) in the form \(y = m x + c\).
  3. Show that the normal to the curve at \(P\) does not intersect the curve again.
OCR S2 2008 January Q1
6 marks Standard +0.3
1 The random variable \(T\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\). It is given that \(\mathrm { P } ( T > 80 ) = 0.05\) and \(\mathrm { P } ( T > 50 ) = 0.75\). Find the values of \(\mu\) and \(\sigma\).
OCR S2 2008 January Q2
5 marks Moderate -0.8
2 A village has a population of 600 people. A sample of 12 people is obtained as follows. A list of all 600 people is obtained and a three-digit number, between 001 and 600 inclusive, is allocated to each name in alphabetical order. Twelve three-digit random numbers, between 001 and 600 inclusive, are obtained and the people whose names correspond to those numbers are chosen.
  1. Find the probability that all 12 of the numbers chosen are 500 or less.
  2. When the selection has been made, it is found that all of the numbers chosen are 500 or less. One of the people in the village says, "The sampling method must have been biased." Comment on this statement.
OCR S2 2008 January Q3
8 marks Standard +0.8
3 The random variable \(G\) has the distribution \(\operatorname { Po } ( \lambda )\). A test is carried out of the null hypothesis \(\mathrm { H } _ { 0 } : \lambda = 4.5\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : \lambda \neq 4.5\), based on a single observation of \(G\). The critical region for the test is \(G \leqslant 1\) and \(G \geqslant 9\).
  1. Find the significance level of the test.
  2. Given that \(\lambda = 5.5\), calculate the probability that the test results in a Type II error.
OCR S2 2008 January Q4
7 marks Moderate -0.8
4 The random variable \(Y\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). The results of 40 independent observations of \(Y\) are summarised by $$\Sigma y = 3296.0 , \quad \Sigma y ^ { 2 } = 286800.40$$
  1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
  2. Use your answers to part (i) to estimate the probability that a single random observation of \(Y\) will be less than 60.0.
  3. Explain whether it is necessary to know that \(Y\) is normally distributed in answering part (i) of this question.
OCR S2 2008 January Q5
9 marks Standard +0.3
5 Over a long period the number of visitors per week to a stately home was known to have the distribution \(\mathrm { N } \left( 500,100 ^ { 2 } \right)\). After higher car parking charges were introduced, a sample of four randomly chosen weeks gave a mean number of visitors per week of 435 . You should assume that the number of visitors per week is still normally distributed with variance \(100 ^ { 2 }\).
  1. Test, at the \(10 \%\) significance level, whether there is evidence that the mean number of visitors per week has fallen.
  2. Explain why it is necessary to assume that the distribution of the number of visitors per week (after the introduction of higher charges) is normal in order to carry out the test.
OCR S2 2008 January Q6
11 marks Standard +0.3
6 The number of house sales per week handled by an estate agent is modelled by the distribution \(\operatorname { Po } ( 3 )\).
  1. Find the probability that, in one randomly chosen week, the number of sales handled is
    1. greater than 4 ,
    2. exactly 4 .
    3. Use a suitable approximation to the Poisson distribution to find the probability that, in a year consisting of 50 working weeks, the estate agent handles more than 165 house sales.
    4. One of the conditions needed for the use of a Poisson model to be valid is that house sales are independent of one another.
      (a) Explain, in non-technical language, what you understand by this condition.
      (b) State another condition that is needed.
OCR S2 2008 January Q7
13 marks Moderate -0.3
7 A continuous random variable \(X _ { 1 }\) has probability density function given by $$f ( x ) = \begin{cases} k x & 0 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 1 } { 2 }\).
  2. Sketch the graph of \(y = \mathrm { f } ( x )\).
  3. Find \(\mathrm { E } \left( X _ { 1 } \right)\) and \(\operatorname { Var } \left( X _ { 1 } \right)\).
  4. Sketch the graph of \(y = \mathrm { f } ( x - 1 )\).
  5. The continuous random variable \(X _ { 2 }\) has probability density function \(\mathrm { f } ( x - 1 )\) for all \(x\). Write down the values of \(\mathrm { E } \left( X _ { 2 } \right)\) and \(\operatorname { Var } \left( X _ { 2 } \right)\).
OCR S2 2008 January Q8
13 marks Standard +0.3
8 Consultations are taking place as to whether a site currently in use as a car park should be developed as a shopping mall. An agency acting on behalf of a firm of developers claims that at least \(65 \%\) of the local population are in favour of the development. In a survey of a random sample of 12 members of the local population, 6 are in favour of the development.
  1. Carry out a test, at the \(10 \%\) significance level, to determine whether the result of the survey is consistent with the claim of the agency.
  2. A local residents' group claims that no more than \(35 \%\) of the local population are in favour of the development. Without further calculations, state with a reason what can be said about the claim of the local residents' group.
  3. A test is carried out, at the \(15 \%\) significance level, of the agency's claim. The test is based on a random sample of size \(2 n\), and exactly \(n\) of the sample are in favour of the development. Find the smallest possible value of \(n\) for which the outcome of the test is to reject the agency's claim.
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