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OCR MEI Further Mechanics Major 2021 November Q3
4 marks Moderate -0.8
3 One end of a light elastic spring of natural length 0.3 m is attached to a fixed point. A mass of 4 kg is attached to the other end of the spring. When the spring hangs vertically in equilibrium the extension of the spring is 0.02 m .
  1. Determine the modulus of elasticity of the spring. A student calculates that if the mass of 4 kg is removed and replaced with a mass of 20 kg the extension of the spring will be 0.1 m .
  2. Suggest a reason why this extension may not be 0.1 m .
OCR MEI Further Mechanics Major 2021 November Q4
6 marks Challenging +1.2
4 In this question you must show detailed reasoning. \includegraphics[max width=\textwidth, alt={}, center]{17e92314-d7df-49b8-a441-8d18c91dbbb0-03_646_812_312_242} The diagram shows parts of the curves \(y = 3 \sqrt { x }\) and \(y = 4 - x ^ { 2 }\), which intersect at the point ( 1,3 ). The shaded region, bounded by the two curves and the \(y\)-axis, is occupied by a uniform lamina. Determine the exact \(x\)-coordinate of the centre of mass of the lamina.
OCR MEI Further Mechanics Major 2021 November Q5
6 marks Standard +0.8
5 Two small uniform smooth spheres A and B , of equal radius, have masses 2 kg and 4 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision, A has speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving along the line of centres, and B has speed \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving along a line which is perpendicular to the line of centres (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{17e92314-d7df-49b8-a441-8d18c91dbbb0-03_389_764_1592_244} The direction of motion of B after the collision makes an angle of \(45 ^ { \circ }\) with the line of centres. Determine the coefficient of restitution between A and B .
OCR MEI Further Mechanics Major 2021 November Q6
11 marks Moderate -0.8
6
  1. Write down the dimensions of force. The force \(F\) of gravitational attraction between two objects with masses \(m _ { 1 }\) and \(m _ { 2 }\), at a distance \(d\) apart, is given by $$F = \frac { G m _ { 1 } m _ { 2 } } { d ^ { 2 } }$$ where \(G\) is the universal gravitational constant.
    In SI units the value of \(G\) is \(6.67 \times 10 ^ { - 11 } \mathrm {~kg} ^ { - 1 } \mathrm {~m} ^ { 3 } \mathrm {~s} ^ { - 2 }\).
  2. Write down the dimensions of \(G\).
  3. Determine the value of \(G\) in imperial units based on pounds, feet, and seconds. Use the facts that 1 pound \(= 0.454 \mathrm {~kg}\) and 1 foot \(= 0.305 \mathrm {~m}\). For a planet of mass \(M\) and radius \(r\), it is suggested that the velocity \(v\) needed for an object to escape the gravitational pull of the planet, the 'escape velocity', is given by the following formula. \(\mathrm { v } = \sqrt { \frac { \mathrm { kGM } } { \mathrm { r } } }\),
    where \(k\) is a dimensionless constant.
  4. Show that this formula is dimensionally consistent. Information regarding the planets Earth and Mars can be found in the table below.
    EarthMars
    Radius (m)63710003389500
    Mass (kg)\(5.97 \times 10 ^ { 24 }\)\(6.39 \times 10 ^ { 23 }\)
    Escape velocity ( \(\mathrm { m } \mathrm { s } ^ { - 1 }\) )11186
  5. Using the formula \(\mathrm { v } = \sqrt { \frac { \mathrm { kGM } } { \mathrm { r } } }\), determine the escape velocity for planet Mars.
OCR MEI Further Mechanics Major 2021 November Q7
12 marks Challenging +1.2
7 A box B of mass \(m \mathrm {~kg}\) is raised vertically by an engine working at a constant rate of \(k m g \mathrm {~W}\). Initially B is at rest. The speed of B when it has been raised a distance \(x \mathrm {~m}\) is denoted by \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that \(v ^ { 2 } \frac { d v } { d x } = ( k - v ) g\).
  2. Verify that \(\mathrm { gx } = \mathrm { k } ^ { 2 } \ln \left( \frac { \mathrm { k } } { \mathrm { k } - \mathrm { v } } \right) - \mathrm { kv } - \frac { 1 } { 2 } \mathrm { v } ^ { 2 }\).
  3. By using the work-energy principle, show that the time taken for B to reach a speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from rest is given by \(\frac { \mathrm { k } } { \mathrm { g } } \ln \left( \frac { \mathrm { k } } { \mathrm { k } - \mathrm { V } } \right) - \frac { \mathrm { V } } { \mathrm { g } }\).
OCR MEI Further Mechanics Major 2021 November Q8
12 marks Challenging +1.8
8 A capsule consists of a uniform hollow right circular cylinder of radius \(r\) and length \(2 h\) attached to two uniform hollow hemispheres of radius \(r\).
The centres of the plane faces of the hemispheres coincide with the centres, A and B , of the ends of the cylinder. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{17e92314-d7df-49b8-a441-8d18c91dbbb0-06_702_684_445_244} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure} Fig. 8 represents a vertical cross-section through a plane of symmetry of the capsule as it rests in limiting equilibrium with a point C of one hemisphere on a rough horizontal floor and a point D of the other hemisphere against a rough vertical wall. The total weight of the capsule is \(W\) and acts at a point midway between A and B . The plane containing \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D is vertical, with AB making an acute angle \(\theta\) with the downward vertical.
  1. Complete the copy of Fig. 8 in the Printed Answer Booklet to show all the remaining forces acting on the capsule. The coefficient of friction at each point of contact is \(\frac { 1 } { 3 }\).
  2. By resolving vertically and horizontally, determine the magnitude of the normal contact force between the floor and the capsule in terms of \(W\).
  3. By determining an expression for \(r\) in terms of \(h\) and \(\theta\), show that \(\tan \theta > \frac { 3 } { 4 }\).
OCR MEI Further Mechanics Major 2021 November Q9
15 marks Challenging +1.2
9 A small ball P is projected with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation of \(( \alpha + \theta )\) from a point O at the bottom of a plane inclined at \(\alpha\) to the horizontal. P subsequently hits the plane at a point R , where OR is a line of greatest slope, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{17e92314-d7df-49b8-a441-8d18c91dbbb0-07_456_862_406_242}
  1. By deriving an expression, in terms of \(\theta\), \(\alpha\) and \(g\), for the time of flight of P , show that the distance OR, in metres, is $$\frac { 50 \sin \theta \cos ( \theta + \alpha ) } { g \cos ^ { 2 } \alpha }$$
  2. By using the identity \(2 \sin \mathrm {~A} \cos \mathrm {~B} \equiv \sin ( \mathrm {~A} + \mathrm { B } ) - \sin ( \mathrm { B } - \mathrm { A } )\), determine, in terms of \(g\) and \(\sin \alpha\), an expression for the maximum range of P up the plane, as \(\theta\) varies.
  3. Given that OR is the maximum range of P up the plane and is equal to 1.8 m , determine the value of \(\theta\). \includegraphics[max width=\textwidth, alt={}, center]{17e92314-d7df-49b8-a441-8d18c91dbbb0-08_625_1180_255_239} A rigid wire ABC is fixed in a vertical plane. The section AB of the wire, of length \(b\), is straight and horizontal. The section BC of the wire is smooth and in the form of a circular arc of radius \(a\) and length \(\frac { 1 } { 2 } a \pi\). The centre of the arc is O , which is vertically above B . A bead P of mass \(m\) is threaded on the wire and projected from B with speed \(u\) towards C . The angle BOP when P is between B and C is denoted by \(\theta\), as shown in the diagram.
OCR MEI Further Mechanics Major 2021 November Q11
16 marks Challenging +1.2
11 Two small uniform smooth spheres A and B , of equal radius, have masses 4 kg and 3 kg respectively. The spheres are placed in a smooth horizontal circular groove. The coefficient of restitution between the spheres is \(e\), where \(e > \frac { 2 } { 5 }\). At a given instant B is at rest and A is set moving along the groove with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It may be assumed that in the subsequent motion the two spheres do not leave the groove.
  1. Determine, in terms of \(e\) and \(V\), the speeds of A and B immediately after the first collision.
  2. Show that the arc through which A moves between the first and second collisions subtends an angle at the centre of the circular groove of $$\frac { 2 \pi ( 4 - 3 e ) } { 7 e } \text { radians. }$$
    1. Determine, in terms of \(e\) and \(V\), the speed of B immediately after the second collision.
    2. What can be said about the motion of A and B if the collisions between A and B are perfectly elastic?
OCR MEI Further Mechanics Major 2021 November Q12
18 marks Challenging +1.2
12 A particle P of mass \(m\) is fixed to one end of a light elastic string of natural length \(l\) and modulus of elasticity 12 mg . The other end of the string is attached to a fixed point O . Particle P is held next to O and then released from rest.
  1. Show that P next comes instantaneously to rest when the length of the string is \(\frac { 3 } { 2 } l\). The string first becomes taut at time \(t = 0\). At time \(t \geqslant 0\), the length of the string is \(l + x\), where \(x\) is the extension in the string.
  2. Show that when the string is taut, \(x\) satisfies the differential equation $$\ddot { \mathrm { x } } + \omega ^ { 2 } \mathrm { x } = \mathrm { g } \text {, where } \omega ^ { 2 } = \frac { 12 \mathrm {~g} } { \mathrm { I } } \text {. }$$
  3. By using the substitution \(x = y + \frac { g } { \omega ^ { 2 } }\), solve the differential equation to show that the time when the string first becomes slack satisfies the equation $$\cos \omega \mathrm { t } - \sqrt { \mathrm { k } } \sin \omega \mathrm { t } = 1$$ where \(k\) is an integer to be determined.
OCR MEI Further Statistics Minor 2019 June Q1
7 marks Easy -1.3
1 In a game at a charity fair, a spinner is spun 4 times.
On each spin the chance that the spinner lands on a score of 5 is 0.2 .
The random variable \(X\) represents the number of spins on which the spinner lands on a score of 5 .
  1. Find \(\mathrm { P } ( X = 3 )\).
  2. Find each of the following.
    • \(\mathrm { E } ( X )\)
    • \(\operatorname { Var } ( X )\)
    One game costs \(\pounds 1\) to play and, for each spin that lands on a score of 5 , the player receives 50 pence.
    1. Find the expected total amount of money gained by a player in one game.
    2. Find the standard deviation of the total amount of money gained by a player in one game.
OCR MEI Further Statistics Minor 2019 June Q2
9 marks Moderate -0.3
2 A market researcher wants to interview people who watched a particular television programme. Audience research data used by the broadcaster indicates that \(12 \%\) of the adult population watched this programme. This figure is used to model the situation.
The researcher asks people in a shopping centre, one at a time, if they watched the programme. You should assume that these people form a random sample of the adult population.
  1. Find the probability that the fifth person the researcher asks is the first to have watched the programme.
  2. Find the probability that the researcher has to ask at least 10 people in order to find one who watched the programme.
  3. Find the probability that the twentieth person the researcher asks is the third to have watched the programme.
  4. Find how many people the researcher would have to ask to ensure that there is a probability of at least 0.95 that at least one of them watched the programme.
OCR MEI Further Statistics Minor 2019 June Q3
4 marks Easy -1.8
3 A company has been commissioned to make 50 very expensive titanium components.
A sample of the components needs to be tested to ensure that they are sufficiently strong. However, this is a test to destruction, so the components which are tested can no longer be used.
  1. Explain why it would not be appropriate to use a census in these circumstances. A manager suggests that the first 5 components to be manufactured should be tested.
  2. Explain why this would not be a sensible method of selecting the sample. A statistician advises the manager that the sample selected should be a random sample.
  3. Give two desirable features (other than randomness) that the sample should have.
OCR MEI Further Statistics Minor 2019 June Q4
17 marks Standard +0.3
4 Zara uses a metal detector to search for coins on a beach.
She wonders if the numbers of coins that she finds in an area of \(10 \mathrm {~m} ^ { 2 }\) can be modelled by a Poisson distribution. The table below shows the numbers of coins that she finds in randomly chosen areas of \(10 \mathrm {~m} ^ { 2 }\) over a period of months.
Number of coins found0123456\(> 6\)
Frequency1328301410230
  1. Software gives the sample mean as 1.98 and the sample standard deviation as 1.4212. Explain how these values suggest that a Poisson distribution may be an appropriate model for the numbers of coins found. Zara decides to carry out a chi-squared test to investigate whether a Poisson distribution is an appropriate model.
    Fig. 4 is a screenshot showing part of the spreadsheet used to analyse the data. Some values in the spreadsheet have been deliberately omitted. \begin{table}[h]
    ABCD
    1Number of coins foundObserved frequencyExpected frequencyChi-squared contribution
    201313.80690.0472
    3128
    423027.06430.3184
    531417.86250.8352
    64108.84190.1517
    7\(\geqslant 5\)50.0015
    \captionsetup{labelformat=empty} \caption{Fig. 4}
    \end{table}
  2. Showing your calculations, find the missing values in each of the following cells.
    For the rest of this question, you should assume that the number of coins that Zara finds in an area of \(10 \mathrm {~m} ^ { 2 }\) can be modelled by a Poisson distribution with mean 1.98.
    Zara also finds pieces of jewellery independently of the coins she finds. The number of pieces of jewellery that she finds per \(10 \mathrm {~m} ^ { 2 }\) area is modelled by a Poisson distribution with mean 0.42 .
  3. Find the probability that Zara finds a total of exactly 3 items (coins and/or jewellery) in an area of \(10 \mathrm {~m} ^ { 2 }\).
  4. Find the probability that Zara finds a total of at least 30 items (coins and/or jewellery) in an area of \(100 \mathrm {~m} ^ { 2 }\).
OCR MEI Further Statistics Minor 2019 June Q5
16 marks Standard +0.3
5 A student wants to know if there is a positive correlation between the amounts of two pollutants, sulphur dioxide and PM10 particulates, on different days in the area of London in which he lives; these amounts, measured in suitable units, are denoted by \(s\) and \(p\) respectively.
He uses a government website to obtain data for a random sample of 15 days on which the amounts of these pollutants were measured simultaneously. Fig. 5.1 is a scatter diagram showing the data. Summary statistics for these 15 values of \(s\) and \(p\) are as follows. \(\sum s _ { 1 } = 155.4 \quad \sum p = 518.9 \quad \sum s ^ { 2 } = 2322.7 \quad \sum p ^ { 2 } = 21270.5 \quad \sum s p = 6009.1\) \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4a4d5816-5b53-49a1-b72f-f8bcf3b4e8bc-4_935_1134_683_260} \captionsetup{labelformat=empty} \caption{Fig. 5.1}
\end{figure}
  1. Explain why the student might come to the conclusion that a test based on Pearson's product moment correlation coefficient may be valid.
  2. Find the value of Pearson's product moment correlation coefficient.
  3. Carry out a test at the \(5 \%\) significance level to investigate whether there is positive correlation between the amounts of sulphur dioxide and PM10 particulates.
  4. Explain why the student made sure that the sample chosen was a random sample. The student also wishes to model the relationship between the amounts of nitrogen dioxide \(n\) and PM10 particulates \(p\).
    He takes a random sample of 54 values of the two variables, both measured at the same times. Fig. 5.2 is a scatter diagram which shows the data, together with the regression line of \(n\) on \(p\), the equation of the regression line and the value of \(r ^ { 2 }\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{4a4d5816-5b53-49a1-b72f-f8bcf3b4e8bc-5_824_1230_495_258} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
    \end{figure}
  5. Predict the value of \(n\) for \(p = 150\).
  6. Discuss the reliability of your prediction in part (e).
OCR MEI Further Statistics Minor 2019 June Q6
7 marks Standard +0.8
6 The discrete random variable \(X\) has a uniform distribution over \(\{ n , n + 1 , \ldots , 2 n \}\).
  1. Given that \(n\) is odd, find \(\mathrm { P } \left( X < \frac { 3 } { 2 } n \right)\).
  2. Given instead that \(n\) is even, find \(\mathrm { P } \left( X < \frac { 3 } { 2 } n \right)\), giving your answer as a single algebraic fraction.
  3. The sum of 6 independent values of \(X\) is denoted by \(Y\). Find \(\operatorname { Var } ( Y )\).
OCR MEI Further Statistics Minor 2022 June Q1
6 marks Moderate -0.3
1 In a quiz a contestant is asked up to four questions. The contestant's turn ends once the contestant gets a question wrong or has answered all four questions. The probability that a particular contestant gets any question correct is 0.6 , independently of other questions. The discrete random variable \(X\) models the number of questions which the contestant gets correct in a turn.
  1. Show that \(\mathrm { P } ( X = 4 ) = 0.1296\). The probability distribution of \(X\) is shown in Fig. 1.1. \begin{table}[h]
    \(r\)01234
    \(\mathrm { P } ( X = r )\)0.40.240.1440.08640.1296
    \captionsetup{labelformat=empty} \caption{Fig. 1.1}
    \end{table}
  2. Find each of the following.
    The number of points that a contestant scores is as shown in Fig. 1.2. \begin{table}[h]
    Number of
    questions correct
    Number of
    points scored
    0 or 10
    22
    33
    45
    \captionsetup{labelformat=empty} \caption{Fig. 1.2}
    \end{table} The discrete random variable \(Y\) models the number of points which the contestant scores.
  3. Without doing any working, explain whether each of the following will be less than, equal to or greater than the corresponding value for \(X\).
OCR MEI Further Statistics Minor 2022 June Q2
13 marks Moderate -0.8
2 A forester is investigating the relationship between the diameter and the height of young beech trees. She selects a random sample of 15 young beech trees in a forest and records their diameters, \(d \mathrm {~cm}\), and their heights, \(h \mathrm {~m}\). The data are illustrated in the scatter diagram. \includegraphics[max width=\textwidth, alt={}, center]{e8624e9b-5143-49d2-9683-cc3a1082694e-3_649_1116_386_230}
  1. State whether either or both of the variables \(d\) and \(h\) are random variables. Summary data for the diameters and heights are as follows. $$\mathrm { n } = 15 \quad \sum \mathrm {~d} = 84.9 \quad \sum \mathrm {~h} = 124.7 \quad \sum \mathrm {~d} ^ { 2 } = 624.55 \quad \sum \mathrm {~h} ^ { 2 } = 1230.57 \quad \sum \mathrm { dh } = 866.63$$
  2. Find the equation of the regression line of \(h\) on \(d\). Give your answer in the form \(h = a d + b\), giving the values of \(a\) and \(b\) correct to \(\mathbf { 2 }\) decimal places.
  3. Use the regression line to predict the heights of beech trees with the following diameters.
    Comment on this in relation to your regression line.
  4. State the coordinates of the point at which the regression line of \(d\) on \(h\) meets the line which you calculated in part (b).
OCR MEI Further Statistics Minor 2022 June Q3
15 marks Standard +0.8
3 Jane wonders whether the number of wasps entering a wasp's nest per 5 second interval can be modelled by a Poisson distribution with mean \(\mu\). She counts the number of wasps entering the nest over 60 randomly selected 5 -second intervals. The results are shown in Fig. 3.1. \begin{table}[h]
Number of wasps0123456789\(\geqslant 10\)
Frequency025512101011140
\captionsetup{labelformat=empty} \caption{Fig. 3.1}
\end{table}
  1. Show that a suitable estimate for the value of \(\mu\) is 5.1. Fig. 3.2 shows part of a screenshot for a \(\chi ^ { 2 }\) test to assess the goodness of fit of a Poisson model. The sample mean has been used as an estimate for the population mean. Some of the values in the spreadsheet have been deliberately omitted. \begin{table}[h]
    ABCDE
    \includegraphics[max width=\textwidth, alt={}]{e8624e9b-5143-49d2-9683-cc3a1082694e-4_132_40_1069_273}Number of waspsObserved frequencyPoisson probabilityExpected frequencyChi-squared contribution
    2\(\leqslant 2\)70.11656.98870.0000
    3358.08741.1786
    44120.2765
    55100.0255
    66100.14908.94000.1257
    77110.10866.51343.0904
    8\(\geqslant 8\)50.14408.6414
    9
    \captionsetup{labelformat=empty} \caption{Fig. 3.2}
    \end{table}
  2. Determine the missing values in each of the following cells, giving your answers correct to 4 decimal places.
    Carry out the hypothesis test at the 5\% significance level.
  3. Jane also carries out a \(\chi ^ { 2 }\) test for the number of wasps leaving another nest. As part of her calculations, she finds that the probability of no wasps leaving the nest in a 5 -second period is 0.0053 . She finds that a Poisson distribution is also an appropriate model in this case. Find a suitable estimate for the value of the mean number of wasps leaving the nest per 5-second period.
OCR MEI Further Statistics Minor 2022 June Q4
5 marks Moderate -0.8
4 Alex is practising bowling at a cricket wicket. Every time she bowls a ball, she has a \(30 \%\) chance of hitting the wicket.
  1. Assuming that successive bowls are independent, determine the probability that Alex first hits the wicket on her third attempt.
  2. Determine the probability that Alex hits the wicket for the fourth time on her tenth attempt.
OCR MEI Further Statistics Minor 2022 June Q5
14 marks Standard +0.3
5 A medical researcher is investigating whether there is any relationship between the age of a person and the level of a particular protein in the person's blood. She measures the levels of the protein (measured in suitable units) in a random sample of 12 hospital patients of various ages (in years). The spreadsheet shows the values obtained, together with a scatter diagram which illustrates the data. \includegraphics[max width=\textwidth, alt={}, center]{e8624e9b-5143-49d2-9683-cc3a1082694e-5_736_1470_1087_246}
  1. The researcher decides that a test based on Pearson's product moment correlation coefficient may not be valid. Explain why she comes to this conclusion.
  2. Calculate the value of Spearman's rank correlation coefficient.
  3. Carry out a test based on this coefficient at the \(5 \%\) significance level to investigate whether there is any association between age and protein level.
  4. Explain why the researcher chose a sample that was random.
  5. The researcher had originally intended to use a sample size of 6 rather than the 12 that she actually used. Explain what advantage there is in using the larger sample size.
OCR MEI Further Statistics Minor 2022 June Q6
7 marks Standard +0.8
6 The random variable \(X\) has a uniform distribution over the values \(\{ 1,4,7 , \ldots , 3 n - 2 \}\), where \(n\) is a positive integer.
  1. Determine \(\operatorname { Var } ( X )\) in terms of \(n\).
  2. Given that \(n = 100\), find the probability that \(X\) is within one standard deviation of the mean.
OCR MEI Further Statistics Minor 2023 June Q1
8 marks Standard +0.3
1 A fair spinner has ten sectors, labelled \(1,2 , \ldots , 10\). In order to start a game, Kofi has to obtain an 8,9 or 10 on the spinner.
  1. Find the probability that Kofi starts the game on the third spin.
  2. Find the probability that Kofi takes at least 5 spins to start the game.
  3. Determine the probability that the number of spins required to start the game is within 1 standard deviation of its mean.
OCR MEI Further Statistics Minor 2023 June Q2
5 marks Easy -1.8
2 A company manufactures batches of twenty thousand tins which are subsequently filled with fruit. The company tests tins from each batch to make sure that they are strong enough. The test is easy and cheap to carry out, but when a tin has been tested it is no longer suitable for filling with fruit.
    1. Explain why a sample size of 5 tins per batch may not be appropriate in this case.
    2. Explain why a sample size of 1000 tins per batch may not be appropriate in this case. The company tests a sample of 30 tins from each batch.
  2. Explain why it would not be sensible for the sample to consist of the final 30 tins produced in a batch.
  3. Give two features that the sample should have.
OCR MEI Further Statistics Minor 2023 June Q3
10 marks Standard +0.3
3 A fair four-sided dice has its faces numbered \(0,1,2,3\). The dice is rolled three times. The discrete random variable \(X\) is the sum of the lowest and highest scores obtained.
  1. Show that \(\mathrm { P } ( X = 1 ) = \frac { 3 } { 32 }\). The table below shows the probability distribution of \(X\).
    \(r\)0123456
    \(\mathrm { P } ( X = r )\)\(\frac { 1 } { 64 }\)\(\frac { 3 } { 32 }\)\(\frac { 13 } { 64 }\)\(\frac { 3 } { 8 }\)\(\frac { 13 } { 64 }\)\(\frac { 3 } { 32 }\)\(\frac { 1 } { 64 }\)
  2. In this question you must show detailed reasoning. Find each of the following.
    • \(\mathrm { E } ( X )\)
    • \(\operatorname { Var } ( X )\)
    • The random variable \(Y\) represents the sum of 10 values of \(X\).
      1. State a property of the 10 values of \(X\) that would make it possible to deduce the standard deviation of \(Y\).
      2. Given that this property holds, determine the standard deviation of \(Y\).
OCR MEI Further Statistics Minor 2023 June Q4
13 marks Standard +0.3
4 Eve lives in a narrow lane in the country. She wonders whether the number of vehicles passing her house per minute can be modelled by a Poisson distribution with mean \(\mu\). She counts the number of vehicles passing her house over 100 randomly selected one-minute intervals. The results are shown in Table 4.1. \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Table 4.1}
Number of vehicles012345678910\(\geqslant 11\)
Frequency3633141041001010
\end{table}
  1. Use the results to find an estimate for \(\mu\). The spreadsheet in Fig. 4.2 shows data for a \(\chi ^ { 2 }\) test to assess the goodness of fit of a Poisson model. The sample mean from part (a) has been used as an estimate for the population mean. Some of the values in the spreadsheet have been deliberately omitted. \begin{table}[h]
    \captionsetup{labelformat=empty} \caption{Fig. 4.2}
    \multirow[b]{2}{*}{1}ABCDE
    Number of vehiclesObserved frequencyPoisson probabilityExpected frequencyChi-squared contribution
    20360.272527.25322.8073
    31330.354335.4291
    42143.5400
    5\(\geqslant 3\)170.5145
    6
    \end{table}
  2. Calculate the missing values in each of the following cells, giving your answers correct to 4 decimal places.
    Carry out the \(\chi ^ { 2 }\) test at the 5\% significance level.
  3. Eve checks her data and notices that the two largest numbers of vehicles per minute (8 and 10) occurred when some horses were being ridden along the lane, causing delays to the vehicles. She therefore repeats the analysis, missing out these two items of data. She finds that the value of the \(\chi ^ { 2 }\) test statistic is now 4.748. The number of degrees of freedom of the test is unchanged. Make two comments about this revised test.