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Edexcel S4 2017 June Q4
12 marks Standard +0.3
4. A coach believes that the average score in the final round of a golf tournament is more than one point below the average score in the first round. To test this belief, the scores of 8 randomly selected players are recorded. The results are given in the table below.
Player\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
First round7680727883888172
Final round7078757579848369
    1. State why a paired \(t\)-test is suitable for use with these data.
    2. State an assumption that needs to be made in order to carry out a paired \(t\)-test in this case.
  2. Test, at the \(5 \%\) level of significance, whether or not there is evidence to support the coach's belief. Show your working clearly.
  3. Explain, in the context of the coach's belief, what a Type II error would be in this case.
Edexcel S4 2017 June Q5
11 marks Challenging +1.2
  1. Jamland and Goodjam are two suppliers of jars of jam. The weights of the jars of jam produced by each supplier can be assumed to be normally distributed with unknown, but equal, variances. A random sample of 20 jars of jam is taken from those supplied by Jamland.
Based on this sample, the 95\% confidence interval for the mean weight of a jar of Jamland jam, in grams, is
[0pt] [ 492, 507 ] A random sample of 10 jars of jam is selected from those supplied by Goodjam. The weight of each jar of Goodjam jam, \(y\) grams, is recorded. The results are summarised as follows $$\bar { y } = 480 \quad s _ { y } ^ { 2 } = 280$$ Find a 90\% confidence interval for the value by which the mean weight of a jar of jam supplied by Jamland exceeds the mean weight of a jar of jam supplied by Goodjam.
Edexcel S4 2017 June Q6
19 marks Challenging +1.2
6. The independent random variables \(X _ { 1 }\) and \(X _ { 2 }\) are each distributed \(\mathrm { B } ( n , p )\), where \(n > 1\) An unbiased estimator for \(p\) is given by $$\hat { p } = \frac { a X _ { 1 } + b X _ { 2 } } { n }$$ where \(a\) and \(b\) are constants.
[0pt] [You may assume that if \(X _ { 1 }\) and \(X _ { 2 }\) are independent then \(\mathrm { E } \left( X _ { 1 } X _ { 2 } \right) = \mathrm { E } \left( X _ { 1 } \right) \mathrm { E } \left( X _ { 2 } \right)\) ]
  1. Show that \(a + b = 1\)
  2. Show that \(\operatorname { Var } ( \hat { p } ) = \frac { \left( 2 a ^ { 2 } - 2 a + 1 \right) p ( 1 - p ) } { n }\)
  3. Hence, justifying your answer, determine the value of \(a\) and the value of \(b\) for which \(\hat { p }\) has minimum variance.
    1. Show that \(\hat { p } ^ { 2 }\) is a biased estimator for \(p ^ { 2 }\)
    2. Show that the bias \(\rightarrow 0\) as \(n \rightarrow \infty\)
  5. By considering \(\mathrm { E } \left[ X _ { 1 } \left( X _ { 1 } - 1 \right) \right]\) find an unbiased estimator for \(p ^ { 2 }\)
Edexcel S4 2018 June Q1
5 marks Moderate -0.3
  1. A machine fills packets with almonds. The weight, in grams, of almonds in a packet is modelled by \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). To check that the machine is working properly, a random sample of 10 packets is selected and unbiased estimates for \(\mu\) and \(\sigma ^ { 2 }\) are
$$\bar { x } = 202 \quad \text { and } \quad s ^ { 2 } = 3.6$$ Stating your hypotheses clearly, test, at the \(1 \%\) level of significance, whether or not the mean weight of almonds in a packet is more than 200 g .
Edexcel S4 2018 June Q2
13 marks Standard +0.3
  1. Jeremiah currently uses a Fruity model of juicer. He agrees to trial a new model of juicer, Zesty. The amounts of juice extracted, \(x \mathrm { ml }\), from each of 9 randomly selected oranges, using the Zesty are summarised as
$$\sum x = 468 \quad \sum x ^ { 2 } = 24560$$ Given that the amounts of juice extracted follow a normal distribution,
  1. calculate a 95\% confidence interval for
    1. the mean amount of juice extracted from an orange using the Zesty,
    2. the standard deviation of the amount of juice extracted from an orange using the Zesty. Jeremiah knows that, for his Fruity, the mean amount of juice extracted from an orange is 38 ml and the standard deviation of juice extracted from an orange is 5 ml . He decides that he will replace his Fruity with a Zesty if both
      • the mean for the Zesty is more than \(20 \%\) higher than the mean for his Fruity and
  2. the standard deviation for the Zesty is less than 5.5 ml .
  3. Using your answers to part (a), explain whether or not Jeremiah should replace his Fruity with the Zesty.
Edexcel S4 2018 June Q3
10 marks Challenging +1.2
  1. A random sample of 8 students is selected from a school database.
Each student's reaction time is measured at the start of the school day and again at the end of the school day. The reaction times, in milliseconds, are recorded below.
StudentA\(B\)CD\(E\)\(F\)G\(H\)
Reaction time at the start of the school day10.87.28.76.89.410.911.17.6
Reaction time at the end of the school day106.18.85.78.78.19.86.8
  1. State one assumption that needs to be made in order to carry out a paired \(t\)-test.
    (1) The random variable \(R\) is the reaction time at the start of the school day minus the reaction time at the end of the school day. The mean of \(R\) is \(\mu\). John uses a paired \(t\)-test to test the hypotheses $$\mathrm { H } _ { 0 } : \mu = m \quad \mathrm { H } _ { 1 } : \mu \neq m$$ Given that \(\mathrm { H } _ { 0 }\) is rejected at the 5\% level of significance but accepted at the 1\% level of significance,
  2. find the ranges of possible values for \(m\).
Edexcel S4 2018 June Q4
17 marks Challenging +1.2
  1. A glue supplier claims that Goglue is stronger than Tackfast. A company is presently using Tackfast but agrees to change to Goglue if, at the 5\% significance level,
  • the standard deviation of the force required for Goglue to fail is not greater than the standard deviation of the force required for Tackfast to fail and
  • the mean force required for Goglue to fail is more than 4 newtons greater than the mean force for Tackfast to fail.
A series of trials is carried out, using Goglue and Tackfast, and the glues are tested to destruction. The force, \(x\) newtons, at which each glue fails is recorded.
Sample size \(( n )\)Sample mean \(( \bar { x } )\)Standard deviation \(( s )\)
Tackfast \(( T )\)65.270.31
Goglue \(( G )\)510.120.66
It can be assumed that the force at which each glue fails is normally distributed.
  1. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the standard deviation of the force required for Goglue to fail is greater than the standard deviation of the force required for the Tackfast to fail. State your hypotheses clearly. The supplier claims that the mean force required for its Goglue to fail is more than 4 newtons greater than the mean force required for Tackfast to fail.
  2. Stating your hypotheses clearly and using a \(5 \%\) level of significance, test the supplier's claim.
  3. Show that, at the \(5 \%\) level of significance, the supplier's claim will be accepted if \(\bar { X } _ { G } - \bar { X } _ { T } > 4.55\), where \(\bar { X } _ { G }\) and \(\bar { X } _ { T }\) are the mean forces required for Goglue to fail and Tackfast to fail respectively. Later, it was found that an error had been made when recording the results for Goglue. This resulted in all the forces recorded for Goglue being 0.5 newtons more than they should have been. The results for Tackfast were correct.
  4. Explain whether or not this information affects the decision about which glue the supplier decides to use.
Edexcel S4 2018 June Q5
11 marks Challenging +1.2
  1. A machine makes posts. The length of a post is normally distributed with unknown mean \(\mu\) and standard deviation 4 cm .
A random sample of size \(n\) is taken to test, at the \(5 \%\) significance level, the hypotheses $$\mathrm { H } _ { 0 } : \mu = 150 \quad \mathrm { H } _ { 1 } : \mu > 150$$
  1. State the probability of a Type I error for this test. The manufacturer requires the probability of a Type II error to be less than 0.1 when the actual value of \(\mu\) is 152
  2. Calculate the minimum value of \(n\).
Edexcel S4 2018 June Q6
19 marks Challenging +1.2
  1. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\)
$$f ( x ) = \left\{ \begin{array} { c c } \frac { x } { 2 \theta ^ { 2 } } & 0 \leqslant x \leqslant 2 \theta \\ 0 & \text { otherwise } \end{array} \right.$$ where \(\theta\) is a constant.
  1. Use integration to show that \(\mathrm { E } \left( X ^ { N } \right) = \frac { 2 ^ { N + 1 } } { N + 2 } \theta ^ { N }\)
  2. Hence
    1. write down an expression for \(\mathrm { E } ( X )\) in terms of \(\theta\)
    2. find \(\operatorname { Var } ( X )\) in terms of \(\theta\) A random sample \(X _ { 1 } , X _ { 2 } , \ldots , X _ { n }\) where \(n \geqslant 2\) is taken to estimate the value of \(\theta\) The random variable \(S _ { 1 } = q \bar { X }\) is an unbiased estimator of \(\theta\)
  3. Write down the value of \(q\) and show that \(S _ { 1 }\) is a consistent estimator of \(\theta\) The continuous random variable \(Y\) is independent of \(X\) and is uniformly distributed over the interval \(\left[ 0 , \frac { 2 \theta } { 3 } \right]\), where \(\theta\) is the same unknown constant as in \(\mathrm { f } ( x )\). The random variable \(S _ { 2 } = a X + b Y\) is an unbiased estimator of \(\theta\) and is based on one observation of \(X\) and one observation of \(Y\).
  4. Find the value of \(a\) and the value of \(b\) for which \(S _ { 2 }\) has minimum variance.
  5. Show that the minimum variance of \(S _ { 2 }\) is \(\frac { \theta ^ { 2 } } { 11 }\)
  6. Explain which of \(S _ { 1 }\) or \(S _ { 2 }\) is the better estimator for \(\theta\)
Edexcel S4 Q1
8 marks Standard +0.3
  1. The weights of the contents of jars of jam are normally distributed with a stated mean of 100 g . A random sample of 7 jars was taken and the contents of each jar, \(x\) grams, was weighed. The results are summarised by the following statistics.
$$\sum x = 710.9 , \sum x ^ { 2 } = 72219.45 .$$ Test at the \(5 \%\) level of significance whether or not there is evidence that the mean weight of the contents of the jars is greater than 100 g . State your hypotheses clearly.
(8 marks)
Edexcel S4 Q2
8 marks Standard +0.3
2. An engineer decided to investigate whether or not the strength of rope was affected by water. A random sample of 9 pieces of rope was taken and each piece was cut in half. One half of each piece was soaked in water over night, and then each piece of rope was tested to find its strength. The results, in coded units, are given in the table below
Rope no.123456789
Dry rope9.78.56.38.37.25.46.88.15.9
Wet rope9.19.58.29.78.54.98.48.77.7
Assuming that the strength of rope follows a normal distribution, test whether or not there is any difference between the mean strengths of dry and wet rope. State your hypotheses clearly and use a \(1 \%\) level of significance.
(8 marks)
Edexcel S4 Q3
13 marks Standard +0.8
3. A certain vaccine is known to be only \(70 \%\) effective against a particular virus; thus \(30 \%\) of those vaccinated will actually catch the virus. In order to test whether or not a new and more expensive vaccine provides better protection against the same virus, a random sample of 30 people were chosen and given the new vaccine. If fewer than 6 people contracted the virus the new vaccine would be considered more effective than the current one.
  1. Write down suitable hypotheses for this test.
  2. Find the probability of making a Type I error.
  3. Find the power of this test if the new vaccine is
    1. \(80 \%\) effective,
    2. \(90 \%\) effective. An independent research organisation decided to test the new vaccine on a random sample of 50 people to see if it could be considered more than \(70 \%\) effective. They required the probability of a Type I error to be as close as possible to 0.05 .
  4. Find the critical region for this test.
  5. State the size of this critical region.
  6. Find the power of this test if the new vaccine is
    1. \(80 \%\) effective,
    2. \(90 \%\) effective.
  7. Give one advantage and one disadvantage of the second test.
Edexcel S4 Q4
14 marks Standard +0.8
4. Gill, a member of the accounts department in a large company, is studying the expenses claims of company employees. She assumes that the claims, in \(\pounds\), follow a normal distribution with mean \(\mu\) and variance \(\sigma ^ { 2 }\). As a first stage in her investigation she took the following random sample of 10 claims. $$30.85,99.75,142.73,223.16,75.43,28.57,53.90,81.43,68.62,43.45 .$$
  1. Find a 95\% confidence interval for \(\mu\). The chief accountant would like a \(95 \%\) confidence interval where the difference between the upper confidence limit and the lower confidence limit is less than 20 .
  2. Show that \(\frac { \sigma ^ { 2 } } { n } < 26.03\) (to 2 decimal places), where \(n\) is the size of the sample required to achieve this. Gill decides to use her original sample of 10 to obtain a value for \(\sigma ^ { 2 }\) so that the chance of her value being an underestimate is 0.01 .
  3. Find such a value for \(\sigma ^ { 2 }\).
  4. Use this value for \(\sigma ^ { 2 }\) to estimate the size of sample the chief accountant requires.
Edexcel S4 Q5
16 marks Standard +0.8
5. An educational researcher is testing the effectiveness of a new method of teaching a topic in mathematics. A random sample of 10 children were taught by the new method and a second random sample of 9 children, of similar age and ability, were taught by the conventional method. At the end of the teaching, the same test was given to both groups of children. The marks obtained by the two groups are summarised in the table below.
New methodConventional method
Mean \(( \bar { x } )\)82.378.2
Standard deviation \(( s )\)3.55.7
Number of students \(( n )\)109
  1. Stating your hypotheses clearly and using a \(5 \%\) level of significance, investigate whether or not
    1. the variance of the marks of children taught by the conventional method is greater than that of children taught by the new method,
    2. the mean score of children taught by the conventional method is lower than the mean score of those taught by the new method.
      [0pt] [In each case you should give full details of the calculation of the test statistics.]
  2. State any assumptions you made in order to carry out these tests.
  3. Find a 95\% confidence interval for the common variance of the marks of the two groups.
Edexcel S4 Q6
18 marks Standard +0.3
6. A statistics student is trying to estimate the probability, \(p\), of rolling a 6 with a particular die. The die is rolled 10 times and the random variable \(X _ { 1 }\) represents the number of sixes obtained. The random variable \(R _ { 1 } = \frac { X _ { 1 } } { 10 }\) is proposed as an estimator of \(p\).
  1. Show that \(R _ { 1 }\) is an unbiased estimator of \(p\). The student decided to roll the die again \(n\) times ( \(n > 10\) ) and the random variable \(X _ { 2 }\) represents the number of sixes in these \(n\) rolls. The random variable \(R _ { 2 } = \frac { X _ { 2 } } { n }\) and the random variable \(Y = \frac { 1 } { 2 } \left( R _ { 1 } + R _ { 2 } \right)\).
  2. Show that both \(R _ { 2 }\) and \(Y\) are unbiased estimators of \(p\).
  3. Find \(\operatorname { Var } \left( R _ { 2 } \right)\) and \(\operatorname { Var } ( Y )\).
  4. State giving a reason which of the 3 estimators \(R _ { 1 } , R _ { 2 }\) and \(Y\) are consistent estimators of \(p\).
  5. For the case \(n = 20\) state, giving a reason, which of the 3 estimators \(R _ { 1 } , R _ { 2 }\) and \(Y\) you would recommend. The student's teacher pointed out that a better estimator could be found based on the random variable \(X _ { 1 } + X _ { 2 }\).
  6. Find a suitable estimator and explain why it is better than \(R _ { 1 } , R _ { 2 }\) and \(Y\). END
Edexcel M1 Q1
6 marks Standard +0.3
1. \includegraphics[max width=\textwidth, alt={}, center]{31efa627-5114-4797-9d46-7f1311c18ff8-1_490_254_354_347} A vertical pole \(X Y\), of length 2.5 m and mass 0.5 kg , has its lower end \(Y\) free to move in a smooth horizontal groove. Forces of magnitude 0.2 N and 0.14 N are applied to the pole horizontally at the points \(V\) and \(W\) respectively, where \(X V = 1.5 \mathrm {~m}\) and \(V W = 0.5 \mathrm {~m}\).
Find, to the nearest cm , the distance from \(X\) at which an opposing horizontal force must be applied to keep the pole at rest in equilibrium, and state the magnitude of this force.
Edexcel M1 Q2
7 marks Standard +0.3
2. A particle passes through a point \(O\) with speed \(9 \mathrm {~ms} ^ { - 1 }\) and moves in a straight line with constant acceleration \(3.6 \mathrm {~ms} ^ { - 2 }\) for \(t\) seconds until it reaches the point \(P\). The acceleration is then reduced to \(2 \mathrm {~ms} ^ { - 2 }\) and this is maintained for another \(t\) seconds until the particle passes the point \(Q\) with speed \(16 \mathrm {~ms} ^ { - 1 }\). Calculate
  1. the time taken by the particle to travel from \(O\) to \(Q\),
  2. the distance \(O Q\).
Edexcel M1 Q3
9 marks Standard +0.3
3. A lump of clay, of mass 0.8 kg , is attached to the end \(A\) of a light \(\operatorname { rod } A B\), which is pivoted at its other end \(B\) so that it can rotate smoothly in a vertical plane. A force is applied to \(A\) at an angle of \(60 ^ { \circ }\) to the vertical, as shown, the magnitude \(F \mathrm {~N}\) of this force being just enough to hold the lump of clay in equilibrium with \(A B\) inclined \includegraphics[max width=\textwidth, alt={}, center]{31efa627-5114-4797-9d46-7f1311c18ff8-1_309_335_1453_1590}
at an angle of \(30 ^ { \circ }\) to the upward vertical.
  1. Find the value of \(F\),
  2. Find the magnitude of the force in the \(\operatorname { rod } A B\).
  3. State the modelling assumption that you have made about the lump of clay.
    (6 marks)
    (2 marks)
    (1 mark)
Edexcel M1 Q4
10 marks Standard +0.3
4. Two particles \(A\) and \(B\), of masses 50 grams and \(y\) grams, are moving in the same straight line, in opposite directions, with speeds \(7 \mathrm {~ms} ^ { - 1 }\) and \(4 \mathrm {~ms} ^ { - 1 }\) respectively, and collide. \includegraphics[max width=\textwidth, alt={}, center]{31efa627-5114-4797-9d46-7f1311c18ff8-1_218_508_2143_1382}
In each of the following separate cases, find the value of \(y\) and the magnitude of the impulse exerted by each particle on the other:
  1. after impact the particles move together with speed \(2.25 \mathrm {~ms} ^ { - 1 }\);
  2. after impact the particles move in opposite directions with speed \(5 \mathrm {~ms} ^ { - 1 }\). \section*{MECHANICS 1 (A) TEST PAPER 6 Page 2}
Edexcel M1 Q5
12 marks Standard +0.3
5.
\includegraphics[max width=\textwidth, alt={}]{31efa627-5114-4797-9d46-7f1311c18ff8-2_262_597_276_356}
A small stone is projected with speed \(7 \mathrm {~ms} ^ { - 1 }\) from \(P\), the bottom of a rough plane inclined at \(25 ^ { \circ }\) to the horizontal, and moves up a line of greatest slope of the plane until it comes to instantaneous rest at \(Q\), where \(P Q = 4 \mathrm {~m}\).
  1. Show that the deceleration of the stone as it moves up the plane has magnitude \(\frac { 49 } { 8 } \mathrm {~ms} ^ { - 2 }\).
  2. Find the coefficient of friction between the stone and the plane,
  3. Find the speed with which the stone returns to \(P\).
  4. Name one force which you have ignored in your mathematical model, and state whether the answer to part (c) would be larger or smaller if that force were taken into account.
Edexcel M1 Q6
14 marks Standard +0.3
6. The points \(A\) and \(B\) have position vectors \(( 30 \mathbf { i } - 60 \mathbf { j } ) \mathrm { m }\) and \(( - 20 \mathbf { i } + 60 \mathbf { j } ) \mathrm { m }\) respectively relative to an origin \(O\), where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors. A cyclist, Chris, starts at \(A\) and cycles towards \(B\) with constant speed \(2.6 \mathrm {~ms} ^ { - 1 }\). Another cyclist, Doug, starts at \(O\) and cycles towards \(B\) with constant speed \(k \sqrt { } 10 \mathrm {~ms} ^ { - 1 }\).
  1. Show that Chris's velocity vector is \(( - \mathbf { i } + 2 \cdot 4 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).
  2. Find Doug's velocity vector in the form \(k ( a \mathbf { i } + b \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). Given that Chris and Doug arrive at \(B\) at the same time,
  3. find the value of \(k\).
Edexcel M1 Q7
17 marks Standard +0.3
7.
[diagram]
A particle \(P\), of mass 4 kg , rests on horizontal ground and is attached by a light, inextensible string to another particle \(Q\) of mass 4.5 kg . The string passes over a smooth pulley whose centre is 3 m above the ground. Initially \(Q\) is 1.1 m below the level of the centre of the pulley. The system is released from rest in this position.
  1. Find the acceleration of the two particles.
  2. Find the speed with which \(Q\) hits the ground. Assuming that \(Q\) does not rebound from the ground while the string is slack,
  3. show that \(P\) does not reach the pulley before \(Q\) starts to move again.
  4. Find the speed with which \(Q\) leaves the ground when the string again becomes taut.
    (3 marks)
Edexcel M1 Q1
5 marks Easy -1.2
A golf ball and a table tennis ball are dropped together from the top of a building. The golf ball hits the ground after 1.7 seconds.
  1. Calculate the height of the top of the building above the ground. According to a simple model, the two balls hit the ground at the same time.
  2. State why this may not be true in practice and describe a refinement to the model which could lead to a more realistic solution.
Edexcel M1 Q2
5 marks Standard +0.3
2. A plank of wood \(X Y\) has length \(5 a\) m and mass 5 kg . It rests on a support at \(Q\), where \(X Q = 3 a\) m . When a kitten of mass 8 kg sits on the plank at \(P\), where \(P Y = a \mathrm {~m}\), the plank just remains horizontal. By modelling the plank as a non-uniform rod and the kitten as a particle, find
  1. the magnitude of the reaction at the support,
  2. the distance from \(X\) to the centre of mass of the plank, in terms of \(a\).
Edexcel M1 Q3
9 marks Moderate -0.3
3. A particle is in equilibrium under the action of three forces \(\mathbf { P } , \mathbf { Q }\) and \(\mathbf { R }\) acting in the same horizontal plane. \(P\) has magnitude 9 N and acts on a bearing of \(030 ^ { \circ } . Q\) has magnitude 12 N . and acts on a bearing of \(225 ^ { \circ }\).
  1. Find the values of \(a\) and \(b\) such that \(\mathbf { R } = ( a \mathbf { i } + b \mathbf { j } ) \mathrm { N }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors in the directions due East and due North respectively.
  2. Calculate the magnitude and direction of \(\mathbf { R }\)