Edexcel 2017 Sample Paper 3 Statistics and Mechanics

Edexcel 2017 Statistics & Mechanics Paper 3 (pdf)

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1. The number of hours of sunshine each day, y, for the month of July at Heathrow are summarised in the table below.
   A histogram was drawn to represent these data. The 8 ≤ y < 11 group was represented by a bar of width 1.5 cm and height 8 cm.
   (a) Find the width and the height of the 0 ≤ y < 5 group.
   (b) Use your calculator to estimate the mean and the standard deviation of the number of hours of sunshine each day, for the month of July at Heathrow.
   Give your answers to 3 significant figures.
   The mean and standard deviation for the number of hours of daily sunshine for the same month in Hurn are 5.98 hours and 4.12 hours respectably.
   Thomas believes that the further south you are the more consistent should be the number of hours of daily sunshine.
   (c) State, giving a reason, whether or not the calculations in part (b) support Thomas’ belief.
   (d) Estimate the number of days in July at Heathrow where the number of hours of sunshine is more than 1 standard deviation above the mean.
   Helen models the number of hours of sunshine each day, for the month of July at Heathrow by N(6.6, 3.7²).
   (e) Use Helen’s model to predict the number of days in July at Heathrow when the number of hours of sunshine is more than 1 standard deviation above the mean.
   (f) Use your answers to part (d) and part (e) to comment on the suitability of Helen’s model.
   
   
2. A meteorologist believes that there is a relationship between the daily mean windspeed, w kn, and the daily mean temperature, t °C. A random sample of 9 consecutive days is taken from past records from a town in the UK in July and the relevant data is given in the table below. The meteorologist calculated the product moment correlation coefficient for the 9 days and obtained r = 0.609
   (a) Explain why a linear regression model based on these data is unreliable on a day when the mean temperature is 24 °C
   (b) State what is measured by the product moment correlation coefficient.
   (c) Stating your hypotheses clearly test, at the 5% significance level, whether or not the product moment correlation coefficient for the population is greater than zero.
   Using the same 9 days a location from the large data set gave t = 27.2 and w = 3.5
   (d) Using your knowledge of the large data set, suggest, giving your reason, the location that gave rise to these statistics.
   
   
3. A machine cuts strips of metal to length L cm, where L is normally distributed with standard deviation 0.5 cm.
   Strips with length either less than 49 cm or greater than 50.75 cm cannot be used.
   Given that 2.5% of the cut lengths exceed 50.98 cm,
   (a) find the probability that a randomly chosen strip of metal can be used.
   Ten strips of metal are selected at random.
   (b) Find the probability fewer than 4 of these strips cannot be used.
   A second machine cuts strips of metal of length X cm, where X is normally distributed with standard deviation 0.6 cm
   A random sample of 15 strips cut by this second machine was found to have a mean length of 50.4 cm
   (c) Stating your hypotheses clearly and using a 1% level of significance, test whether or not the mean length of all the strips, cut by the second machine, is greater than 50.1 cm
   
   

4. Given that
   P(A) = 0.35
   P(B) = 0.45
   P(A ∩ B) 0.13
   find
   (a) P(A' | B')
   (b) Explain why the events A and B are not independent.
   The event C has P(C) = 0.20
   The events A and C are mutually exclusive and the events B and C are statistically independent.
   (c) Draw a Venn diagram to illustrate the events A, B and C, giving the probabilities for each region.
   (d) Find P((B U C)')
   
   
5. A company sells seeds and claims that 55% of its pea seeds germinate.
   (a) Write down a reason why the company should not justify their claim by testing all the pea seeds they produce.
   A random selection of the pea seeds is planted in 10 trays with 24 seeds in each tray.
   (b) Assuming that the company’s claim is correct, calculate the probability that in at least half of the trays 15 or more of the seeds germinate.
   (c) Write down two conditions under which the normal distribution may be used as an approximation to the binomial distribution.
   A random sample of 240 pea seeds was planted and 150 of these seeds germinated.
   (d) Assuming that the company’s claim is correct, use a normal approximation to find the probability that at least 150 pea seeds germinate.
   (e) Using your answer to part (d), comment on whether or not the proportion of the company’s pea seeds that germinate is different from the company’s claim of 55%
   
   
6. At time t seconds, where t ≥ 0, a particle P moves so that its acceleration a m s^–2 is given by
   a = 5ti – 15t^1/2 j
   When t = 0, the velocity of P is 20i m s^–1
   Find the speed of P when t = 4
   
   
7. A rough plane is inclined to the horizontal at an angle α, where tan α = 3/4.
   A particle of mass m is placed on the plane and then projected up a line of greatest slope of the plane.
   The coefficient of friction between the particle and the plane is μ.
   The particle moves up the plane with a constant deceleration of 4/5 g.
   (a) Find the value of μ.
   The particle comes to rest at the point A on the plane.
   (b) Determine whether the particle will remain at A, carefully justifying your answer.
   
   
8. (In this question i and j are horizontal unit vectors due east and due north respectively)
   A radio controlled model boat is placed on the surface of a large pond.
   The boat is modelled as a particle.
   At time t = 0, the boat is at the fixed point O and is moving due north with speed 0.6 m s^–1.
   Relative to O, the position vector of the boat at time t seconds is r metres.
   At time t = 15, the velocity of the boat is (10.5i – 0.9j) m s^–1.
   The acceleration of the boat is constant.
   (a) Show that the acceleration of the boat is (0.7i – 0.1j) m s^–2.
   (b) Find r in terms of t.
   (c) Find the value of t when the boat is north-east of O.
   (d) Find the value of t when the boat is moving in a north-east direction.
   
   
9. A uniform ladder AB, of length 2a and weight W, has its end A on rough horizontal ground.
   The coefficient of friction between the ladder and the ground is 1/4.
   The end B of the ladder is resting against a smooth vertical wall, as shown in Figure 1.
   A builder of weight 7W stands at the top of the ladder.
   To stop the ladder from slipping, the builder’s assistant applies a horizontal force of magnitude P to the ladder at A, towards the wall.
   The force acts in a direction which is perpendicular to the wall.
   The ladder rests in equilibrium in a vertical plane perpendicular to the wall and makes an angle α with the horizontal ground, where tan α = 5/2
   The builder is modelled as a particle and the ladder is modelled as a uniform rod.
   (a) Show that the reaction of the wall on the ladder at B has magnitude 3W.
   (b) Find, in terms of W, the range of possible values of P for which the ladder remains in equilibrium.
   Often in practice, the builder’s assistant will simply stand on the bottom of the ladder.
   (c) Explain briefly how this helps to stop the ladder from slipping.
   
   
10. A boy throws a stone with speed Ums⁻¹ from a point O at the top of a vertical cliff. The point O is 18m above sea level.
    The stone is thrown at an angle α above the horizontal, where tan α = 3/4.
    The stone hits the sea at the point S which is at a horizontal distance of 36m from the foot of the cliff, as shown in Figure 2.
    The stone is modelled as a particle moving freely under gravity with g = 10ms⁻²
    Find
    (a) the value of U,
    (b) the speed of the stone when it is 10.8m above sea level, giving your answer to 2 significant figures.
    (c) Suggest two improvements that could be made to the model.
    

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