Questions — Edexcel S2 (494 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
Edexcel S2 2001 June Q7
7. In a computer game, a star moves across the screen, with constant speed, taking 1 s to travel from one side to the other. The player can stop the star by pressing a key. The object of the game is to stop the star in the middle of the screen by pressing the key exactly 0.5 s after the star first appears. Given that the player actually presses the key \(T\) s after the star first appears, a simple model of the game assumes that \(T\) is a continuous uniform random variable defined over the interval \([ 0,1 ]\).
  1. Write down \(\mathrm { P } ( \mathrm { T } < 0.2 )\).
  2. Write down E(T).
  3. Use integration to find \(\operatorname { Var } ( T )\). A group of 20 children each play this game once.
  4. Find the probability that no more than 4 children stop the star in less than 0.2 s . The children are allowed to practise.this game so that this continuous uniform model is no longer applicable.
  5. Explain how you would expect the mean and variance of T to change. It is found that a more appropriate model of the game when played by experienced players assumes that \(T\) has a probability density function \(\mathrm { g } ( t )\) given by $$g ( t ) = \begin{cases} 4 t , & 0 \leq t \leq 0.5
    4 - 4 t , & 0.5 \leq t \leq 1 ,
    0 , & \text { otherwise } . \end{cases}$$
  6. Using this model show that \(\mathrm { P } ( T < 0.2 ) = 0.08\). A group of 75 experienced players each played this game once.
  7. Using a suitable approximation, find the probability that more than 7 of them stop the star in less than 0.2 s .
    (4) END
Edexcel S2 2002 June Q1
\begin{enumerate} \item The manager of a leisure club is considering a change to the club rules. The club has a large membership and the manager wants to take the views of the members into consideration before deciding whether or not to make the change.
  1. Explain briefly why the manager might prefer to use a sample survey rather than a census to obtain the views.
  2. Suggest a suitable sampling frame.
  3. Identify the sampling units. \item A random sample \(X _ { 1 } , X _ { 2 } , \ldots , X _ { n }\) is taken from a finite population. A statistic \(Y\) is based on this sample.
Edexcel S2 2002 June Q5
5. A garden centre sells canes of nominal length 150 cm . The canes are bought from a supplier who uses a machine to cut canes of length \(L\) where \(L \sim \mathrm {~N} \left( \mu , 0.3 ^ { 2 } \right)\).
  1. Find the value of \(\mu\), to the nearest 0.1 cm , such that there is only a \(5 \%\) chance that a cane supplied to the garden centre will have length less than 150 cm . A customer buys 10 of these canes from the garden centre.
  2. Find the probability that at most 2 of the canes have length less than 150 cm . Another customer buys 500 canes.
  3. Using a suitable approximation, find the probability that fewer than 35 of the canes will have length less than 150 cm .
    (6)
Edexcel S2 2002 June Q6
6. From past records, a manufacturer of twine knows that faults occur in the twine at random and at a rate of 1.5 per 25 m .
  1. Find the probability that in a randomly chosen 25 m length of twine there will be exactly 4 faults. The twine is usually sold in balls of length 100 m . A customer buys three balls of twine.
  2. Find the probability that only one of them will have fewer than 6 faults. As a special order a ball of twine containing 500 m is produced.
  3. Using a suitable approximation, find the probability that it will contain between 23 and 33 faults inclusive.
    (6)
Edexcel S2 2002 June Q7
7. The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { x } { 15 } , & 0 \leq x \leq 2
\frac { 2 } { 15 } , & 2 < x < 7
\frac { 4 } { 9 } - \frac { 2 x } { 45 } , & 7 \leq x \leq 10
0 , & \text { otherwise } \end{cases}$$
  1. Sketch \(\mathrm { f } ( x )\) for all values of \(x\).
    1. Find expressions for the cumulative distribution function, \(\mathrm { F } ( x )\), for \(0 \leq x \leq 2\) and for \(7 \leq x \leq 10\).
    2. Show that for \(2 < x < 7 , \mathrm {~F} ( x ) = \frac { 2 x } { 15 } - \frac { 2 } { 15 }\).
    3. Specify \(\mathrm { F } ( x )\) for \(x < 0\) and for \(x > 10\).
  3. Find \(\mathrm { P } ( X \leq 8.2 )\).
  4. Find, to 3 significant figures, \(\mathrm { E } ( X )\).
Edexcel S2 2003 June Q1
  1. Explain briefly what you understand by
    1. a statistic,
    2. a sampling distribution.
    3. (a) Write down the condition needed to approximate a Poisson distribution by a Normal distribution.
    The random variable \(Y \sim \operatorname { Po } ( 30 )\).
  2. Estimate \(\mathrm { P } ( Y > 28 )\).
Edexcel S2 2003 June Q3
3. In a town, \(30 \%\) of residents listen to the local radio station. Four residents are chosen at random.
  1. State the distribution of the random variable \(X\), the number of these four residents that listen to local radio.
  2. On graph paper, draw the probability distribution of \(X\).
  3. Write down the most likely number of these four residents that listen to the local radio station.
  4. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
Edexcel S2 2003 June Q4
4. (a) Write down the conditions under which the binomial distribution may be a suitable model to use in statistical work. A six-sided die is biased. When the die is thrown the number 5 is twice as likely to appear as any other number. All the other faces are equally likely to appear. The die is thrown repeatedly. Find the probability that
(b) (i) the first 5 will occur on the sixth throw,
(ii) in the first eight throws there will be exactly three 5 s .
Edexcel S2 2003 June Q5
5. A drinks machine dispenses lemonade into cups. It is electronically controlled to cut off the flow of lemonade randomly between 180 ml and 200 ml . The random variable \(X\) is the volume of lemonade dispensed into a cup.
  1. Specify the probability density function of \(X\) and sketch its graph.
  2. Find the probability that the machine dispenses
    1. less than 183 ml ,
    2. exactly 183 ml .
  3. Calculate the inter-quartile range of \(X\).
  4. Determine the value of \(x\) such that \(\mathrm { P } ( X \geq x ) = 2 \mathrm { P } ( X \leq x )\).
  5. Interpret in words your value of \(x\).
Edexcel S2 2003 June Q6
6. A doctor expects to see, on average, 1 patient per week with a particular disease.
  1. Suggest a suitable model for the distribution of the number of times per week that the doctor sees a patient with the disease. Give a reason for your answer.
  2. Using your model, find the probability that the doctor sees more than 3 patients with the disease in a 4 week period. The doctor decides to send information to his patients to try to reduce the number of patients he sees with the disease. In the first 6 weeks after the information is sent out, the doctor sees 2 patients with the disease.
  3. Test, at the \(5 \%\) level of significance, whether or not there is reason to believe that sending the information has reduced the number of times the doctor sees patients with the disease. State your hypotheses clearly. Medical research into the nature of the disease discovers that it can be passed from one patient to another.
  4. Explain whether or not this research supports your choice of model. Give a reason for your answer.
Edexcel S2 2003 June Q7
7. A continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \begin{cases} k \left( x ^ { 2 } + 2 x + 1 \right) & - 1 \leq x \leq 0
0 , & \text { otherwise } \end{cases}$$ where \(k\) is a positive integer.
  1. Show that \(k = 3\). Find
  2. \(\mathrm { E } ( X )\),
  3. the cumulative distribution function \(\mathrm { F } ( x )\),
  4. \(\mathrm { P } ( - 0.3 < X < 0.3 )\). END
Edexcel S2 2004 June Q1
  1. Explain briefly what you understand by
    1. a sampling frame,
    2. a statistic.
    3. The continuous random variable \(X\) is uniformly distributed over the interval \([ - 1,4 ]\).
    Find
  2. \(\mathrm { P } ( X < 2.7 )\),
  3. \(\mathrm { E } ( X )\),
  4. \(\operatorname { Var } ( X )\).
Edexcel S2 2004 June Q3
3. Brad planted 25 seeds in his greenhouse. He has read in a gardening book that the probability of one of these seeds germinating is 0.25 . Ten of Brad's seeds germinated. He claimed that the gardening book had underestimated this probability. Test, at the \(5 \%\) level of significance, Brad's claim. State your hypotheses clearly.
Edexcel S2 2004 June Q4
4. (a) State two conditions under which a random variable can be modelled by a binomial distribution. In the production of a certain electronic component it is found that \(10 \%\) are defective.
The component is produced in batches of 20 .
(b) Write down a suitable model for the distribution of defective components in a batch. Find the probability that a batch contains
(c) no defective components,
(d) more than 6 defective components.
(e) Find the mean and the variance of the defective components in a batch. A supplier buys 100 components. The supplier will receive a refund if there are more than 15 defective components.
(f) Using a suitable approximation, find the probability that the supplier will receive a refund.
Edexcel S2 2004 June Q5
5. (a) Explain what you understand by a critical region of a test statistic. The number of breakdowns per day in a large fleet of hire cars has a Poisson distribution with mean \(\frac { 1 } { 7 }\).
(b) Find the probability that on a particular day there are fewer than 2 breakdowns.
(c) Find the probability that during a 14-day period there are at most 4 breakdowns. The cars are maintained at a garage. The garage introduced a weekly check to try to decrease the number of cars that break down. In a randomly selected 28-day period after the checks are introduced, only 1 hire car broke down.
(d) Test, at the \(5 \%\) level of significance, whether or not the mean number of breakdowns has decreased. State your hypotheses clearly.
Edexcel S2 2004 June Q6
6. Minor defects occur in a particular make of carpet at a mean rate of 0.05 per \(\mathrm { m } ^ { 2 }\).
  1. Suggest a suitable model for the distribution of the number of defects in this make of carpet. Give a reason for your answer. A carpet fitter has a contract to fit this carpet in a small hotel. The hotel foyer requires \(30 \mathrm {~m} ^ { 2 }\) of this carpet. Find the probability that the foyer carpet contains
  2. exactly 2 defects,
  3. more than 5 defects. The carpet fitter orders a total of \(355 \mathrm {~m} ^ { 2 }\) of the carpet for the whole hotel.
  4. Using a suitable approximation, find the probability that this total area of carpet contains 22 or more defects.
    (6)
Edexcel S2 2004 June Q7
7. A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 3 } , & 0 \leq x \leq 1
\frac { 8 x ^ { 3 } } { 45 } , & 1 \leq x \leq 2
0 , & \text { otherwise } \end{cases}$$
  1. Calculate the mean of \(X\).
  2. Specify fully the cumulative distribution function \(\mathrm { F } ( x )\).
  3. Find the median of \(X\).
  4. Comment on the skewness of the distribution of \(X\). END
Edexcel S2 2005 June Q1
  1. It is estimated that \(4 \%\) of people have green eyes. In a random sample of size \(n\), the expected number of people with green eyes is 5 .
    1. Calculate the value of \(n\).
    The expected number of people with green eyes in a second random sample is 3 .
  2. Find the standard deviation of the number of people with green eyes in this second sample. expected number of people with green eyes is 5 .
  3. Calculate the value of \(n\) - The expected number of people with green eyes in a second random sample is 3 .
  4. sample. C) T. " D
Edexcel S2 2005 June Q2
2. The continuous random variable \(X\) is uniformly distributed over the interval \([ 2,6 ]\).
  1. Write down the probability density function \(\mathrm { f } ( x )\). Find
  2. \(\mathrm { E } ( X )\),
  3. \(\operatorname { Var } ( X )\),
  4. the cumulative distribution function of \(X\), for all \(x\),
  5. \(\mathrm { P } ( 2.3 < X < 3.4 )\).
Edexcel S2 2005 June Q3
3. The random variable \(X\) is the number of misprints per page in the first draft of a novel.
  1. State two conditions under which a Poisson distribution is a suitable model for \(X\). The number of misprints per page has a Poisson distribution with mean 2.5. Find the probability that
  2. a randomly chosen page has no misprints,
  3. the total number of misprints on 2 randomly chosen pages is more than 7 . The first chapter contains 20 pages.
  4. Using a suitable approximation find, to 2 decimal places, the probability that the chapter will contain less than 40 misprints.
Edexcel S2 2005 June Q4
4. Explain what you understand by
  1. a sampling unit,
  2. a sampling frame,
  3. a sampling distribution.
Edexcel S2 2005 June Q5
5. In a manufacturing process, \(2 \%\) of the articles produced are defective. A batch of 200 articles is selected.
  1. Giving a justification for your choice, use a suitable approximation to estimate the probability that there are exactly 5 defective articles.
  2. Estimate the probability there are less than 5 defective articles.
Edexcel S2 2005 June Q6
6. A continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) where $$f ( x ) = \begin{cases} k \left( 4 x - x ^ { 3 } \right) , & 0 \leqslant x \leqslant 2
0 , & \text { otherwise } \end{cases}$$ where \(k\) is a positive integer.
  1. Show that \(k = \frac { 1 } { 4 }\). Find
  2. \(\mathrm { E } ( X )\),
  3. the mode of \(X\),
  4. the median of \(X\).
  5. Comment on the skewness of the distribution.
  6. Sketch f(x).
Edexcel S2 2005 June Q7
7. A drugs company claims that \(75 \%\) of patients suffering from depression recover when treated with a new drug. A random sample of 10 patients with depression is taken from a doctor's records.
  1. Write down a suitable distribution to model the number of patients in this sample who recover when treated with the new drug. Given that the claim is correct,
  2. find the probability that the treatment will be successful for exactly 6 patients. The doctor believes that the claim is incorrect and the percentage who will recover is lower. From her records she took a random sample of 20 patients who had been treated with the new drug. She found that 13 had recovered.
  3. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, the doctor's belief.
  4. From a sample of size 20, find the greatest number of patients who need to recover for the test in part (c) to be significant at the \(1 \%\) level.
Edexcel S2 2006 June Q1
  1. Before introducing a new rule the secretary of a golf club decided to find out how members might react to this rule.
    1. Explain why the secretary decided to take a random sample of club members rather than ask all the members.
    2. Suggest a suitable sampling frame.
    3. Identify the sampling units.
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    4. The continuous random variable \(L\) represents the error, in mm , made when a machine cuts rods to a target length. The distribution of \(L\) is continuous uniform over the interval [-4.0, 4.0].
    Find
  2. \(\mathrm { P } ( L < - 2.6 )\),
  3. \(\mathrm { P } ( L < - 3.0\) or \(L > 3.0 )\). A random sample of 20 rods cut by the machine was checked.
  4. Find the probability that more than half of them were within 3.0 mm of the target length.