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 2018 Specimen Q2
2. A continuous random variable \(X\) has cumulative distribution function $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 1
\frac { 1 } { 5 } ( x - 1 ) & 1 \leqslant x \leqslant 6
1 & x > 6 \end{array} \right.$$
  1. Find \(\mathrm { P } ( X > 4 )\)
  2. Write down the value of \(\mathrm { P } ( X \neq 4 )\)
  3. Find the probability density function of \(X\), specifying it for all values of \(x\)
  4. Write down the value of \(\mathrm { E } ( X )\)
  5. Find \(\operatorname { Var } ( X )\)
  6. Hence or otherwise find \(\mathrm { E } \left( 3 X ^ { 2 } + 1 \right)\)
Edexcel S2 2018 Specimen Q3
3. Explain what you understand by
  1. a statistic,
  2. a sampling distribution. A factory stores screws in packets. A small packet contains 100 screws and a large packet contains 200 screws. The factory keeps small and large packets in the ratio 4:3 respectively.
  3. Find the mean and the variance of the number of screws in the packets stored at the factory. A random sample of 3 packets is taken from the factory and \(Y _ { 1 } , Y _ { 2 }\) and \(Y _ { 3 }\) denote the number of screws in each of these packets.
  4. List all the possible samples.
  5. Find the sampling distribution of \(\bar { Y }\)
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Edexcel S2 2018 Specimen Q4
4. Accidents occur randomly at a crossroads at a rate of 0.5 per month. A researcher records the number of accidents, \(X\), which occur at the crossroads in a year.
  1. Find \(\mathrm { P } ( 5 \leqslant X < 7 )\) A new system is introduced at the crossroads. In the first 18 months, 4 accidents occur at the crossroads.
  2. Test, at the \(5 \%\) level of significance, whether or not there is reason to believe that the new system has led to a reduction in the mean number of accidents per month. State your hypotheses clearly.
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Edexcel S2 2018 Specimen Q5
5. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by $$f ( x ) = \left\{ \begin{array} { c c } k \left( x ^ { 2 } + a \right) & - 1 < x \leqslant 2
3 k & 2 < x \leqslant 3
0 & \text { otherwise } \end{array} \right.$$ where \(k\) and \(a\) are constants.
Given that \(\mathrm { E } ( X ) = \frac { 17 } { 12 }\)
  1. find the value of \(k\) and the value of \(a\)
  2. Write down the mode of \(X\)
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Edexcel S2 2018 Specimen Q6
6. The Headteacher of a school claims that \(30 \%\) of parents do not support a new curriculum. In a survey of 20 randomly selected parents, the number, \(X\), who do not support the new curriculum is recorded. Assuming that the Headteacher's claim is correct, find
  1. the probability that \(X = 5\)
  2. the mean and the standard deviation of \(X\) The Director of Studies believes that the proportion of parents who do not support the new curriculum is greater than \(30 \%\). Given that in the survey of 20 parents 8 do not support the new curriculum,
  3. test, at the \(5 \%\) level of significance, the Director of Studies' belief. State your hypotheses clearly. The teachers believe that the sample in the original survey was biased and claim that only \(25 \%\) of the parents are in support of the new curriculum. A second random sample, of size \(2 n\), is taken and exactly half of this sample supports the new curriculum. A test is carried out at a \(10 \%\) level of significance of the teachers' belief using this sample of size \(2 n\) Using the hypotheses \(\mathrm { H } _ { 0 } : p = 0.25\) and \(\mathrm { H } _ { 1 } : p > 0.25\)
  4. find the minimum value of \(n\) for which the outcome of the test is that the teachers' belief is rejected.
Edexcel S2 2018 Specimen Q7
  1. A multiple choice examination paper has \(n\) questions where \(n > 30\)
Each question has 5 answers of which only 1 is correct. A pass on the paper is obtained by answering 30 or more questions correctly. The probability of obtaining a pass by randomly guessing the answer to each question should not exceed 0.0228 Use a normal approximation to work out the greatest number of questions that could be used.
7.
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Edexcel S2 Specimen Q1
  1. Explain what you understand by
    1. a population,
    2. a statistic.
    A researcher took a sample of 100 voters from a certain town and asked them who they would vote for in an election. The proportion who said they would vote for Dr Smith was \(35 \%\).
  2. State the population and the statistic in this case.
  3. Explain what you understand by the sampling distribution of this statistic.
Edexcel S2 Specimen Q2
2. Bhim and Joe play each other at badminton and for each game, independently of all others, the probability that Bhim loses is 0.2 Find the probability that, in 9 games, Bhim loses
  1. exactly 3 of the games,
  2. fewer than half of the games. Bhim attends coaching sessions for 2 months. After completing the coaching, the probability that he loses each game, independently of all others, is 0.05 Bhim and Joe agree to play a further 60 games.
  3. Calculate the mean and variance for the number of these 60 games that Bhim loses.
  4. Using a suitable approximation calculate the probability that Bhim loses more than 4 games.
Edexcel S2 Specimen Q3
  1. A rectangle has a perimeter of 20 cm . The length, \(X \mathrm {~cm}\), of one side of this rectangle is uniformly distributed between 1 cm and 7 cm .
Find the probability that the length of the longer side of the rectangle is more than 6 cm long.
Edexcel S2 Specimen Q4
4. The lifetime, \(X\), in tens of hours, of a battery has a cumulative distribution function \(\mathrm { F } ( x )\) given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 1
\frac { 4 } { 9 } \left( x ^ { 2 } + 2 x - 3 \right) & 1 \leqslant x \leqslant 1.5
1 & x > 1.5 \end{array} \right.$$
  1. Find the median of \(X\), giving your answer to 3 significant figures.
  2. Find, in full, the probability density function of the random variable \(X\).
  3. Find \(\mathrm { P } ( X \geqslant 1.2 )\) A camping lantern runs on 4 batteries, all of which must be working. Four new batteries are put into the lantern.
  4. Find the probability that the lantern will still be working after 12 hours.
Edexcel S2 Specimen Q5
  1. A company has a large number of regular users logging onto its website. On average 4 users every hour fail to connect to the company's website at their first attempt.
    1. Explain why the Poisson distribution may be a suitable model in this case.
    Find the probability that, in a randomly chosen \(\mathbf { 2 }\) hour period,
    1. all users connect at their first attempt,
    2. at least 4 users fail to connect at their first attempt. The company suffered from a virus infecting its computer system. During this infection it was found that the number of users failing to connect at their first attempt, over a 12 hour period, was 60 .
  3. Using a suitable approximation, test whether or not the mean number of users per hour who failed to connect at their first attempt had increased. Use a \(5 \%\) level of significance and state your hypotheses clearly.
Edexcel S2 Specimen Q6
6. A company claims that a quarter of the bolts sent to them are faulty. To test this claim the number of faulty bolts in a random sample of 50 is recorded.
  1. Give two reasons why a binomial distribution may be a suitable model for the number of faulty bolts in the sample.
  2. Using a 5\% significance level, find the critical region for a two-tailed test of the hypothesis that the probability of a bolt being faulty is \(\frac { 1 } { 4 }\). The probability of rejection in either tail should be as close as possible to 0.025
  3. Find the actual significance level of this test. In the sample of 50 the actual number of faulty bolts was 8 .
  4. Comment on the company's claim in the light of this value. Justify your answer. The machine making the bolts was reset and another sample of 50 bolts was taken. Only 5 were found to be faulty.
  5. Test at the \(1 \%\) level of significance whether or not the probability of a faulty bolt has decreased. State your hypotheses clearly.
Edexcel S2 Specimen Q7
  1. The random variable \(Y\) has probability density function \(\mathrm { f } ( y )\) given by
$$\mathrm { f } ( y ) = \left\{ \begin{array} { c c } k y ( a - y ) & 0 \leqslant y \leqslant 3
0 & \text { otherwise } \end{array} \right.$$ where \(k\) and \(a\) are positive constants.
    1. Explain why \(a \geqslant 3\)
    2. Show that \(k = \frac { 2 } { 9 ( a - 2 ) }\) Given that \(\mathrm { E } ( Y ) = 1.75\)
  2. show that \(a = 4\) and write down the value of \(k\). For these values of \(a\) and \(k\),
  3. sketch the probability density function,
  4. write down the mode of \(Y\).
Edexcel S2 2002 January Q1
  1. Explain what you understand by
    1. a population,
    2. a statistic.
    A questionnaire concerning attitudes to classes in a college was completed by a random sample of 50 students. The students gave the college a mean approval rating of 75\%.
  2. Identify the population and the statistic in this situation.
  3. Explain what you understand by the sampling distribution of this statistic.
Edexcel S2 2002 January Q2
2. The number of houses sold per week by a firm of estate agents follows a Poisson distribution with mean 2.5. The firm appoints a new salesman and wants to find out whether or not house sales increase as a result. After the appointment of the salesman, the number of house sales in a randomly chosen 4-week period is 14. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not the new salesman has increased house sales.
Edexcel S2 2002 January Q3
3. An airline knows that overall \(3 \%\) of passengers do not turn up for flights. The airline decides to adopt a policy of selling more tickets than there are seats on a flight. For an aircraft with 196 seats, the airline sold 200 tickets for a particular flight.
  1. Write down a suitable model for the number of passengers who do not turn up for this flight after buying a ticket. By using a suitable approximation, find the probability that
  2. more than 196 passengers turn up for this flight,
  3. there is at least one empty seat on this flight.
Edexcel S2 2002 January Q4
4. Jean catches a bus to work every morning. According to the timetable the bus is due at 8 a.m., but Jean knows that the bus can arrive at a random time between five minutes early and 9 minutes late. The random variable \(X\) represents the time, in minutes, after 7.55 a.m. when the bus arrives.
  1. Suggest a suitable model for the distribution of \(X\) and specify it fully.
  2. Calculate the mean time of arrival of the bus.
  3. Find the cumulative distribution function of \(X\). Jean will be late for work if the bus arrives after 8.05 a.m.
  4. Find the probability that Jean is late for work.
Edexcel S2 2002 January Q5
5. An Internet service provider has a large number of users regularly connecting to its computers. On average only 3 users every hour fail to connect to the Internet at their first attempt.
  1. Give 2 reasons why a Poisson distribution might be a suitable model for the number of failed connections every hour.
  2. Find the probability that in a randomly chosen hour
    1. all Internet users connect at their first attempt,
    2. more than 4 users fail to connect at their first attempt.
  3. Write down the distribution of the number of users failing to connect at their first attempt in an 8-hour period.
  4. Using a suitable approximation, find the probability that 12 or more users fail to connect at their first attempt in a randomly chosen 8-hour period.
Edexcel S2 2002 January Q6
6. The owner of a small restaurant decides to change the menu. A trade magazine claims that \(40 \%\) of all diners choose organic foods when eating away from home. On a randomly chosen day there are 20 diners eating in the restaurant.
  1. Assuming the claim made by the trade magazine to be correct, suggest a suitable model to describe the number of diners \(X\) who choose organic foods.
  2. Find \(\mathrm { P } ( 5 < X < 15 )\).
  3. Find the mean and standard deviation of \(X\). The owner decides to survey her customers before finalising the new menu. She surveys 10 randomly chosen diners and finds 8 who prefer eating organic foods.
  4. Test, at the \(5 \%\) level of significance, whether or not there is reason to believe that the proportion of diners in her restaurant who prefer to eat organic foods is higher than the trade magazine's claim. State your hypotheses clearly.
    (5)
Edexcel S2 2002 January Q7
7. A continuous random variable \(X\) has cumulative distribution function \(\mathrm { F } ( x )\) given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 , & x < 0
k x ^ { 2 } + 2 k x , & 0 \leq x \leq 2
8 k , & x > 2 \end{array} \right.$$
  1. Show that \(k = \frac { 1 } { 8 }\).
  2. Find the median of \(X\).
  3. Find the probability density function \(\mathrm { f } ( x )\).
  4. Sketch \(\mathrm { f } ( x )\) for all values of \(x\).
  5. Write down the mode of \(X\).
  6. Find \(\mathrm { E } ( X )\).
  7. Comment on the skewness of this distribution.
Edexcel S2 2003 January Q1
  1. An engineer measures, to the nearest cm , the lengths of metal rods.
    1. Suggest a suitable model to represent the difference between the true lengths and the measured lengths.
    2. Find the probability that for a randomly chosen rod the measured length will be within 0.2 cm of the true length.
    Two rods are chosen at random.
  2. Find the probability that for both rods the measured lengths will be within 0.2 cm of their true lengths.
Edexcel S2 2003 January Q2
2. A single observation \(x\) is to be taken from a Poisson distribution with parameter \(\lambda\). This observation is to be used to test \(\mathrm { H } _ { 0 } : \lambda = 7\) against \(\mathrm { H } _ { 1 } : \lambda \neq 7\).
  1. Using a \(5 \%\) significance level, find the critical region for this test assuming that the probability of rejection in either tail is as close as possible to \(2.5 \%\).
  2. Write down the significance level of this test. The actual value of \(x\) obtained was 5 .
  3. State a conclusion that can be drawn based on this value.
Edexcel S2 2003 January Q3
3. A botanist suggests that the number of a particular variety of weed growing in a meadow can be modelled by a Poisson distribution.
  1. Write down two conditions that must apply for this model to be applicable. Assuming this model and a mean of 0.7 weeds per \(\mathrm { m } ^ { 2 }\), find
  2. the probability that in a randomly chosen plot of size \(4 \mathrm {~m} ^ { 2 }\) there will be fewer than 3 of these weeds.
  3. Using a suitable approximation, find the probability that in a plot of \(100 \mathrm {~m} ^ { 2 }\) there will be more than 66 of these weeds.
    (6)
Edexcel S2 2003 January Q4
4. The continuous random variable \(X\) has cumulative distribution function $$\mathrm { F } ( x ) = \begin{cases} 0 , & x < 0
\frac { 1 } { 3 } x ^ { 2 } \left( 4 - x ^ { 2 } \right) , & 0 \leq x \leq 1
1 & x > 1 \end{cases}$$
  1. Find \(\mathrm { P } ( X > 0.7 )\).
  2. Find the probability density function \(\mathrm { f } ( x )\) of \(X\).
  3. Calculate \(\mathrm { E } ( X )\) and show that, to 3 decimal places, \(\operatorname { Var } ( X ) = 0.057\). One measure of skewness is $$\frac { \text { Mean - Mode } } { \text { Standard deviation } } .$$
  4. Evaluate the skewness of the distribution of \(X\).
Edexcel S2 2003 January Q5
5. A farmer noticed that some of the eggs laid by his hens had double yolks. He estimated the probability of this happening to be 0.05 . Eggs are packed in boxes of 12 . Find the probability that in a box, the number of eggs with double yolks will be
  1. exactly one,
  2. more than three. A customer bought three boxes.
  3. Find the probability that only 2 of the boxes contained exactly 1 egg with a double yolk. The farmer delivered 10 boxes to a local shop.
  4. Using a suitable approximation, find the probability that the delivery contained at least 9 eggs with double yolks. The weight of an individual egg can be modelled by a normal distribution with mean 65 g and standard deviation 2.4 g .
  5. Find the probability that a randomly chosen egg weighs more than 68 g .