Questions S2 (1597 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 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
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Edexcel S2 Specimen Q1
5 marks Easy -1.8
  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
10 marks Moderate -0.3
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
5 marks Standard +0.3
  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
10 marks Standard +0.3
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
15 marks Standard +0.3
  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
15 marks Standard +0.3
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
15 marks Standard +0.3
  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
7 marks Easy -1.8
  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
7 marks Standard +0.3
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
7 marks Moderate -0.3
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
11 marks Moderate -0.8
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
13 marks Standard +0.3
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
14 marks Moderate -0.3
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
16 marks Standard +0.3
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
6 marks Moderate -0.8
  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
8 marks Standard +0.8
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
12 marks Easy -1.2
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
14 marks Standard +0.3
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
15 marks Standard +0.3
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 .
Edexcel S2 2003 January Q6
20 marks Moderate -0.8
6. A magazine has a large number of subscribers who each pay a membership fee that is due on January 1st each year. Not all subscribers pay their fee by the due date. Based on correspondence from the subscribers, the editor of the magazine believes that \(40 \%\) of subscribers wish to change the name of the magazine. Before making this change the editor decides to carry out a sample survey to obtain the opinions of the subscribers. He uses only those members who have paid their fee on time.
  1. Define the population associated with the magazine.
  2. Suggest a suitable sampling frame for the survey.
  3. Identify the sampling units.
  4. Give one advantage and one disadvantage that would have resulted from the editor using a census rather than a sample survey. As a pilot study the editor took a random sample of 25 subscribers.
  5. Assuming that the editor's belief is correct, find the probability that exactly 10 of these subscribers agreed with changing the name. In fact only 6 subscribers agreed to the name being changed.
  6. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not the percentage agreeing to the change is less that the editor believes. The full survey is to be carried out using 200 randomly chosen subscribers.
  7. Again assuming the editor's belief to be correct and using a suitable approximation, find the probability that in this sample there will be least 71 but fewer than 83 subscribers who agree to the name being changed. \section*{END}
Edexcel S2 2004 January Q1
5 marks Easy -1.8
  1. A large dental practice wishes to investigate the level of satisfaction of its patients.
    1. Suggest a suitable sampling frame for the investigation.
    2. Identify the sampling units.
    3. State one advantage and one disadvantage of using a sample survey rather than a census.
    4. Suggest a problem that might arise with the sampling frame when selecting patients.
    5. The random variable \(R\) has the binomial distribution \(\mathrm { B } ( 12,0.35 )\).
    6. Find \(\mathrm { P } ( R \geq 4 )\).
    The random variable \(S\) has the Poisson distribution with mean 2.71.
  2. Find \(\mathrm { P } ( S \leq 1 )\). The random variable \(T\) has the normal distribution \(\mathrm { N } \left( 25,5 ^ { 2 } \right)\).
  3. Find \(\mathrm { P } ( T \leq 18 )\).
Edexcel S2 2004 January Q3
9 marks Moderate -0.8
3. The discrete random variable \(X\) is distributed \(\mathrm { B } ( n , p )\).
  1. Write down the value of \(p\) that will give the most accurate estimate when approximating the binomial distribution by a normal distribution.
  2. Give a reason to support your value.
  3. Given that \(n = 200\) and \(p = 0.48\), find \(\mathrm { P } ( 90 \leq X < 105 )\).
Edexcel S2 2004 January Q4
10 marks Moderate -0.8
4. (a) Write down two conditions needed to be able to approximate the binomial distribution by the Poisson distribution. A researcher has suggested that 1 in 150 people is likely to catch a particular virus.
Assuming that a person catching the virus is independent of any other person catching it,
(b) find the probability that in a random sample of 12 people, exactly 2 of them catch the virus.
(c) Estimate the probability that in a random sample of 1200 people fewer than 7 catch the virus.
Edexcel S2 2004 January Q5
13 marks Standard +0.3
5. Vehicles pass a particular point on a road at a rate of 51 vehicles per hour.
  1. Give two reasons to support the use of the Poisson distribution as a suitable model for the number of vehicles passing this point. Find the probability that in any randomly selected 10 minute interval
  2. exactly 6 cars pass this point,
  3. at least 9 cars pass this point. After the introduction of a roundabout some distance away from this point it is suggested that the number of vehicles passing it has decreased. During a randomly selected 10 minute interval 4 vehicles pass the point.
  4. Test, at the \(5 \%\) level of significance, whether or not there is evidence to support the suggestion that the number of vehicles has decreased. State your hypotheses clearly.
    (6)
Edexcel S2 2004 January Q6
13 marks Standard +0.3
6. From past records a manufacturer of ceramic plant pots knows that \(20 \%\) of them will have defects. To monitor the production process, a random sample of 25 pots is checked each day and the number of pots with defects is recorded.
  1. Find the critical regions for a two-tailed test of the hypothesis that the probability that a plant pot has defects is 0.20 . The probability of rejection in either tail should be as close as possible to \(2.5 \%\).
  2. Write down the significance level of the above test. A garden centre sells these plant pots at a rate of 10 per week. In an attempt to increase sales, the price was reduced over a six-week period. During this period a total of 74 pots was sold.
  3. Using a \(5 \%\) level of significance, test whether or not there is evidence that the rate of sales per week has increased during this six-week period.