Questions — Edexcel S2 (494 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure 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 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 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 Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
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 2003 January Q6
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
  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
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
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
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
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.
Edexcel S2 2004 January Q7
7. The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} k x ( 5 - x ) , & 0 \leq x \leq 4
0 , & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 3 } { 56 }\).
  2. Find the cumulative distribution function \(\mathrm { F } ( x )\) for all values of \(x\).
  3. Evaluate \(\mathrm { E } ( X )\).
  4. Find the modal value of \(X\).
  5. Verify that the median value of \(X\) lies between 2.3 and 2.5.
  6. Comment on the skewness of \(X\). Justify your answer. \section*{END}
Edexcel S2 2005 January Q1
  1. The random variables \(R , S\) and \(T\) are distributed as follows
$$R \sim \mathrm {~B} ( 15,0.3 ) , \quad S \sim \mathrm { Po } ( 7.5 ) , \quad T \sim \mathrm {~N} \left( 8,2 ^ { 2 } \right) .$$ Find
  1. \(\mathrm { P } ( R = 5 )\),
  2. \(\mathrm { P } ( S = 5 )\),
  3. \(\mathrm { P } ( T = 5 )\).
Edexcel S2 2005 January Q2
2. (a) Explain what you understand by (i) a population and (ii) a sampling frame. The population and the sampling frame may not be the same.
(b) Explain why this might be the case.
(c) Give an example, justifying your choices, to illustrate when you might use
  1. a census,
  2. a sample.
Edexcel S2 2005 January Q3
3. A rod of length \(2 l\) was broken into 2 parts. The point at which the rod broke is equally likely to be anywhere along the rod. The length of the shorter piece of rod is represented by the random variable \(X\).
  1. Write down the name of the probability density function of \(X\), and specify it fully.
  2. Find \(\mathrm { P } \left( X < \frac { 1 } { 3 } l \right)\).
  3. Write down the value of \(\mathrm { E } ( X )\). Two identical rods of length \(2 l\) are broken.
  4. Find the probability that both of the shorter pieces are of length less than \(\frac { 1 } { 3 } l\).
Edexcel S2 2005 January Q4
4. In an experiment, there are 250 trials and each trial results in a success or a failure.
  1. Write down two other conditions needed to make this into a binomial experiment. It is claimed that \(10 \%\) of students can tell the difference between two brands of baked beans. In a random sample of 250 students, 40 of them were able to distinguish the difference between the two brands.
  2. Using a normal approximation, test at the \(1 \%\) level of significance whether or not the claim is justified. Use a one-tailed test.
  3. Comment on the acceptability of the assumptions you needed to carry out the test.
Edexcel S2 2005 January Q5
5. From company records, a manager knows that the probability that a defective article is produced by a particular production line is 0.032 . A random sample of 10 articles is selected from the production line.
  1. Find the probability that exactly 2 of them are defective. On another occasion, a random sample of 100 articles is taken.
  2. Using a suitable approximation, find the probability that fewer than 4 of them are defective. At a later date, a random sample of 1000 is taken.
  3. Using a suitable approximation, find the probability that more than 42 are defective.
    (6)
Edexcel S2 2005 January Q6
6. Over a long period of time, accidents happened on a stretch of road at random at a rate of 3 per month. Find the probability that
  1. in a randomly chosen month, more than 4 accidents occurred,
  2. in a three-month period, more than 4 accidents occurred. At a later date, a speed restriction was introduced on this stretch of road. During a randomly chosen month only one accident occurred.
  3. Test, at the \(5 \%\) level of significance, whether or not there is evidence to support the claim that this speed restriction reduced the mean number of road accidents occurring per month. The speed restriction was kept on this road. Over a two-year period, 55 accidents occurred.
  4. Test, at the \(5 \%\) level of significance, whether or not there is now evidence that this speed restriction reduced the mean number of road accidents occurring per month.
Edexcel S2 2005 January Q7
7. The random variable \(X\) has probability density function $$\mathrm { f } ( x ) = \begin{cases} k \left( - x ^ { 2 } + 5 x - 4 \right) , & 1 \leq x \leq 4
0 , & \text { otherwise } \end{cases}$$
  1. Show that \(k = \frac { 2 } { 9 }\). Find
  2. \(\mathrm { E } ( X )\),
  3. the mode of \(X\).
  4. the cumulative distribution function \(\mathrm { F } ( x )\) for all \(x\).
  5. Evaluate \(\mathrm { P } ( X \leq 2.5 )\),
  6. Deduce the value of the median and comment on the shape of the distribution.
Edexcel S2 2006 January Q1
  1. A fair coin is tossed 4 times.
Find the probability that
  1. an equal number of head and tails occur
  2. all the outcomes are the same,
  3. the first tail occurs on the third throw.
Edexcel S2 2006 January Q2
2. Accidents on a particular stretch of motorway occur at an average rate of 1.5 per week.
  1. Write down a suitable model to represent the number of accidents per week on this stretch of motorway. Find the probability that
  2. there will be 2 accidents in the same week,
  3. there is at least one accident per week for 3 consecutive weeks,
  4. there are more than 4 accidents in a 2 week period.
Edexcel S2 2006 January Q3
3. The random variable \(X\) is uniformly distributed over the interval \([ - 1,5 ]\).
  1. Sketch the probability density function \(\mathrm { f } ( x )\) of \(X\). Find
  2. \(\mathrm { E } ( X )\),
  3. \(\operatorname { Var } ( \mathrm { X } )\),
  4. \(\mathrm { P } ( - 0.3 < X < 3.3 )\).
Edexcel S2 2006 January Q4
4. The random variable \(X \sim \mathrm {~B} ( 150,0.02 )\). Use a suitable approximation to estimate \(\mathrm { P } ( X > 7 )\).
Edexcel S2 2006 January Q5
5. A continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) where $$f ( x ) = \begin{cases} k x ( x - 2 ) , & 2 \leq x \leq 3
0 , & \text { otherwise } \end{cases}$$ where \(k\) is a positive constant.
  1. Show that \(k = \frac { 3 } { 4 }\). Find
  2. \(\mathrm { E } ( X )\),
  3. the cumulative distribution function \(\mathrm { F } ( x )\).
  4. Show that the median value of \(X\) lies between 2.70 and 2.75.
Edexcel S2 2006 January Q6
6. A bag contains a large number of coins. Half of them are 1 p coins, one third are 2 p coins and the remainder are 5p coins.
  1. Find the mean and variance of the value of the coins. A random sample of 2 coins is chosen from the bag.
  2. List all the possible samples that can be drawn.
  3. Find the sampling distribution of the mean value of these samples.
Edexcel S2 2006 January Q7
7. A teacher thinks that \(20 \%\) of the pupils in a school read the Deano comic regularly. He chooses 20 pupils at random and finds 9 of them read the Deano.
    1. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the percentage of pupils that read the Deano is different from 20\%. State your hypotheses clearly.
    2. State all the possible numbers of pupils that read the Deano from a sample of size 20 that will make the test in part (a)(i) significant at the \(5 \%\) level.
      (9) The teacher takes another 4 random samples of size 20 and they contain 1, 3, 1 and 4 pupils that read the Deano.
  2. By combining all 5 samples and using a suitable approximation test, at the \(5 \%\) level of significance, whether or not this provides evidence that the percentage of pupils in the school that read the Deano is different from 20\%.
  3. Comment on your results for the tests in part (a) and part (b).
Edexcel S2 2007 January Q1
  1. (a) Define a statistic.
A random sample \(X _ { 1 } , X _ { 2 } , \ldots , X _ { \mathrm { n } }\) is taken from a population with unknown mean \(\mu\).
(b) For each of the following state whether or not it is a statistic.
  1. \(\frac { X _ { 1 } + X _ { 4 } } { 2 }\),
  2. \(\frac { \sum X ^ { 2 } } { n } - \mu ^ { 2 }\).
Edexcel S2 2007 January Q2
2. The random variable \(J\) has a Poisson distribution with mean 4.
  1. Find \(\mathrm { P } ( J \geqslant 10 )\). The random variable \(K\) has a binomial distribution with parameters \(n = 25 , p = 0.27\).
  2. Find \(\mathrm { P } ( K \leqslant 1 )\).
Edexcel S2 2007 January Q3
3. For a particular type of plant \(45 \%\) have white flowers and the remainder have coloured flowers. Gardenmania sells plants in batches of 12. A batch is selected at random. Calculate the probability that this batch contains
  1. exactly 5 plants with white flowers,
  2. more plants with white flowers than coloured ones. Gardenmania takes a random sample of 10 batches of plants.
  3. Find the probability that exactly 3 of these batches contain more plants with white flowers than coloured ones. Due to an increasing demand for these plants by large companies, Gardenmania decides to sell them in batches of 50 .
  4. Use a suitable approximation to calculate the probability that a batch of 50 plants contains more than 25 plants with white flowers.
Edexcel S2 2007 January Q4
4. (a) State the condition under which the normal distribution may be used as an approximation to the Poisson distribution.
(b) Explain why a continuity correction must be incorporated when using the normal distribution as an approximation to the Poisson distribution. A company has yachts that can only be hired for a week at a time. All hiring starts on a Saturday.
During the winter the mean number of yachts hired per week is 5 .
(c) Calculate the probability that fewer than 3 yachts are hired on a particular Saturday in winter. During the summer the mean number of yachts hired per week increases to 25 . The company has only 30 yachts for hire.
(d) Using a suitable approximation find the probability that the demand for yachts cannot be met on a particular Saturday in the summer. In the summer there are 16 Saturdays on which a yacht can be hired.
(e) Estimate the number of Saturdays in the summer that the company will not be able to meet the demand for yachts.