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 Q7
7. In a certain field, daisies are randomly distributed, at an average density of 0.8 daisies per \(\mathrm { cm } ^ { 2 }\). One particular patch, of area \(1 \mathrm {~cm} ^ { 2 }\), is selected at random. Assuming that the number of daisies per \(\mathrm { cm } ^ { 2 }\) has a Poisson distribution,
  1. find the probability that the chosen patch contains
    1. no daisies,
    2. one daisy. Ten such patches are chosen. Using your answers to part (a),
  2. find the probability that the total number of daisies is less than two.
  3. By considering the distribution of daisies over patches of \(10 \mathrm {~cm} ^ { 2 }\), use the Poisson distribution to find the probability that a particular area of \(10 \mathrm {~cm} ^ { 2 }\) contains no more than one daisy.
  4. Compare your answers to parts (b) and (c).
  5. Use a suitable approximation to find the probability that a patch of area \(1 \mathrm {~m} ^ { 2 }\) contains more than 8100 daisies.
Edexcel S2 Q1
  1. (a) Explain why it is often useful to take samples as a means of obtaining information.
    (b) Briefly define the term sampling frame.
    (c) Suggest a suitable sampling frame for a sample survey on a proposal to install speed humps on a road.
  2. An insurance company conducts its business by using a Call Centre. The average number of calls per minute is \(3 \cdot 5\). In the first minute after a TV advertisement is shown, the number of calls received is 7 .
    (a) Stating your hypotheses carefully, and working at the \(5 \%\) significance level, test whether the advertisement has had an effect.
    (b) Find the number of calls that would be required in the first minute for the null hypothesis to be rejected at the \(0.1 \%\) significance level.
  3. On average, \(35 \%\) of the candidates in a certain subject get an A or B grade in their exam. In a class of 20 students, find the probability that
    (a) less than 5 get A or B grades,
    (b) exactly 8 get A or B grades.
Five such classes of 20 students are combined to sit the exam.
(c) Use a suitable approximation to find the probability that less than a quarter of the total get A or B grades.
Edexcel S2 Q4
4. Light bulbs produced in a certain factory have lifetimes, in 100 s of hours, whose distribution is modelled by the random variable \(X\) with probability density function $$\begin{array} { l l } \mathrm { f } ( x ) = \frac { 2 x ( 3 - x ) } { 9 } , & 0 \leq x \leq 3
\mathrm { f } ( x ) = 0 & \text { otherwise } \end{array}$$
  1. Sketch \(\mathrm { f } ( x )\).
  2. Write down the mean lifetime of a bulb.
  3. Show that ten times as many bulbs fail before 200 hours as survive beyond 250 hours.
  4. Given that a bulb lasts for 200 hours, find the probability that it will then last for at least another 50 hours.
  5. State, with a reason, whether you consider that the density function \(f\) is a realistic model for the lifetimes of light bulbs. \section*{STATISTICS 2 (A) TEST PAPER 2 Page 2}
Edexcel S2 Q5
  1. In a packet of 40 biscuits, the number of currants in each biscuit is as follows
Number of currants, \(x\)0123456
Number of biscuits49118431
  1. Find the mean and variance of the random variable \(X\) representing the number of currants per biscuit.
  2. State an appropriate model for the distribution of \(X\), giving two reasons for your answer. Another machine produces biscuits with a mean of 1.9 currants per biscuit.
  3. Determine which machine is more likely to produce a biscuit with at least two currants.
Edexcel S2 Q6
6. A greengrocer sells apples from a barrel in his shop. He claims that no more than \(5 \%\) of the apples are of poor quality. When he takes 10 apples out for a customer, 2 of them are bad.
  1. Stating your hypotheses clearly, test his claim at the \(1 \%\) significance level.
  2. State an assumption that has been made about the selection of the apples.
  3. When five other customers also buy 10 apples each, the numbers of bad apples they get are \(1,3,1,2\) and 1 respectively. By combining all six customers' results, and using a suitable approximation, test at the \(1 \%\) significance level whether the combined results provide evidence that the proportion of bad apples in the barrel is greater than \(5 \%\).
  4. Comment briefly on your results in parts (a) and (c).
Edexcel S2 Q7
7. Some children are asked to mark the centre of a scale 10 cm long. The position they choose is indicated by the variable \(X\), where \(0 \leq X \leq 10\). Initially, \(X\) is modelled as a random variable with a continuous uniform distribution.
  1. Find the mean and the standard deviation of \(X\). It is suggested that a better model would be the distribution with probability density function $$f ( x ) = c x , 0 \leq x \leq 5 , \quad f ( x ) = c ( 10 - x ) , 5 < x \leq 10 , \quad f ( x ) = 0 \text { otherwise. }$$
  2. Write down the mean of \(X\).
  3. Find \(c\), and hence find the standard deviation of \(X\) in this model.
  4. Find \(\mathrm { P } ( 4 < X < 6 )\). It is then proposed that an even better model for \(X\) would be a Normal distribution with the mean and standard deviation found in parts (b) and (c).
  5. Use these results to find \(\mathrm { P } ( 4 < X < 6 )\) in the third model.
  6. Compare your answer with (d). Which model do you think is most appropriate? (1 mark)
Edexcel S2 Q1
  1. Explain what is meant by
    1. a population,
    2. a sampling unit.
    Suggest suitable sampling frames for surveys of
  2. families who have holidays in Greece,
  3. mothers with children under two years old.
Edexcel S2 Q2
2. A continuous random variable \(X\) has the probability density function $$\begin{array} { l l } \mathrm { f } ( x ) = k & 5 \leq x \leq 15 ,
\mathrm { f } ( x ) = 0 & \text { otherwise. } \end{array}$$
  1. Find \(k\) and specify the cumulative density function \(\mathrm { F } ( x )\).
  2. Write down the value of \(\mathrm { P } ( X < 8 )\).
Edexcel S2 Q3
3. A coin is tossed 20 times, giving 16 heads.
  1. Test at the \(1 \%\) significance level whether the coin is fair, stating your hypotheses clearly.
  2. Find the critical region for the same test at the \(0.1 \%\) significance level.
Edexcel S2 Q4
4. Alison and Gemma play table tennis. Alison starts by serving for the first five points. The probability that she wins a point when serving is \(p\).
  1. Show that the probability that Alison is ahead at the end of her five serves is given by $$p ^ { 3 } \left( 6 p ^ { 2 } - 15 p + 10 \right) .$$
  2. Evaluate this probability when \(p = 0.6\).
Edexcel S2 Q5
5. In a certain school, \(32 \%\) of Year 9 pupils are left-handed. A random sample of 10 Year 9 pupils is chosen.
  1. Find the probability that none are left-handed.
  2. Find the probability that at least two are left-handed.
  3. Use a suitable approximation to find the probability of getting more than 5 but less than 15 left-handed pupils in a group of 35 randomly selected Year 9 pupils.
    Explain what adjustment is necessary when using this approximation. \section*{STATISTICS 2 (A) TEST PAPER 3 Page 2}
Edexcel S2 Q6
  1. A sample of radioactive material decays randomly, with an approximate mean of 1.5 counts per minute.
    1. Name a distribution that would be suitable for modelling the number of counts per minute.
    Give any parameters required for the model.
  2. Find the probability of at least 4 counts in a randomly chosen minute.
  3. Find the probability of 3 counts or fewer in a random interval lasting 5 minutes. More careful measurements, over 50 one-minute intervals, give the following data for \(x\), the number of counts per minute: $$\sum x = 84 , \quad \sum x ^ { 2 } = 226$$
  4. Decide whether these data support your answer to part (a).
  5. Use the improved data to find probability of exactly two counts in a given one-minute interval.
Edexcel S2 Q7
7. Each day on the way to work, a commuter encounters a similar traffic jam. The length of time, in 10-minute units, spent waiting in the traffic jam is modelled by the random variable \(T\) with the cumulative distribution function: $$\begin{array} { l l } \mathrm { F } ( t ) = 0 & t < 0 ,
\mathrm {~F} ( t ) = \frac { t ^ { 2 } \left( 3 t ^ { 2 } - 16 t + 24 \right) } { 16 } & 0 \leq t \leq 2 ,
\mathrm {~F} ( t ) = 1 & t > 2 . \end{array}$$
  1. Show that 0.77 is approximately the median value of \(T\).
  2. Given that he has already waited for 12 minutes, find the probability that he will have to wait another 3 minutes.
  3. Find, and sketch, the probability density function of \(T\).
  4. Hence find the modal value of \(T\).
  5. Comment on the validity of this model.
Edexcel S2 Q1
  1. A random sample is to be taken from the A-level results obtained by the final-year students in a Sixth Form College. Suggest
    1. suitable sampling units,
    2. a suitable sampling frame.
    3. Would it be advisable simply to use the results of all those doing A-level Maths?
    Explain your answer.
Edexcel S2 Q2
2. The random variable \(X\), which can take any value in the interval \(1 \leq X \leq n\), is modelled by the continuous uniform distribution with mean 12.
  1. Show that \(n = 23\) and find the variance of \(X\).
  2. Find \(\mathrm { P } ( 10 < X < 14 )\).
Edexcel S2 Q3
3. The Driving Theory Test includes 30 questions which require one answer to be selected from four options.
  1. Phil ticks answers at random. Find how many of the 30 he should expect to get right.
  2. If he gets 15 correct, decide whether this is evidence that he has actually done some revision. Use a \(5 \%\) significance level. Another candidate, Sarah, has revised and has a 0.9 probability of getting each question right.
  3. Determine the expected number of answers that Sarah will get right.
  4. Find the probability that Sarah gets more than 25 correct answers out of 30.
Edexcel S2 Q4
4. A continuous random variable \(X\) has probability density function $$\begin{array} { l l } \mathrm { f } ( x ) = 0 & x < 1 ,
\mathrm { f } ( x ) = k x & 1 \leq x \leq 4 ,
\mathrm { f } ( x ) = 0 & x > 4 . \end{array}$$
  1. Sketch a graph of \(\mathrm { f } ( x )\), and hence find the value of \(k\).
  2. Calculate the mean and the variance of \(X\). \section*{STATISTICS 2 (A)TEST PAPER 4 Page 2}
Edexcel S2 Q5
  1. In World War II, the number of V2 missiles that landed on each square mile of London was, on average, \(3 \cdot 5\). Assuming that the hits were randomly distributed throughout London,
    1. suggest a suitable model for the number of hits on each square mile, giving a suitable value for any parameters.
    2. calculate the probability that a particular square mile received
      1. no hits,
      2. more than 7 hits.
    3. State, with a reason, whether the model is likely to be accurate.
    In contrast, the number of bombs weighing more than 1 ton landing on each square mile was 45 .
  2. Use a suitable approximation to find the probability that a randomly selected square mile received more than 60 such bombs. Explain what adjustment must be made when using this approximation.
Edexcel S2 Q6
6. In a fruit packing plant, apples are packed on to trays of 10 , and then checked for blemishes. The chance of any particular apple having a blemish is \(5 \%\). If a tray is selected at random, find
  1. the probability that at least two of the apples in it are blemished,
  2. the probability that exactly two are blemished. Trays are now packed in boxes of 50 trays each. In one such box, find
  3. the probability that at most one tray contains at least two blemished apples,
  4. the expected number of trays containing at least two blemished apples.
  5. Use a suitable approximation to find the probability that in a random selection of 20 trays there are more than 10 blemished apples.
Edexcel S2 Q7
7. The time, in hours, taken to run the London marathon is modelled by a continuous random variable \(T\) with the probability density function $$f ( t ) = \begin{cases} c ( t - 2 ) & 2 \leq t < 4
\frac { 2 c ( 7 - t ) } { 3 } & 4 \leq t \leq 7
0 & \text { otherwise } \end{cases}$$
  1. Sketch the function \(\mathrm { f } ( t )\), and show that \(c = \frac { 1 } { 5 }\).
  2. Calculate the median value of \(T\).
  3. Make two critical comments about the model.
Edexcel S2 Q1
  1. (a) Explain briefly why it is often useful to take a sample from a population.
    (b) Suggest a suitable sampling frame for a local council to use to survey attitudes towards a proposed new shopping centre.
  2. A certain type of lettuce seed has a \(12 \%\) failure rate for germination. In a new sample of 25 genetically modified seeds, only 1 did not germinate.
    Clearly stating your hypotheses, test, at the \(1 \%\) significance level, whether the GM seeds are better.
  3. A random variable \(X\) has a Poisson distribution with a mean, \(\lambda\), which is assumed to equal 5 .
    (a) Find \(\mathrm { P } ( X = 0 )\).
    (b) In 100 measurements, the value 0 occurs three times. Find the highest significance level at which you should reject the original hypothesis in favour of \(\lambda < 5\).
  4. The waiting time, in minutes, at a dentist is modelled by the continuous random variable \(T\) with probability density function
$$\begin{array} { l l } \mathrm { f } ( t ) = k ( 10 - t ) & 0 \leq t \leq 10
\mathrm { f } ( t ) = 0 & \text { otherwise. } \end{array}$$ (a) Sketch the graph of \(\mathrm { f } ( t )\) and find the value of \(k\).
(b) Find the mean value of \(T\).
(c) Find the 95th percentile of \(T\).
(d) State whether you consider this function to be a sensible model for \(T\) and suggest how it could be modified to provide a better model.
Edexcel S2 Q5
5. A textbook contains, on average, 1.2 misprints per page. Assuming that the misprints are randomly distributed throughout the book,
  1. specify a suitable model for \(X\), the random variable representing the number of misprints on a given page.
  2. Find the probability that a particular page has more than 2 misprints.
  3. Find the probability that Chapter 1, with 8 pages, has no misprints at all. Chapter 2 is longer, at 20 pages.
  4. Use a suitable approximation to find the probability that Chapter 2 has less than ten misprints altogether. Explain what adjustment is necessary when making this approximation. \section*{STATISTICS 2 (A) TEST PAPER 5 Page 2}
Edexcel S2 Q6
  1. On a production line, bags are filled with cement and weighed as they emerge. It is found that \(20 \%\) of the bags are underweight. In a random sample consisting of \(n\) bags, the variance of the number of underweight bags is found to be \(2 \cdot 4\).
    1. Show that \(n = 15\).
    2. Use cumulative binomial probability tables to find the probability that, in a further random sample of 15 bags, the number that are underweight is
      1. less than 3 ,
      2. at least 5 .
    Ten samples of 15 bags each are tested. Find the probability that
  2. all these batches contain less than 5 underweight bags,
  3. the fourth batch tested is the first to contain less than 5 underweight bags.
Edexcel S2 Q7
7. A continuous random variable \(X\) has a probability density function given by $$\begin{array} { l l } \mathrm { f } ( x ) = \frac { x ^ { 2 } } { 312 } & 4 \leq x \leq 10
\mathrm { f } ( x ) = 0 & \text { otherwise. } \end{array}$$
  1. Find \(\mathrm { E } ( X )\).
  2. Find the variance of \(X\).
  3. Find the cumulative distribution function \(\mathrm { F } ( x )\), for all values of \(x\).
  4. Hence find the median value of \(X\).
  5. Write down the modal value of \(X\). It is sometimes suggested that, for most distributions, $$2 \times ( \text { median } - \text { mean } ) \approx \text { mode } - \text { median } .$$
  6. Show that this result is not satisfied in this case, and suggest a reason why.
Edexcel S2 Q1
\begin{enumerate} \item An insurance company is investigating how often its customers crash their cars.
  1. Suggest an appropriate sampling frame.
  2. Describe the sampling units.
  3. State the advantage of a sample survey over a census in this case. \item A searchlight is rotating in a horizontal circle. It is assumed that that, at any moment, the centre of its beam is equally likely to be pointing in any direction. The random variable \(X\) represents this direction, expressed as a bearing in the range \(000 ^ { \circ }\) to \(360 ^ { \circ }\).