Questions S2 (1597 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 Q5
5. The continuous random variable \(X\) has the following cumulative distribution function: $$\mathrm { F } ( x ) = \begin{cases} 0 , & x < 0
\frac { 1 } { 432 } x ^ { 2 } \left( x ^ { 2 } - 16 x + 72 \right) , & 0 \leq x \leq 6
1 , & x > 6 \end{cases}$$
  1. Find \(\mathrm { P } ( X < 2 )\).
  2. Find and specify fully the probability density function \(\mathrm { f } ( x )\) of \(X\).
  3. Show that the mode of \(X\) is 2 .
  4. State, with a reason, whether the median of \(X\) is higher or lower than the mode of \(X\).
Edexcel S2 Q6
6. A shop receives weekly deliveries of 120 eggs from a local farm. The proportion of eggs received from the farm that are broken is 0.008
  1. Explain why it is reasonable to use the binomial distribution to model the number of eggs that are broken in each delivery.
  2. Use the binomial distribution to calculate the probability that at most one egg in a delivery will be broken.
  3. State the conditions under which the binomial distribution can be approximated by the Poisson distribution.
  4. Using the Poisson approximation to the binomial, find the probability that at most one egg in a delivery will be broken. Comment on your answer.
Edexcel S2 Q7
7. The random variable \(X\) follows a continuous uniform distribution over the interval [2,11].
  1. Write down the mean of \(X\).
  2. Find \(\mathrm { P } ( X \geq 8.6 )\).
  3. Find \(\mathrm { P } ( | X - 5 | < 2 )\). The random variable \(Y\) follows a continuous uniform distribution over the interval \([ a , b ]\).
  4. Show by integration that $$\mathrm { E } \left( Y ^ { 2 } \right) = \frac { 1 } { 3 } \left( b ^ { 2 } + a b + a ^ { 2 } \right)$$
  5. Hence, prove that $$\operatorname { Var } ( Y ) = \frac { 1 } { 12 } ( b - a ) ^ { 2 }$$ You may assume that \(\mathrm { E } ( Y ) = \frac { 1 } { 2 } ( a + b )\).
Edexcel S2 Q1
  1. The continuous random variable \(X\) has the following cumulative distribution function:
$$F ( x ) = \begin{cases} 0 , & x < 2
k \left( 19 x - x ^ { 2 } - 34 \right) , & 2 \leq x \leq 5
1 , & x > 5 \end{cases}$$
  1. Show that \(k = \frac { 1 } { 36 }\).
  2. Find \(\mathrm { P } ( X > 4 )\).
  3. Find and specify fully the probability density function \(\mathrm { f } ( x )\) of \(X\).
Edexcel S2 Q2
2. Suggest, with reasons, suitable distributions for modelling each of the following:
  1. the number of times the letter J occurs on each page of a magazine,
  2. the length of string left over after cutting as many 3 metre long pieces as possible from partly used balls of string,
  3. the number of heads obtained when spinning a coin 15 times.
Edexcel S2 Q3
3. A primary school teacher finds that exactly half of his year 6 class have mobile phones. He decides to investigate whether the proportion of pupils with mobile phones is different from 0.5 in the year 5 class at his school. There are 25 pupils in the year 5 class.
  1. State the hypotheses that he should use.
  2. Find the largest critical region for this test such that the probability in each "tail" is less than \(2.5 \%\).
  3. Determine the significance level of this test. He finds that eight of the year 5 pupils have mobile phones and concludes that there is not sufficient evidence of the proportion being different from 0.5
  4. Stating the new hypotheses clearly, find if the number of year 5 pupils with mobile phones would have been significant if he had tested whether or not the proportion was less than 0.5 and used the largest critical region with a probability of less than \(5 \%\).
    (3 marks)
Edexcel S2 Q4
4. A hardware store is open on six days each week. On average the store sells 8 of a particular make of electric drill each week. Find the probability that the store sells
  1. no more than 4 of the drills in a week,
  2. more than 2 of the drills in one day. The store receives one delivery of drills at the same time each week.
  3. Find the number of drills that need to be in stock after a delivery for there to be at most a 5\% chance of the store not having sufficient drills to meet demand before the next delivery.
    (3 marks)
    [0pt]
Edexcel S2 Q5
5. In a party game, a bottle is spun and whoever it points to when it stops has to play next. The acute angle, in degrees, that the bottle makes with the side of the room is modelled by a rectangular distribution over the interval [0,90]. Find the probability that on one spin this angle is
  1. between \(25 ^ { \circ }\) and \(38 ^ { \circ }\),
  2. \(45 ^ { \circ }\) to the nearest degree. The bottle is spun ten times.
  3. Find the probability that the acute angle it makes with the side of the room is less than \(10 ^ { \circ }\) more than twice.
Edexcel S2 Q6
6. A teacher is monitoring attendance at lessons in her department. She believes that the number of students absent from each lesson follows a Poisson distribution and wished to test the null hypothesis that the mean is 2.5 against the alternative hypothesis that it is greater than 2.5 She visits one lesson and decides on a critical region of 6 or more students absent.
  1. Find the significance level of this test.
  2. State any assumptions made in carrying out this test and comment on their validity. The teacher decides to undertake a wider study by looking at a sample of all the lessons that have taken place in the department during the previous four weeks.
  3. Suggest a suitable sampling frame. She finds that there have been 96 pupils absent from the 30 lessons in her sample.
  4. Using a suitable approximation, test at the \(5 \%\) level of significance the null hypothesis that the mean is 2.5 students absent per lesson against the alternative hypothesis that it is greater than 2.5. You may assume that the number of absences follows a Poisson distribution.
    (6 marks)
Edexcel S2 Q7
7. In a competition at a funfair, participants have to stay on a log being rotated in a pool of water for as long as possible. The length of time, in tens of seconds, that the competitors stay on the log is modelled by the random variable \(T\) with the following probability density function: $$\mathrm { f } ( t ) = \begin{cases} k ( t - 3 ) ^ { 2 } , & 0 \leq t \leq 3
0 , & \text { otherwise } \end{cases}$$
  1. Show that \(k = \frac { 1 } { 9 }\).
  2. Sketch f \(( t )\) for all values of \(t\).
  3. Show that the mean time that competitors stay on the \(\log\) is 7.5 seconds. When the competition is next run the organisers decide to make it easier at first by spinning the log more slowly and then increasing the speed of rotation. The length of time, in tens of seconds, that the competitors now stay on the log is modelled by the random variable \(S\) with the following probability density function: $$f ( s ) = \begin{cases} \frac { 1 } { 12 } \left( 8 - s ^ { 3 } \right) , & 0 \leq s \leq 2
    0 , & \text { otherwise } \end{cases}$$
  4. Find the change in the mean time that competitors stay on the log.
Edexcel S2 Q1
  1. (a) State one advantage and one disadvantage in using a census rather than a sample survey in statistical work.
    (b) Give an example of a situation in which you would choose to use a census rather than a sample survey and explain why.
    (2 marks)
  2. An advert for Tatty's Crisps claims that 1 in 10 bags contain a free scratchcard game.
Tatty's Crisps can be bought in a Family Pack containing 10 bags. Find the probability that the bags in one of these Family Packs contain
(a) no scratchcards,
(b) more than 2 scratchcards. Tatty's Crisps can also be bought wholesale in boxes containing 50 bags. A pub Landlord notices that her customers only found 2 scratchcards in the crisps from one of these boxes.
(c) Stating your hypotheses clearly, test at the \(10 \%\) level of significance whether or not this gives evidence of there being fewer free scratchcards than is claimed by the advert.
(4 marks)
Edexcel S2 Q3
3. A class of children are each asked to draw a line that they think is 10 cm long without using a ruler. The teacher models how many centimetres each child's line is longer than 10 cm by the random variable \(X\) and believes that \(X\) has the following probability density function: $$f ( x ) = \left\{ \begin{array} { l c } \frac { 1 } { 8 } , & - 4 \leq x \leq 4
0 , & \text { otherwise } \end{array} \right.$$
  1. Write down the name of this distribution.
  2. Define fully the cumulative distribution function \(\mathrm { F } ( x )\) of \(X\).
  3. Calculate the proportion of children making an error of less than \(15 \%\) according to this model.
  4. Give two reasons why this may not be a very suitable model.
Edexcel S2 Q4
4. A bag contains 40 beads of the same shape and size. The ratio of red to green to blue beads is \(1 : 3 : 4\) and there are no beads of any other colour. In an experiment, a bead is picked at random, its colour noted and the bead replaced in the bag. This is done ten times.
  1. Suggest a suitable distribution for modelling the number of times a blue bead is picked out and give the value of any parameters needed.
  2. Explain why this distribution would not be suitable if the beads were not replaced in the bag.
  3. Find the probability that of the ten beads picked out
    1. five are blue,
    2. at least one is red. The experiment is repeated, but this time a bead is picked out and replaced \(n\) times.
  4. Find in the form \(a ^ { n } < b\), where \(a\) and \(b\) are exact fractions, the condition which \(n\) must satisfy in order to have at least a \(99 \%\) chance of picking out at least one red bead.
Edexcel S2 Q5
5. A charity receives donations of more than \(\pounds 10000\) at an average rate of 25 per year. Find the probability that the charity receives
  1. exactly 30 such donations in one year,
  2. less than 3 such donations in one month.
  3. Using a suitable approximation, find the probability that the charity receives more than 45 donations of more than \(\pounds 10000\) in the next two years.
Edexcel S2 Q6
6. The length of time, in tens of minutes, that patients spend waiting at a doctor's surgery is modelled by the continuous random variable \(T\), with the following cumulative distribution function: $$\mathrm { F } ( t ) = \begin{cases} 0 , & t < 0
\frac { 1 } { 135 } \left( 54 t + 9 t ^ { 2 } - 4 t ^ { 3 } \right) , & 0 \leq t \leq 3
1 , & t > 3 \end{cases}$$
  1. Find the probability that a patient waits for more than 20 minutes.
  2. Show that the median waiting time is between 11 and 12 minutes.
  3. Define fully the probability density function \(\mathrm { f } ( t )\) of \(T\).
  4. Find the modal waiting time in minutes.
  5. Give one reason why this model may need to be refined.
Edexcel S2 Q7
7. A student collects data on the number of bicycles passing outside his house in 5-minute intervals during one morning.
  1. Suggest, with reasons, a suitable distribution for modelling this situation. The student's data is shown in the table below.
    Number of bicycles0123456 or more
    Frequency714102120
  2. Show that the mean and variance of these data are 1.5 and 1.58 (to 3 significant figures) respectively and explain how these values support your answer to part (a). An environmental organisation declares a "car free day" encouraging the public to leave their cars at home. The student wishes to test whether or not there are more bicycles passing along his road on this day and records 16 bicycles in a half-hour period during the morning.
  3. Stating your hypotheses clearly, test at the \(5 \%\) level of significance whether or not there are more than 1.5 bicycles passing along his road per 5-minute interval that morning.
Edexcel S2 Q1
  1. (a) The random variable \(X\) follows a Poisson distribution with a mean of 1.4
Find \(\mathrm { P } ( X \leq 3 )\).
(b) The random variable \(Y\) follows a binomial distribution such that \(Y \sim \mathrm {~B} ( 20,0.6 )\). Find \(\mathrm { P } ( Y \leq 12 )\).
(4 marks)
Edexcel S2 Q2
2. A driving instructor keeps records of all the learners she has taught. In order to analyse her success rate she wishes to take a random sample of 120 of these learners.
  1. Suggest a suitable sampling frame and identify the sampling units. She believes that only 1 in 20 of the people she teaches fail to pass their test in their first two attempts. She decides to use her sample to test whether or not the proportion is different from this.
  2. Using a suitable approximation and stating clearly the hypotheses she should use, find the largest critical region for this test such that the probability in each "tail" is less than \(2.5 \%\).
  3. State the significance level of this test.
Edexcel S2 Q3
3. In an old computer game a white square representing a ball appears at random at the top of the playing area, which is 24 cm wide, and moves down the screen. The continuous random variable \(X\) represents the distance, in centimetres, of the dot from the left-hand edge of the screen when it appears. The distribution of \(X\) is rectangular over the interval [4,28].
  1. Find the mean and variance of \(X\).
  2. Find \(\mathrm { P } ( | X - 16 | < 3 )\). During a single game, a player receives 12 "balls".
  3. Find the probability that the ball appears within 3 cm of the middle of the top edge of the playing area more than four times in a single game.
    (3 marks)
Edexcel S2 Q4
4. A music website is visited by an average of 30 different people per hour on a weekday evening. The site's designer believes that the number of visitors to the site per hour can be modelled by a Poisson distribution.
  1. State the conditions necessary for a Poisson distribution to be applicable and comment on their validity in this case. Assuming that the number of visitors does follow a Poisson distribution, find the probability that there will be
  2. less than two visitors in a 10 -minute interval,
  3. at least ten visitors in a 15-minute interval.
  4. Using a suitable approximation, find the probability of the site being visited by more than 100 people between 6 pm and 9 pm on a Thursday evening.
    (5 marks)
Edexcel S2 Q5
5. Four coins are flipped together and the random variable \(H\) represents the number of heads obtained. Assuming that the coins are fair,
  1. suggest with reasons a suitable distribution for modelling \(H\) and give the value of any parameters needed,
  2. show that the probability of obtaining more heads than tails is \(\frac { 5 } { 16 }\). The four coins are flipped 5 times and more heads are obtained than tails 4 times.
  3. Stating your hypotheses clearly, test at the \(5 \%\) level of significance whether or not there is evidence of the probability of getting more heads than tails being more than \(\frac { 5 } { 16 }\). Given that the four coins are all biased such that the chance of each one showing a head is 50\% more than the chance of it showing a tail,
  4. find the probability of obtaining more heads than tails when the four coins are flipped together.
Edexcel S2 Q6
6. The continuous random variable \(X\) has the following probability density function: $$f ( x ) = \begin{cases} \frac { 1 } { 16 } x , & 2 \leq x \leq 6
0 , & \text { otherwise } \end{cases}$$
  1. Sketch \(\mathrm { f } ( x )\) for all values of \(x\).
  2. Find \(\mathrm { E } ( X )\).
  3. Show that \(\operatorname { Var } ( X ) = \frac { 11 } { 9 }\).
  4. Define fully the cumulative distribution function \(\mathrm { F } ( x )\) of \(X\).
  5. Show that the interquartile range of \(X\) is \(2 ( \sqrt { } 7 - \sqrt { 3 } )\). END
CAIE S2 2016 June Q6
  1. Find \(\mathrm { P } ( X + Y = 4 )\). A random sample of 75 values of \(X\) is taken.
  2. State the approximate distribution of the sample mean, \(\bar { X }\), including the values of the parameters.
  3. Hence find the probability that the sample mean is more than 1.7.
  4. Explain whether the Central Limit theorem was needed to answer part (ii).
CAIE S2 2019 June Q6
  1. Show that \(b = \frac { a } { a - 1 }\).
  2. Given that the median of \(X\) is \(\frac { 3 } { 2 }\), find the values of \(a\) and \(b\).
  3. Use your values of \(a\) and \(b\) from part (ii) to find \(\mathrm { E } ( X )\).
Edexcel S2 2002 June Q2
  1. Explain what you understand by the statistic \(Y\).
  2. Give an example of a statistic.
  3. Explain what you understand by the sampling distribution of \(Y\). \item The continuous random variable \(R\) is uniformly distributed on the interval \(\alpha \leq R \leq \beta\). Given that \(\mathrm { E } ( R ) = 3\) and \(\operatorname { Var } ( R ) = \frac { 25 } { 3 }\), find
  4. the value of \(\alpha\) and the value of \(\beta\),
  5. \(\mathrm { P } ( R < 6.6 )\). \item Past records show that \(20 \%\) of customers who buy crisps from a large supermarket buy them in single packets. During a particular day a random sample of 25 customers who had bought crisps was taken and 2 of them had bought them in single packets.
  6. Use these data to test, at the \(5 \%\) level of significance, whether or not the percentage of customers who bought crisps in single packets that day was lower than usual. State your hypotheses clearly.
    (6) \end{enumerate} At the same supermarket, the manager thinks that the probability of a customer buying a bumper pack of crisps is 0.03 . To test whether or not this hypothesis is true the manager decides to take a random sample of 300 customers.
  7. Stating your hypotheses clearly, find the critical region to enable the manager to test whether or not there is evidence that the probability is different from 0.03 . The probability for each tail of the region should be as close as possible to \(2.5 \%\).
  8. Write down the significance level of this test.