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Edexcel S2 2022 October Q6
9 marks Standard +0.8
  1. A bag contains a large number of counters with one of the numbers 5 , 10 or 20 written on each of them in the ratio \(5 : 2 : a\)
A jar contains a large number of counters with one of the numbers 5 or 10 written on each of them in the ratio \(1 : 3\) One counter is selected at random from the bag and then two counters are selected at random from the jar.
The random variable \(R\) represents the range of the numbers on the 3 counters.
Given that \(\mathrm { P } ( R = 15 ) = \frac { 63 } { 256 }\)
  1. by forming and solving an equation in \(a\), show that \(a = 9\)
  2. find the sampling distribution of \(R\)
Edexcel S2 2022 October Q7
12 marks Standard +0.3
  1. (i) The continuous random variable \(X\) is uniformly distributed over the interval \([ a , b ]\)
Given that \(\mathrm { P } ( 5 < X < 13 ) = \frac { 1 } { 5 }\) and \(\mathrm { E } ( X ) = 9\), find \(\mathrm { P } ( 3 X > a + b )\) (ii) The continuous random variable \(Y\) is uniformly distributed over the interval \([ 1 , c ]\) Given that \(\operatorname { Var } ( Y ) = 0.48\), find the exact value of \(\mathrm { E } \left( Y ^ { 2 } \right)\) (iii) A wire of length 20 cm is cut into 2 pieces at a random point. The longest piece of wire is then cut into 2 pieces, equal in length, giving 3 pieces of wire altogether. Find the probability that the length of the shortest piece of wire is less than 6 cm .
Edexcel S2 2023 October Q1
10 marks Moderate -0.8
  1. Sam is a telephone sales representative.
For each call to a customer
  • Sam either makes a sale or does not make a sale
  • sales are made independently
Past records show that, for each call to a customer, the probability that Sam makes a sale is 0.2
  1. Find the probability that Sam makes
    1. exactly 2 sales in 14 calls,
    2. more than 3 sales in 25 calls. Sam makes \(n\) calls each day.
  2. Find the minimum value of \(n\)
    1. so that the expected number of sales each day is at least 6
    2. so that the probability of at least 1 sale in a randomly selected day exceeds 0.95
Edexcel S2 2023 October Q2
8 marks Standard +0.8
  1. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by
$$\mathrm { f } ( x ) = \begin{cases} a x ^ { 3 } & 0 \leqslant x \leqslant 4 \\ b x + c & 4 < x \leqslant d \\ 0 & \text { otherwise } \end{cases}$$ where \(a\), \(b\), \(c\) and \(d\) are constants such that
  • \(b x + c = a x ^ { 3 }\) at \(x = 4\)
  • \(b x + c\) is a straight line segment with end coordinates ( \(4,64 a\) ) and ( \(d , 0\) )
    1. State the mode of \(X\)
Given that the mode of \(X\) is equal to the median of \(X\)
  • use algebraic integration to show that \(a = \frac { 1 } { 128 }\)
  • Find the value of \(d\)
  • Hence find the value of \(b\) and the value of \(c\)
    •
    Edexcel S2 2023 October Q3
    9 marks Moderate -0.8
    1. Every morning Navtej travels from home to work. Navtej leaves home at a random time between 08:00 and 08:15
    • It always takes Navtej 3 minutes to walk to the bus stop
    • Buses run every 15 minutes and Navtej catches the first bus that arrives
    • Once Navtej has caught the bus it always takes a further 29 minutes for Navtej to reach work
    The total time, \(T\) minutes, for Navtej's journey from home to work is modelled by a continuous uniform distribution over the interval \([ \alpha , \beta ]\)
      1. Show that \(\alpha = 32\)
      2. Show that \(\beta = 47\)
    2. State fully the probability density function for this distribution.
    3. Find the value of
      1. \(\mathrm { E } ( T )\)
      2. \(\operatorname { Var } ( T )\)
    4. Find the probability that the time for Navtej's journey is within 5 minutes of 35 minutes.
    Edexcel S2 2023 October Q4
    10 marks Standard +0.3
    1. A manufacturer makes t -shirts in 3 sizes, small, medium and large.
    20\% of the t -shirts made by the manufacturer are small and sell for \(\pounds 10 30 \%\) of the t -shirts made by the manufacturer are medium and sell for \(\pounds 12\) The rest of the t -shirts made by the manufacturer are large and sell for \(\pounds 15\)
    1. Find the mean value of the t -shirts made by the manufacturer. A random sample of 3 t -shirts made by the manufacturer is taken.
    2. List all the possible combinations of the individual selling prices of these 3 t-shirts.
    3. Find the sampling distribution of the median selling price of these 3 t-shirts.
    Edexcel S2 2023 October Q5
    16 marks Standard +0.3
    1. A supermarket receives complaints at a mean rate of 6 per week.
      1. State one assumption necessary, in order for a Poisson distribution to be used to model the number of complaints received by the supermarket.
      2. Find the probability that, in a given week, there are
        1. fewer than 3 complaints received by the supermarket,
        2. at least 6 complaints received by the supermarket.
      In a randomly selected week, the supermarket received 12 complaints.
    2. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the mean number of complaints is greater than 6 per week.
      State your hypotheses clearly. Following changes made by the supermarket, it received 26 complaints over a 6-week period.
    3. Use a suitable approximation to test whether or not there is evidence that, following the changes, the mean number of complaints received is less than 6 per week. You should state your hypotheses clearly and use a 5\% significance level.
    Edexcel S2 2023 October Q6
    12 marks Challenging +1.2
    1. The continuous random variable \(Y\) has cumulative distribution function given by
    $$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 0 \\ \frac { 1 } { 21 } y ^ { 2 } & 0 \leqslant y \leqslant k \\ \frac { 2 } { 15 } \left( 6 y - \frac { y ^ { 2 } } { 2 } \right) - \frac { 7 } { 5 } & k < y \leqslant 6 \\ 1 & y > 6 \end{array} \right.$$
    1. Find \(\mathrm { P } \left( \left. Y < \frac { 1 } { 4 } k \right\rvert \, Y < k \right)\)
    2. Find the value of \(k\)
    3. Use algebraic calculus to find \(\mathrm { E } ( Y )\)
    Edexcel S2 2023 October Q20
    Moderate -0.3
    20\% of the t -shirts made by the manufacturer are small and sell for \(\pounds 10 30 \%\) of the t -shirts made by the manufacturer are medium and sell for \(\pounds 12\) The rest of the t -shirts made by the manufacturer are large and sell for \(\pounds 15\)
    1. Find the mean value of the t -shirts made by the manufacturer. A random sample of 3 t -shirts made by the manufacturer is taken.
    2. List all the possible combinations of the individual selling prices of these 3 t-shirts.
    3. Find the sampling distribution of the median selling price of these 3 t-shirts.
      1. A supermarket receives complaints at a mean rate of 6 per week.
      2. State one assumption necessary, in order for a Poisson distribution to be used to model the number of complaints received by the supermarket.
      3. Find the probability that, in a given week, there are
        1. fewer than 3 complaints received by the supermarket,
        2. at least 6 complaints received by the supermarket.
      In a randomly selected week, the supermarket received 12 complaints.
    4. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the mean number of complaints is greater than 6 per week.
      State your hypotheses clearly. Following changes made by the supermarket, it received 26 complaints over a 6-week period.
    5. Use a suitable approximation to test whether or not there is evidence that, following the changes, the mean number of complaints received is less than 6 per week. You should state your hypotheses clearly and use a 5\% significance level.
      1. The continuous random variable \(Y\) has cumulative distribution function given by
      $$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 0 \\ \frac { 1 } { 21 } y ^ { 2 } & 0 \leqslant y \leqslant k \\ \frac { 2 } { 15 } \left( 6 y - \frac { y ^ { 2 } } { 2 } \right) - \frac { 7 } { 5 } & k < y \leqslant 6 \\ 1 & y > 6 \end{array} \right.$$
    6. Find \(\mathrm { P } \left( \left. Y < \frac { 1 } { 4 } k \right\rvert \, Y < k \right)\)
    7. Find the value of \(k\)
    8. Use algebraic calculus to find \(\mathrm { E } ( Y )\)
      1. The discrete random variable \(X\) is given by
      $$X \sim \mathrm {~B} ( n , p )$$ The value of \(n\) and the value of \(p\) are such that \(X\) can be approximated by a normal random variable \(Y\) where $$Y \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)$$ Given that when using a normal approximation $$\mathrm { P } ( X < 86 ) = 0.2266 \text { and } \mathrm { P } ( X > 97 ) = 0.1056$$
    9. show that \(\sigma = 6\)
    10. Hence find the value of \(n\) and the value of \(p\)
    Edexcel S2 2018 Specimen Q1
    16 marks Standard +0.8
    1. The number of cars caught speeding per day, by a particular camera, has a Poisson distribution with mean 0.8
      1. Find the probability that in a given 4 day period exactly 3 cars will be caught speeding by this camera.
      A car has been caught speeding by this camera.
    2. Find the probability that the period of time that elapses before the next car is caught speeding by this camera is less than 48 hours. Given that 4 cars were caught speeding by this camera in a two day period,
    3. find the probability that 1 was caught on the first day and 3 were caught on the second day. Each car that is caught speeding by this camera is fined £60
    4. Using a suitable approximation, find the probability that, in 90 days, the total amount of fines issued will be more than \(\pounds 5000\)
    Edexcel S2 2018 Specimen Q2
    11 marks Moderate -0.3
    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
    11 marks Moderate -0.3
    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
    7 marks Standard +0.3
    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
    9 marks Standard +0.8
    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
    13 marks Standard +0.3
    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
    8 marks Standard +0.8
    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
    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.