Questions — Edexcel (9685 questions)

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Edexcel S2 2016 June Q2
12 marks Moderate -0.3
  1. In a region of the UK, \(5 \%\) of people have red hair. In a random sample of size \(n\), taken from this region, the expected number of people with red hair is 3
    1. Calculate the value of \(n\).
    A random sample of 20 people is taken from this region. Find the probability that
    1. exactly 4 of these people have red hair,
    2. at least 4 of these people have red hair. Patrick claims that Reddman people have a probability greater than \(5 \%\) of having red hair. In a random sample of 50 Reddman people, 4 of them have red hair.
  3. Stating your hypotheses clearly, test Patrick's claim. Use a \(1 \%\) level of significance.
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Edexcel S2 2016 June Q3
6 marks Moderate -0.8
  1. The random variable \(R\) has a continuous uniform distribution over the interval [5,9]
    1. Specify fully the probability density function of \(R\).
    2. Find \(\mathrm { P } ( 7 < R < 10 )\)
    The random variable \(A\) is the area of a circle radius \(R \mathrm {~cm}\).
  2. Find \(\mathrm { E } ( \mathrm { A } )\)
Edexcel S2 2016 June Q4
10 marks Standard +0.8
4. A continuous random variable \(X\) has cumulative distribution function \(\mathrm { F } ( x )\) given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 2 \\ k \left( a x + b x ^ { 2 } - x ^ { 3 } \right) & 2 \leqslant x \leqslant 3 \\ 1 & x > 3 \end{array} \right.$$ Given that the mode of \(X\) is \(\frac { 8 } { 3 }\)
  1. show that \(b = 8\)
  2. find the value of \(k\).
Edexcel S2 2016 June Q5
8 marks Standard +0.8
5. In a large school, \(20 \%\) of students own a touch screen laptop. A random sample of \(n\) students is chosen from the school. Using a normal approximation, the probability that more than 55 of these \(n\) students own a touch screen laptop is 0.0401 correct to 3 significant figures. Find the value of \(n\).
(8)
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Edexcel S2 2016 June Q6
10 marks Standard +0.3
6. A bag contains a large number of counters with one of the numbers 4,6 or 8 written on each of them in the ratio \(5 : 3 : 2\) respectively. A random sample of 2 counters is taken from the bag.
  1. List all the possible samples of size 2 that can be taken. The random variable \(M\) represents the mean value of the 2 counters.
    Given that \(\mathrm { P } ( M = 4 ) = \frac { 1 } { 4 }\) and \(\mathrm { P } ( M = 8 ) = \frac { 1 } { 25 }\)
  2. find the sampling distribution for \(M\). A sample of \(n\) sets of 2 counters is taken. The random variable \(Y\) represents the number of these \(n\) sets that have a mean of 8
  3. Calculate the minimum value of \(n\) such that \(\mathrm { P } ( Y \geqslant 1 ) > 0.9\)
Edexcel S2 2016 June Q7
15 marks Standard +0.3
7. The weight, \(X \mathrm {~kg}\), of staples in a bin full of paper has probability density function $$f ( x ) = \left\{ \begin{array} { c c } \frac { 9 x - 3 x ^ { 2 } } { 10 } & 0 \leqslant x < 2 \\ 0 & \text { otherwise } \end{array} \right.$$ Use integration to find
  1. \(\mathrm { E } ( X )\)
  2. \(\operatorname { Var } ( X )\)
  3. \(\mathrm { P } ( X > 1.5 )\) Peter raises money by collecting paper and selling it for recycling. A bin full of paper is sold for \(\pounds 50\) but if the weight of the staples exceeds 1.5 kg it sells for \(\pounds 25\)
  4. Find the expected amount of money Peter raises per bin full of paper. Peter could remove all the staples before the paper is sold but the time taken to remove the staples means that Peter will have \(20 \%\) fewer bins full of paper to sell.
  5. Decide whether or not Peter should remove all the staples before selling the bins full of paper. Give a reason for your answer.
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Edexcel S2 2017 June Q1
9 marks Moderate -0.3
  1. A potter believes that \(20 \%\) of pots break whilst being fired in a kiln. Pots are fired in batches of 25 .
    1. Let \(X\) denote the number of broken pots in a batch. A batch is selected at random. Using a 10\% significance level, find the critical region for a two tailed test of the potter's belief. You should state the probability in each tail of your critical region.
    The potter aims to reduce the proportion of pots which break in the kiln by increasing the size of the batch fired. He now fires pots in batches of 50 . He then chooses a batch at random and discovers there are 6 pots which broke whilst being fired in the kiln.
  2. Test, at the \(5 \%\) level of significance, whether or not there is evidence that increasing the number of pots in a batch has reduced the percentage of pots that break whilst being fired in the kiln. State your hypotheses clearly.
Edexcel S2 2017 June Q2
13 marks Standard +0.3
  1. A company receives telephone calls at random at a mean rate of 2.5 per hour.
    1. Find the probability that the company receives
      1. at least 4 telephone calls in the next hour,
      2. exactly 3 telephone calls in the next 15 minutes.
    2. Find, to the nearest minute, the maximum length of time the telephone can be left unattended so that the probability of missing a telephone call is less than 0.2
    The company puts an advert in the local newspaper. The number of telephone calls received in a randomly selected 2 hour period after the paper is published is 10
  2. Test at the 5\% level of significance whether or not the mean rate of telephone calls has increased. State your hypotheses clearly.
Edexcel S2 2017 June Q3
12 marks Standard +0.3
3. The lifetime, \(X\), in tens of hours, of a battery is modelled by the probability density function $$f ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { 9 } x ( 4 - x ) & 1 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{array} \right.$$ Use algebraic integration to find
  1. \(\mathrm { E } ( X )\)
  2. \(\mathrm { P } ( X > 2.5 )\) A radio runs using 2 of these batteries, both of which must be working. Two fully-charged batteries are put into the radio.
  3. Find the probability that the radio will be working after 25 hours of use. Given that the radio is working after 16 hours of use,
  4. find the probability that the radio will be working after being used for another 9 hours.
Edexcel S2 2017 June Q4
11 marks Moderate -0.3
4. The continuous random variable \(X\) is uniformly distributed over the interval \([ \alpha , \beta ]\) Given that \(\mathrm { E } ( X ) = 3.5\) and \(\mathrm { P } ( X > 5 ) = \frac { 2 } { 5 }\)
  1. find the value of \(\alpha\) and the value of \(\beta\) Given that \(\mathrm { P } ( X < c ) = \frac { 2 } { 3 }\)
    1. find the value of \(c\)
    2. find \(\mathrm { P } ( c < X < 9 )\) A rectangle has a perimeter of 200 cm . The length, \(S \mathrm {~cm}\), of one side of this rectangle is uniformly distributed between 30 cm and 80 cm .
  3. Find the probability that the length of the shorter side of the rectangle is less than 45 cm .
Edexcel S2 2017 June Q5
14 marks Standard +0.3
5. The time taken for a randomly selected person to complete a test is \(M\) minutes, where \(M \sim \mathrm {~N} \left( 14 , \sigma ^ { 2 } \right)\) Given that \(10 \%\) of people take less than 12 minutes to complete the test,
  1. find the value of \(\sigma\) Graham selects 15 people at random.
  2. Find the probability that fewer than 2 of these people will take less than 12 minutes to complete the test. Jovanna takes a random sample of \(n\) people. Using a normal approximation, the probability that fewer than 9 of these \(n\) people will take less than 12 minutes to complete the test is 0.3085 to 4 decimal places.
  3. Find the value of \(n\).
Edexcel S2 2017 June Q6
16 marks Standard +0.3
6. The continuous random variable \(X\) has a probability density function $$\mathrm { f } ( x ) = \left\{ \begin{array} { c c } k ( x - 2 ) & 2 \leqslant x \leqslant 3 \\ k & 3 < x < 5 \\ k ( 6 - x ) & 5 \leqslant x \leqslant 6 \\ 0 & \text { otherwise } \end{array} \right.$$ where \(k\) is a positive constant.
  1. Sketch the graph of \(\mathrm { f } ( x )\).
  2. Show that the value of \(k\) is \(\frac { 1 } { 3 }\)
  3. Define fully the cumulative distribution function \(\mathrm { F } ( x )\).
  4. Hence find the 90th percentile of the distribution.
  5. Find \(\mathrm { P } [ \mathrm { E } ( X ) < X < 5.5 ]\)
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Edexcel S2 2018 June Q1
12 marks Standard +0.3
  1. In a call centre, the number of telephone calls, \(X\), received during any 10 -minute period follows a Poisson distribution with mean 9
    1. Find
      1. \(\mathrm { P } ( X > 5 )\)
      2. \(\mathrm { P } ( 4 \leqslant X < 10 )\)
    The length of a working day is 7 hours.
  2. Using a suitable approximation, find the probability that there are fewer than 370 telephone calls in a randomly selected working day. A week, consisting of 5 working days, is selected at random.
  3. Find the probability that in this week at least 4 working days have fewer than 370 telephone calls.
Edexcel S2 2018 June Q2
9 marks Moderate -0.8
2. A fair coin is spun 6 times and the random variable \(T\) represents the number of tails obtained.
  1. Give two reasons why a binomial model would be a suitable distribution for modelling \(T\).
  2. Find \(\mathrm { P } ( T = 5 )\)
  3. Find the probability of obtaining more tails than heads. A second coin is biased such that the probability of obtaining a head is \(\frac { 1 } { 4 }\) This second coin is spun 6 times.
  4. Find the probability that, for the second coin, the number of heads obtained is greater than or equal to the number of tails obtained.
Edexcel S2 2018 June Q3
18 marks Standard +0.3
  1. The length of time, \(T\), minutes, spent completing a particular task has probability density function
$$f ( t ) = \left\{ \begin{array} { c c } \frac { 1 } { 2 } ( t - 1 ) & 1 < t \leqslant 2 \\ \frac { 1 } { 16 } \left( 14 t - 3 t ^ { 2 } - 8 \right) & 2 < t \leqslant 4 \\ 0 & \text { otherwise } \end{array} \right.$$
  1. Use algebraic integration to find \(\mathrm { E } ( T )\) Given that \(\mathrm { E } \left( T ^ { 2 } \right) = \frac { 267 } { 40 }\)
  2. find \(\operatorname { Var } ( T )\)
  3. Find the cumulative distribution function \(\mathrm { F } ( t )\)
  4. Find the 20th percentile of the time taken to complete the task.
  5. Find the probability that the time spent completing the task is more than 1.5 minutes. Given that a person has already spent 1.5 minutes on the task,
  6. find the probability that this person takes more than 3 minutes to complete the task.
Edexcel S2 2018 June Q4
10 marks Moderate -0.3
  1. David aims to catch the train to work each morning. The scheduled departure time of the train is 0830
The number of minutes after 0830 that the train departs may be modelled by the random variable \(X\). Given that \(X\) has a continuous uniform distribution over \([ \alpha , \beta ]\) and that \(\mathrm { E } ( X ) = 4\) and \(\operatorname { Var } ( X ) = 12\)
  1. find the value of \(\alpha\) and the value of \(\beta\). Each morning, the probability that David oversleeps is 0.05 If David oversleeps he will be late for work. If he does not oversleep he will be in time to catch the train, but will be late for work if the train departs after 0835
  2. Find the probability that David will be late for work. Given that David is late for work,
  3. find the probability that he overslept.
Edexcel S2 2018 June Q5
16 marks Standard +0.3
5. Past records show that the proportion of customers buying organic vegetables from Tesson supermarket is 0.35 During a particular day, a random sample of 40 customers from Tesson supermarket was taken and 18 of them bought organic vegetables.
  1. Test, at the \(5 \%\) level of significance, whether or not this provides evidence that the proportion of customers who bought organic vegetables has increased. State your hypotheses clearly. The manager of Tesson supermarket claims that the proportion of customers buying organic eggs is different from the proportion of those buying organic vegetables. To test this claim the manager decides to take a random sample of 50 customers.
  2. Using a \(5 \%\) level of significance, find the critical region to enable the Tesson supermarket manager to test her claim. The probability for each tail of the region should be as close as possible to \(2.5 \%\) During a particular day, a random sample of 50 customers from Tesson supermarket is taken and 8 of them bought organic eggs.
  3. Using your answer to part (b), state whether or not this sample supports the manager's claim. Use a \(5 \%\) level of significance.
  4. State the actual significance level of this test. The proportion of customers who buy organic fruit from Tesson supermarket is 0.2 During a particular day, a random sample of 200 customers from Tesson supermarket is taken. Using a suitable approximation, the probability that fewer than \(n\) of these customers bought organic fruit is 0.0465 correct to 4 decimal places.
  5. Find the value of \(n\).
Edexcel S2 2018 June Q6
10 marks Challenging +1.2
  1. The continuous random variable \(X\) has the following cumulative distribution function
$$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x \leqslant 1 \\ \frac { 4 } { 15 } ( x - 1 ) & 1 < x \leqslant 2 \\ k \left( \frac { a x ^ { 3 } } { 3 } - \frac { x ^ { 4 } } { 4 } \right) + b & 2 < x \leqslant 4 \\ 1 & x > 4 \end{array} \right.$$ where \(k , a\) and \(b\) are constants.
Given that the mode of \(X\) is \(\frac { 8 } { 3 }\)
  1. show that \(a = 4\)
  2. Find \(\mathrm { P } ( X < 2.5 )\) giving your answer to 3 significant figures.
Edexcel S2 Q1
9 marks Standard +0.3
  1. The lifetime, in tens of hours, of a certain delicate electrical component can be modelled by the random variable \(X\) with probability density function
$$f ( x ) = \begin{cases} \frac { 1 } { 42 } x , & 0 \leq x < 6 \\ \frac { 1 } { 7 } & 6 \leq x \leq 10 \\ 0 , & \text { otherwise } \end{cases}$$
  1. Sketch \(\mathrm { f } ( x )\) for all values of \(x\).
  2. Find the probability that a component lasts at least 50 hours. A particular device requires two of these components and it will not operate if one or more of the components fail. The device has just been fitted with two new components and the lifetimes of these two components are independent.
  3. Find the probability that the device breaks down within the next 50 hours.
Edexcel S2 Q2
11 marks Standard +0.3
2. The continuous random variable \(X\) represents the error, in mm, made when a machine cuts piping to a target length. The distribution of \(X\) is rectangular over the interval \([ - 5.0,5.0 ]\). Find
  1. \(\mathrm { P } ( X < - 4.2 )\),
  2. \(\mathrm { P } ( | X | < 1.5 )\). A supervisor checks a random sample of 10 lengths of piping cut by the machine.
  3. Find the probability that more than half of them are within 1.5 cm of the target length.
    (3 marks)
    If \(X < - 4.2\), the length of piping cannot be used. At the end of each day the supervisor checks a random sample of 60 lengths of piping.
  4. Use a suitable approximation to estimate the probability that no more than 2 of these lengths of piping cannot be used.
    (5 marks)
Edexcel S2 Q3
11 marks Moderate -0.8
3. An athletics teacher has kept careful records over the past 20 years of results from school sports days. There are always 10 competitors in the javelin competition. Each competitor is allowed 3 attempts and the teacher has a record of the distances thrown by each competitor at each attempt. The random variable \(D\) represents the greatest distance thrown by each competitor and the random variable \(A\) represents the number of the attempt in which the competitor achieved their greatest distance.
  1. State which of the two random variables \(D\) or \(A\) is continuous. A new athletics coach wishes to take a random sample of the records of 36 javelin competitors.
  2. Specify a suitable sampling frame and explain how such a sample could be taken.
    (2 marks)
    The coach assumes that \(\mathrm { P } ( A = 2 ) = \frac { 1 } { 3 }\), and is therefore surprised to find that 20 of the 36 competitors in the sample achieved their greatest distance on their second attempt. Using a suitable approximation, and assuming that \(\mathrm { P } ( A = 2 ) = \frac { 1 } { 3 }\),
  3. find the probability that at least 20 of the competitors achieved their greatest distance on their second attempt.
    (6 marks)
  4. Comment on the assumption that \(\mathrm { P } ( A = 2 ) = \frac { 1 } { 3 }\).
Edexcel S2 Q4
12 marks Standard +0.3
4. From past records a manufacturer of glass vases knows that \(15 \%\) of the production have slight defects. To monitor the production, a random sample of 20 vases is checked each day and the number of vases with slight defects is recorded.
  1. Using a 5\% significance level, find the critical regions for a two-tailed test of the hypothesis that the probability of a vase with slight defects is 0.15 . The probability of rejecting, in either tail, should be as close as possible to \(2.5 \%\).
  2. State the actual significance level of the test described in part (a). A shop sells these vases at a rate of 2.5 per week. In the 4 weeks of December the shop sold 15 vases.
  3. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not there is evidence that the rate of sales per week had increased in December.
    (6 marks)
Edexcel S2 Q5
14 marks Standard +0.3
5. The continuous random variable \(T\) represents the time in hours that students spend on homework. The cumulative distribution function of \(T\) is $$\mathrm { F } ( t ) = \begin{cases} 0 , & t < 0 \\ k \left( 2 t ^ { 3 } - t ^ { 4 } \right) & 0 \leq t \leq 1.5 \\ 1 , & t > 1.5 \end{cases}$$ where \(k\) is a positive constant.
  1. Show that \(k = \frac { 16 } { 27 }\).
  2. Find the proportion of students who spend more than 1 hour on homework.
  3. Find the probability density function \(\mathrm { f } ( t )\) of \(T\).
  4. Show that \(\mathrm { E } ( T ) = 0.9\).
  5. Show that \(\mathrm { F } ( \mathrm { E } ( T ) ) = 0.4752\). A student is selected at random. Given that the student spent more than the mean amount of time on homework,
  6. find the probability that this student spent more than 1 hour on homework.
Edexcel S2 Q6
18 marks Standard +0.8
6. On a typical weekday morning customers arrive at a village post office independently and at a rate of 3 per 10 minute period. Find the probability that
  1. at least 4 customers arrive in the next 10 minutes,
  2. no more than 7 customers arrive between 11.00 a.m. and 11.30 a.m. The period from 11.00 a.m. to 11.30 a.m. next Tuesday morning will be divided into 6 periods of 5 minutes each.
  3. Find the probability that no customers arrive in at most one of these periods. The post office is open for \(3 \frac { 1 } { 2 }\) hours on Wednesday mornings.
  4. Using a suitable approximation, estimate the probability that more than 49 customers arrive at the post office next Wednesday morning. END
Edexcel S2 Specimen Q1
4 marks Easy -1.2
  1. A school held a disco for years 9,10 and 11 which was attended by 500 pupils. The pupils were registered as they entered the disco. The disco organisers were keen to assess the success of the event. They designed a questionnaire to obtain information from those who attended.
    1. State one advantage and one disadvantage of using a sample survey rather than a census.
    2. Suggest a suitable sampling frame.
    3. Identify the sampling units.
    4. A piece of string \(A B\) has length 12 cm . A child cuts the string at a randomly chosen point \(P\), into two pieces. The random variable \(X\) represents the length, in cm, of the piece \(A P\).
    5. Suggest a suitable model for the distribution of \(X\) and specify it fully
    6. Find the cumulative distribution function of \(X\).
    7. Write down \(\mathrm { P } ( X < 4 )\).
    8. A manufacturer of chocolates produces 3 times as many soft centred chocolates as hard centred ones.
    Assuming that chocolates are randomly distributed within boxes of chocolates, find the probability that in a box containing 20 chocolates there are
  2. equal numbers of soft centred and hard centred chocolates,
  3. fewer than 5 hard centred chocolates. A large box of chocolates contains 100 chocolates.
  4. Write down the expected number of hard centred chocolates in a large box.