Questions S2 (1597 questions)

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Edexcel S2 2014 June Q5
5. (a) State the conditions under which the normal distribution may be used as an approximation to the binomial distribution. A company sells seeds and claims that 55\% of its pea seeds germinate.
(b) Write down a reason why the company should not justify their claim by testing all the pea seeds they produce. To test the company's claim, a random sample of 220 pea seeds was planted.
(c) State the hypotheses for a two-tailed test of the company's claim. Given that 135 of the 220 pea seeds germinated,
(d) use a normal approximation to test, at the \(5 \%\) level of significance, whether or not the company’s claim is justified.
Edexcel S2 2014 June Q6
6. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by $$f ( x ) = \left\{ \begin{array} { c c } \frac { 2 x } { 9 } & 0 \leqslant x \leqslant 1
\frac { 2 } { 9 } & 1 < x < 4
\frac { 2 } { 3 } - \frac { x } { 9 } & 4 \leqslant x \leqslant 6
0 & \text { otherwise } \end{array} \right.$$
  1. Find \(\mathrm { E } ( X )\).
  2. Find the cumulative distribution function \(\mathrm { F } ( x )\) for all values of \(x\).
  3. Find the median of \(X\).
  4. Describe the skewness. Give a reason for your answer.
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Edexcel S2 2015 June Q1
  1. In a survey it is found that barn owls occur randomly at a rate of 9 per \(1000 \mathrm {~km} ^ { 2 }\).
    1. Find the probability that in a randomly selected area of \(1000 \mathrm {~km} ^ { 2 }\) there are at least 10 barn owls.
    2. Find the probability that in a randomly selected area of \(200 \mathrm {~km} ^ { 2 }\) there are exactly 2 barn owls.
    3. Using a suitable approximation, find the probability that in a randomly selected area of \(50000 \mathrm {~km} ^ { 2 }\) there are at least 470 barn owls.
    4. The proportion of houses in Radville which are unable to receive digital radio is \(25 \%\). In a survey of a random sample of 30 houses taken from Radville, the number, \(X\), of houses which are unable to receive digital radio is recorded.
    5. Find \(\mathrm { P } ( 5 \leqslant X < 11 )\)
    A radio company claims that a new transmitter set up in Radville will reduce the proportion of houses which are unable to receive digital radio. After the new transmitter has been set up, a random sample of 15 houses is taken, of which 1 house is unable to receive digital radio.
  2. Test, at the \(10 \%\) level of significance, the radio company's claim. State your hypotheses clearly.
Edexcel S2 2015 June Q3
3. A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} k x ^ { 2 } & 0 \leqslant x \leqslant 2
k \left( 1 - \frac { x } { 6 } \right) & 2 < x \leqslant 6
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 1 } { 4 }\)
  2. Write down the mode of \(X\).
  3. Specify fully the cumulative distribution function \(\mathrm { F } ( x )\).
  4. Find the upper quartile of \(X\).
Edexcel S2 2015 June Q4
  1. The continuous random variable \(L\) represents the error, in metres, made when a machine cuts poles to a target length. The distribution of \(L\) is a continuous uniform distribution over the interval \([ 0,0.5 ]\)
    1. Find \(\mathrm { P } ( L < 0.4 )\).
    2. Write down \(\mathrm { E } ( L )\).
    3. Calculate \(\operatorname { Var } ( L )\).
    A random sample of 30 poles cut by this machine is taken.
  2. Find the probability that fewer than 4 poles have an error of more than 0.4 metres from the target length. When a new machine cuts poles to a target length, the error, \(X\) metres, is modelled by the cumulative distribution function \(\mathrm { F } ( x )\) where $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 0
    4 x - 4 x ^ { 2 } & 0 \leqslant x \leqslant 0.5
    1 & \text { otherwise } \end{array} \right.$$
  3. Using this model, find \(\mathrm { P } ( X > 0.4 )\) A random sample of 100 poles cut by this new machine is taken.
  4. Using a suitable approximation, find the probability that at least 8 of these poles have an error of more than 0.4 metres.
Edexcel S2 2015 June Q5
  1. Liftsforall claims that the lift they maintain in a block of flats breaks down at random at a mean rate of 4 times per month. To test this, the number of times the lift breaks down in a month is recorded.
    1. Using a 5\% level of significance, find the critical region for a two-tailed test of the null hypothesis that 'the mean rate at which the lift breaks down is 4 times per month'. The probability of rejection in each of the tails should be as close to \(2.5 \%\) as possible.
    Over a randomly selected 1 month period the lift broke down 3 times.
  2. Test, at the \(5 \%\) level of significance, whether Liftsforall's claim is correct. State your hypotheses clearly.
  3. State the actual significance level of this test.
    ! The residents in the block of flats have a maintenance contract with Liftsforall. The residents pay Liftsforall \(\pounds 500\) for every quarter ( 3 months) in which there are at most 3 breakdowns. If there are 4 or more breakdowns in a quarter then the residents do not pay for that quarter. Liftsforall installs a new lift in the block of flats.
    Given that the new lift breaks down at a mean rate of 2 times per month,
  4. find the probability that the residents do not pay more than \(\pounds 500\) to Liftsforall in the next year.
Edexcel S2 2015 June Q6
  1. A continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) where
$$f ( x ) = \left\{ \begin{array} { c c } k x ^ { n } & 0 \leqslant x \leqslant 1
0 & \text { otherwise } \end{array} \right.$$ where \(k\) and \(n\) are positive integers.
  1. Find \(k\) in terms of \(n\).
  2. Find \(\mathrm { E } ( X )\) in terms of \(n\).
  3. Find \(\mathrm { E } \left( X ^ { 2 } \right)\) in terms of \(n\). Given that \(n = 2\)
  4. find \(\operatorname { Var } ( 3 X )\).
Edexcel S2 2015 June Q7
  1. A bag contains a large number of \(10 \mathrm { p } , 20 \mathrm { p }\) and 50 p coins in the ratio \(1 : 2 : 2\)
A random sample of 3 coins is taken from the bag.
Find the sampling distribution of the median of these samples.
Edexcel S2 2016 June Q1
  1. A student is investigating the numbers of cherries in a Rays fruit cake. A random sample of Rays fruit cakes is taken and the results are shown in the table below.
Number of cherries012345\(\geqslant 6\)
Frequency24372112420
  1. Calculate the mean and the variance of these data.
  2. Explain why the results in part (a) suggest that a Poisson distribution may be a suitable model for the number of cherries in a Rays fruit cake. The number of cherries in a Rays fruit cake follows a Poisson distribution with mean 1.5 A Rays fruit cake is to be selected at random. Find the probability that it contains
    1. exactly 2 cherries,
    2. at least 1 cherry. Rays fruit cakes are sold in packets of 5
  4. Show that the probability that there are more than 10 cherries, in total, in a randomly selected packet of Rays fruit cakes, is 0.1378 correct to 4 decimal places. Twelve packets of Rays fruit cakes are selected at random.
  5. Find the probability that exactly 3 packets contain more than 10 cherries. \href{http://PhysicsAndMathsTutor.com}{PhysicsAndMathsTutor.com}
Edexcel S2 2016 June Q2
  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
  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
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
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
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
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
  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
  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
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
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
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
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
  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
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
  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
  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.