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Edexcel S2 2013 June Q3
13 marks Standard +0.3
  1. An online shop sells a computer game at an average rate of 1 per day.
    1. Find the probability that the shop sells more than 10 games in a 7 day period.
    Once every 7 days the shop has games delivered before it opens.
  2. Find the least number of games the shop should have in stock immediately after a delivery so that the probability of running out of the game before the next delivery is less than 0.05 In an attempt to increase sales of the computer game, the price is reduced for six months. A random sample of 28 days is taken from these six months. In the sample of 28 days, 36 computer games are sold.
  3. Using a suitable approximation and a \(5 \%\) level of significance, test whether or not the average rate of sales per day has increased during these six months. State your hypotheses clearly.
Edexcel S2 2013 June Q4
9 marks Moderate -0.3
  1. A continuous random variable \(X\) is uniformly distributed over the interval [ \(b , 4 b\) ] where \(b\) is a constant.
    1. Write down \(\mathrm { E } ( X )\).
    2. Use integration to show that \(\operatorname { Var } ( X ) = \frac { 3 b ^ { 2 } } { 4 }\).
    3. Find \(\operatorname { Var } ( 3 - 2 X )\).
    Given that \(b = 1\) find
  2. the cumulative distribution function of \(X , \mathrm {~F} ( x )\), for all values of \(x\),
  3. the median of \(X\).
Edexcel S2 2013 June Q5
12 marks Standard +0.3
  1. The continuous random variable \(X\) has a cumulative distribution function
$$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < 1 \\ \frac { x ^ { 3 } } { 10 } + \frac { 3 x ^ { 2 } } { 10 } + a x + b & 1 \leqslant x \leqslant 2 \\ 1 & x > 2 \end{array} \right.$$ where \(a\) and \(b\) are constants.
  1. Find the value of \(a\) and the value of \(b\).
  2. Show that \(\mathrm { f } ( x ) = \frac { 3 } { 10 } \left( x ^ { 2 } + 2 x - 2 \right) , \quad 1 \leqslant x \leqslant 2\)
  3. Use integration to find \(\mathrm { E } ( X )\).
  4. Show that the lower quartile of \(X\) lies between 1.425 and 1.435
Edexcel S2 2013 June Q6
10 marks Standard +0.3
6. In a manufacturing process \(25 \%\) of articles are thought to be defective. Articles are produced in batches of 20
  1. A batch is selected at random. Using a \(5 \%\) significance level, find the critical region for a two tailed test that the probability of an article chosen at random being defective is 0.25
    You should state the probability in each tail which should be as close as possible to 0.025 The manufacturer changes the production process to try to reduce the number of defective articles. She then chooses a batch at random and discovers there are 3 defective articles.
  2. Test at the \(5 \%\) level of significance whether or not there is evidence that the changes to the process have reduced the percentage of defective articles. State your hypotheses clearly.
Edexcel S2 2013 June Q7
10 marks Moderate -0.3
  1. A telesales operator is selling a magazine. Each day he chooses a number of people to telephone. The probability that each person he telephones buys the magazine is 0.1
    1. Suggest a suitable distribution to model the number of people who buy the magazine from the telesales operator each day.
    2. On Monday, the telesales operator telephones 10 people. Find the probability that he sells at least 4 magazines.
    3. Calculate the least number of people he needs to telephone on Tuesday, so that the probability of selling at least 1 magazine, on that day, is greater than 0.95
    A call centre also sells the magazine. The probability that a telephone call made by the call centre sells a magazine is 0.05 The call centre telephones 100 people every hour.
  2. Using a suitable approximation, find the probability that more than 10 people telephoned by the call centre buy a magazine in a randomly chosen hour.
Edexcel S2 2014 June Q1
5 marks Moderate -0.3
  1. Before Roger will use a tennis ball he checks it using a "bounce" test. The probability that a ball from Roger's usual supplier fails the bounce test is 0.2 . A new supplier claims that the probability of one of their balls failing the bounce test is less than 0.2 . Roger checks a random sample of 40 balls from the new supplier and finds that 3 balls fail the bounce test.
Stating your hypotheses clearly, use a \(5 \%\) level of significance to test the new supplier's claim.
Edexcel S2 2014 June Q2
7 marks Moderate -0.8
2. A bag contains a large number of counters. Each counter has a single digit number on it and the mean of all the numbers in the bag is the unknown parameter \(\mu\). The number 2 is on \(40 \%\) of the counters and the number 5 is on \(25 \%\) of the counters. All the remaining counters have numbers greater than 5 on them. A random sample of 10 counters is taken from the bag.
  1. State whether or not each of the following is a statistic
    1. \(S =\) the sum of the numbers on the counters in the sample,
    2. \(D =\) the difference between the highest number in the sample and \(\mu\),
    3. \(F =\) the number of counters in the sample with a number 5 on them. The random variable \(T\) represents the number of counters in a random sample of 10 with the number 2 on them.
  2. Specify the sampling distribution of \(T\). The counters are selected one by one.
  3. Find the probability that the third counter selected is the first counter with the number 2 on it.
Edexcel S2 2014 June Q3
10 marks Standard +0.3
3. Accidents occur randomly at a road junction at a rate of 18 every year. The random variable \(X\) represents the number of accidents at this road junction in the next 6 months.
  1. Write down the distribution of \(X\).
  2. Find \(\mathrm { P } ( X > 7 )\).
  3. Show that the probability of at least one accident in a randomly selected month is 0.777 (correct to 3 decimal places).
  4. Find the probability that there is at least one accident in exactly 4 of the next 6 months.
Edexcel S2 2014 June Q4
14 marks Standard +0.3
4. The random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by $$\mathrm { f } ( x ) = \left\{ \begin{array} { c c } 3 k & 0 \leqslant x < 1 \\ k x ( 4 - x ) & 1 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{array} \right.$$ where \(k\) is a constant.
  1. Sketch f (x).
  2. Write down the mode of \(X\). Given that \(\mathrm { E } ( X ) = \frac { 29 } { 16 }\)
  3. describe, giving a reason, the skewness of the distribution.
  4. Use integration to find the value of \(k\).
  5. Write down the lower quartile of \(X\). Given also that \(\mathrm { P } ( 2 < X < 3 ) = \frac { 11 } { 36 }\)
  6. find the exact value of \(\mathrm { P } ( X > 3 )\).
Edexcel S2 2014 June Q5
13 marks Standard +0.3
  1. Sammy manufactures wallpaper. She knows that defects occur randomly in the manufacturing process at a rate of 1 every 8 metres. Once a week the machinery is cleaned and reset. Sammy then takes a random sample of 40 metres of wallpaper from the next batch produced to test if there has been any change in the rate of defects.
    1. Stating your hypotheses clearly and using a \(10 \%\) level of significance, find the critical region for this test. You should choose your critical region so that the probability of rejection is less than 0.05 in each tail.
    2. State the actual significance level of this test.
    Thomas claims that his new machine would reduce the rate of defects and invites Sammy to test it. Sammy takes a random sample of 200 metres of wallpaper produced on Thomas' machine and finds 19 defects.
  2. Using a suitable approximation, test Thomas' claim. You should use a \(5 \%\) level of significance and state your hypotheses clearly.
Edexcel S2 2014 June Q6
12 marks Standard +0.3
6. In an experiment some children were asked to estimate the position of the centre of a circle. The random variable \(D\) represents the distance, in centimetres, between the child's estimate and the actual position of the centre of the circle. The cumulative distribution function of \(D\) is given by $$\mathrm { F } ( d ) = \left\{ \begin{array} { c c } 0 & d < 0 \\ \frac { d ^ { 2 } } { 2 } - \frac { d ^ { 4 } } { 16 } & 0 \leqslant d \leqslant 2 \\ 1 & d > 2 \end{array} \right.$$
  1. Find the median of \(D\).
  2. Find the mode of \(D\). Justify your answer. The experiment is conducted on 80 children.
  3. Find the expected number of children whose estimate is less than 1 cm from the actual centre of the circle.
Edexcel S2 2014 June Q7
14 marks Standard +0.3
7. A piece of string \(A B\) has length 9 cm . The string is cut at random at a point \(P\) and the random variable \(X\) represents the length of the piece of string \(A P\).
  1. Write down the distribution of \(X\).
  2. Find the probability that the length of the piece of string \(A P\) is more than 6 cm . The two pieces of string \(A P\) and \(P B\) are used to form two sides of a rectangle. The random variable \(R\) represents the area of the rectangle.
  3. Show that \(R = a X ^ { 2 } + b X\) and state the values of the constants \(a\) and \(b\).
  4. Find \(\mathrm { E } ( R )\).
  5. Find the probability that \(R\) is more than twice the area of a square whose side has the length of the piece of string \(A P\).
Edexcel S2 2014 June Q1
8 marks Moderate -0.8
  1. Patients arrive at a hospital accident and emergency department at random at a rate of 6 per hour.
    1. Find the probability that, during any 90 minute period, the number of patients arriving at the hospital accident and emergency department is
      1. exactly 7
      2. at least 10
    A patient arrives at 11.30 a.m.
  2. Find the probability that the next patient arrives before 11.45 a.m.
Edexcel S2 2014 June Q2
14 marks Moderate -0.3
2. The length of time, in minutes, that a customer queues in a Post Office is a random variable, \(T\), with probability density function $$\mathrm { f } ( t ) = \left\{ \begin{array} { c c } c \left( 81 - t ^ { 2 } \right) & 0 \leqslant t \leqslant 9 \\ 0 & \text { otherwise } \end{array} \right.$$ where \(c\) is a constant.
  1. Show that the value of \(c\) is \(\frac { 1 } { 486 }\)
  2. Show that the cumulative distribution function \(\mathrm { F } ( t )\) is given by $$\mathrm { F } ( t ) = \left\{ \begin{array} { c c } 0 & t < 0 \\ \frac { t } { 6 } - \frac { t ^ { 3 } } { 1458 } & 0 \leqslant t \leqslant 9 \\ 1 & t > 9 \end{array} \right.$$
  3. Find the probability that a customer will queue for longer than 3 minutes. A customer has been queueing for 3 minutes.
  4. Find the probability that this customer will be queueing for at least 7 minutes. Three customers are selected at random.
  5. Find the probability that exactly 2 of them had to queue for longer than 3 minutes.
Edexcel S2 2014 June Q3
13 marks Standard +0.3
  1. A company claims that it receives emails at a mean rate of 2 every 5 minutes.
    1. Give two reasons why a Poisson distribution could be a suitable model for the number of emails received.
    2. Using a \(5 \%\) level of significance, find the critical region for a two-tailed test of the hypothesis that the mean number of emails received in a 10 minute period is 4 . The probability of rejection in each tail should be as close as possible to 0.025
    3. Find the actual level of significance of this test.
    To test this claim, the number of emails received in a random 10 minute period was recorded. During this period 8 emails were received.
  2. Comment on the company's claim in the light of this value. Justify your answer. During a randomly selected 15 minutes of play in the Wimbledon Men's Tennis Tournament final, 2 emails were received by the company.
  3. Test, at the \(10 \%\) level of significance, whether or not the mean rate of emails received by the company during the Wimbledon Men’s Tennis Tournament final is lower than the mean rate received at other times. State your hypotheses clearly.
Edexcel S2 2014 June Q4
14 marks Standard +0.3
  1. A cadet fires shots at a target at distances ranging from 25 m to 90 m . The probability of hitting the target with a single shot is \(p\). When firing from a distance \(d \mathrm {~m} , p = \frac { 3 } { 200 } ( 90 - d )\). Each shot is fired independently.
The cadet fires 10 shots from a distance of 40 m .
    1. Find the probability that exactly 6 shots hit the target.
    2. Find the probability that at least 8 shots hit the target. The cadet fires 20 shots from a distance of \(x \mathrm {~m}\).
  2. Find, to the nearest integer, the value of \(x\) if the cadet has an \(80 \%\) chance of hitting the target at least once. The cadet fires 100 shots from 25 m .
  3. Using a suitable approximation, estimate the probability that at least 95 of these shots hit the target.
Edexcel S2 2014 June Q5
11 marks Moderate -0.3
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
15 marks Standard +0.3
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. \includegraphics[max width=\textwidth, alt={}, center]{caaa0133-5b13-4ca7-8e65-8543327c33fd-12_104_61_2412_1884}
Edexcel S2 2015 June Q1
11 marks Moderate -0.3
  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
14 marks Standard +0.3
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
12 marks Standard +0.3
  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
12 marks Standard +0.3
  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
11 marks Moderate -0.3
  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
7 marks Standard +0.8
  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
14 marks Standard +0.3
  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}