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Edexcel S2 2017 June Q3
12 marks Standard +0.3
3. The random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} a x + b & 1 \leqslant x < 4 \\ \frac { 3 } { 2 } - \frac { 1 } { 4 } x & 4 \leqslant x \leqslant 6 \\ 0 & \text { otherwise } \end{cases}$$ as shown in Figure 1, where \(a\) and \(b\) are constants. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1a1534ea-4c62-4945-850a-9460ea87fd64-08_634_1132_694_397} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure}
  1. Show that the median of \(X\) is 4
  2. Find the value of \(a\) and the value of \(b\)
  3. Specify fully the cumulative distribution function of \(X\)
Edexcel S2 2017 June Q4
13 marks Challenging +1.2
4. In a large population, past records show that 1 in 200 adults has a particular allergy. In a random sample of 700 adults selected from the population, estimate
    1. the mean number of adults with the allergy,
    2. the standard deviation of the number of adults with the allergy. Give your answer to 3 decimal places. A doctor claims that the past records are out of date and the proportion of adults with the allergy is higher than the records indicate. A random sample of 500 adults is taken from the population and 5 are found to have the allergy. A test of the doctor's claim is to be carried out at the 5\% level of significance.
    1. State the hypotheses for this test.
    2. Using a suitable approximation, carry out the test.
      (6) It is also claimed that \(30 \%\) of those with the allergy take medication for it daily. To test this claim, a random sample of \(n\) people with the allergy is taken. The random variable \(Y\) represents the number of people in the sample who take medication for the allergy daily. A two-tailed test, at the \(1 \%\) level of significance, is carried out to see if the proportion differs from 30\% The critical region for the test is \(Y = 0\) or \(Y \geqslant w\)
  3. Find the smallest possible value of \(n\) and the corresponding value of \(w\)
Edexcel S2 2017 June Q5
11 marks Standard +0.3
5. A call centre records the length of time, \(T\) minutes, its customers wait before being connected to an agent. The random variable \(T\) has a cumulative distribution function given by $$\mathrm { F } ( t ) = \left\{ \begin{array} { l r } 0 & t < 0 \\ 0.3 t - 0.004 t ^ { 3 } & 0 \leqslant t \leqslant 5 \\ 1 & t > 5 \end{array} \right.$$
  1. Find the proportion of customers waiting more than 4 minutes to be connected to an agent.
  2. Given that a customer waits more than 2 minutes to be connected to an agent, find the probability that the customer waits more than 4 minutes.
  3. Show that the upper quartile lies between 2.7 and 2.8 minutes.
  4. Find the mean length of time a customer waits to be connected to an agent.
Edexcel S2 2017 June Q6
7 marks Standard +0.8
6. At a men's tennis tournament there are 3 , 4 or 5 sets in a match. Over many years, data collected show that 50\% of matches last for exactly 3 sets, 30\% of matches last for exactly 4 sets and 20\% of matches last for exactly 5 sets. A random sample of 3 tennis matches is taken. The number of sets in each match is recorded as \(S _ { 1 } , S _ { 2 }\) and \(S _ { 3 }\) respectively. The random variable \(M\) represents the maximum value of \(S _ { 1 } , S _ { 2 }\) and \(S _ { 3 }\)
  1. List all the samples where \(M \neq 5\)
  2. Find the sampling distribution of \(M\)
  3. Write down the mode of \(S _ { 1 }\) and the mode of \(M\)
Edexcel S2 2017 June Q7
9 marks Moderate -0.3
7. The continuous random variable \(X\) is uniformly distributed over the interval \([ a , b ]\)
  1. Find an expression, in terms of \(a\) and \(b\), for \(\mathrm { E } ( 3 - 2 X )\)
  2. Find \(\mathrm { P } \left( X > \frac { 1 } { 3 } b + \frac { 2 } { 3 } a \right)\) Given that \(\mathrm { E } ( X ) = 0\)
  3. find an expression, in terms of \(b\) only, for \(\mathrm { E } \left( 3 X ^ { 2 } \right)\) Given also that the range of \(X\) is 18
  4. find \(\operatorname { Var } ( X )\)
    \includegraphics[max width=\textwidth, alt={}]{1a1534ea-4c62-4945-850a-9460ea87fd64-24_2630_1828_121_121}
Edexcel S2 2018 June Q1
9 marks Moderate -0.8
  1. A salesman sells insurance to people. Each day he chooses a number of people to contact. The probability that the salesman sells insurance to a person he contacts is 0.05
On Monday he chooses to contact 10 people.
  1. Find the probability that on Monday the salesman sells insurance to
    1. exactly 1 person,
    2. at least 3 people.
  2. Find the number of people he should contact each day in order to sell insurance, on average, to 3 people per day.
  3. Calculate the least number of people he must choose to contact on Friday, so that the probability of selling insurance to at least 1 person on Friday exceeds 0.99
Edexcel S2 2018 June Q2
12 marks Standard +0.3
2. John weaves cloth by hand. Emma believes that faults are randomly distributed in John's cloth at a rate of more than 4 per 50 metres of cloth. To check her belief, Emma takes a random sample of 100 metres of the cloth and finds that it contains 14 faults.
  1. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, Emma's belief. Armani also weaves cloth by hand. He knows that faults are randomly distributed in his cloth at a rate of 4 per 50 metres of cloth. Emma decides to buy a large amount of Armani's cloth to sell in pieces of length \(l\) metres. She chooses \(l\) so that the probability of no faults in a piece is exactly 0.9
  2. Show that \(l = 1.3\) to 2 significant figures. Emma sells 5000 of these pieces of cloth of length 1.3 metres. She makes a profit of \(\pounds 2.50\) on each piece of cloth that does not contain any faults but a loss of \(\pounds 0.50\) on any piece that contains at least one fault.
  3. Find Emma's expected profit.
Edexcel S2 2018 June Q3
11 marks Moderate -0.8
3. A machine pours oil into bottles. It is electronically controlled to cut off the flow of oil randomly between 100 ml and \(k \mathrm { ml }\), where \(k > 100\). It is equally likely to cut off the flow at any point in this range. The random variable \(X\) is the volume of oil poured into a bottle. Given that \(\mathrm { P } ( 102 \leqslant X \leqslant k ) = \frac { 2 } { 3 }\)
  1. show that \(k = 106\)
  2. Find the probability that the volume of oil poured into a bottle is
    1. less than 105 ml ,
    2. exactly 105 ml .
  3. Write down the value of \(\mathrm { E } ( X )\)
  4. Find the 15th percentile of this distribution.
  5. Determine the value of \(x\) such that \(3 \mathrm { P } ( X \leqslant x - 1.5 ) = \mathrm { P } ( X \geqslant x + 1.5 )\)
Edexcel S2 2018 June Q4
6 marks Moderate -0.8
4. The volume of milk, \(M\) litres, in cartons produced by a dairy, has distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\), where \(\mu\) and \(\sigma\) are unknown. A random sample of 12 cartons is taken and the volume of milk in each carton is measured ( \(M _ { 1 } , M _ { 2 } , \ldots , M _ { 12 }\) ). A statistic \(X\) is based on this sample.
  1. Explain what is meant by "a random sample" in this case.
  2. State the population in this case.
  3. Write down the distribution of \(\frac { M _ { 12 } - \mu } { \sigma }\)
  4. Explain what you understand by the sampling distribution of \(X\).
  5. State, giving a reason, which of the following is not a statistic based on this sample.
    (I) \(3 M _ { 1 } + \frac { 2 M _ { 11 } } { 6 }\) (II) \(\sum _ { i = 1 } ^ { 12 } \left( \frac { M _ { i } - \mu } { \sigma } \right) ^ { 2 }\) (III) \(\sum _ { i = 1 } ^ { 12 } \left( 2 M _ { i } - 3 \right)\)
Edexcel S2 2018 June Q5
13 marks Standard +0.8
5. Cars stop at a service station randomly at a rate of 3 every 5 minutes.
  1. Calculate the probability that in a randomly selected 10 minute period,
    1. exactly 7 cars will stop at the service station,
    2. more than 7 cars will stop at the service station. Using a normal approximation, the probability that more than 40 cars will stop at the service station during a randomly selected \(n\) minute period is 0.2266 correct to 4 significant figures.
  2. Find the value of \(n\).
Edexcel S2 2018 June Q6
14 marks Standard +0.3
6. A random variable \(X\) has probability density function given by $$f ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { 4 } & 0 \leqslant x < 1 \\ \frac { x ^ { 3 } } { 5 } & 1 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{array} \right.$$
  1. Use algebraic integration to find \(\mathrm { E } ( X )\)
  2. Use algebraic integration to find \(\operatorname { Var } ( X )\)
  3. Define the cumulative distribution function \(\mathrm { F } ( x )\) for all values of \(x\).
  4. Find the median of \(X\), giving your answer to 3 significant figures.
  5. Comment on the skewness of the distribution, justifying your answer.
Edexcel S2 2018 June Q7
10 marks Standard +0.3
7. A manufacturer produces packets of sweets. Each packet contains 25 sweets. The manufacturer claims that, on average, 40\% of the sweets in each packet are red. A packet is selected at random.
  1. Using a \(1 \%\) level of significance, find the critical region for a two-tailed test that the proportion of red sweets is 0.40 You should state the probability in each tail, which should be as close as possible to 0.005
  2. Find the actual significance level of this test. The manufacturer changes the production process to try to reduce the number of red sweets. She chooses 2 packets at random and finds that 8 of the sweets are red.
  3. Test, at the \(1 \%\) level of significance, whether or not there is evidence that the manufacturer's changes to the production process have been successful. State your hypotheses clearly.
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Edexcel S2 2021 June Q1
14 marks Standard +0.3
  1. Spany sells seeds and claims that \(5 \%\) of its pansy seeds do not germinate. A packet of pansy seeds contains 20 seeds. Each seed germinates independently of the other seeds.
    1. Find the probability that in a packet of Spany's pansy seeds
      1. more than 2 but fewer than 5 seeds do not germinate,
      2. more than 18 seeds germinate.
    Jem buys 5 packets of Spany’s pansy seeds.
  2. Calculate the probability that all of these packets contain more than 18 seeds that germinate. Jem believes that Spany's claim is incorrect. She believes that the percentage of pansy seeds that do not germinate is greater than 5\%
  3. Write down the hypotheses for a suitable test to examine Jem's belief. Jem planted all of the 100 seeds she bought from Spany and found that 8 did not germinate.
  4. Using a suitable approximation, carry out the test using a \(5 \%\) level of significance.
Edexcel S2 2021 June Q2
15 marks Standard +0.3
  1. Luis makes and sells rugs. He knows that faults occur randomly in his rugs at a rate of 3 every \(4 \mathrm {~m} ^ { 2 }\)
    1. Find the probability of there being exactly 5 faults in one of his rugs that is \(4 \mathrm {~m} ^ { 2 }\) in size.
    2. Find the probability that there are more than 5 faults in one of his rugs that is \(6 \mathrm {~m} ^ { 2 }\) in size.
    Luis makes a rug that is \(4 \mathrm {~m} ^ { 2 }\) in size and finds it has exactly 5 faults in it.
  2. Write down the probability that the next rug that Luis makes, which is \(4 \mathrm {~m} ^ { 2 }\) in size, will have exactly 5 faults. Give a reason for your answer. A small rug has dimensions 80 cm by 150 cm . Faults still occur randomly at a rate of 3 every \(4 \mathrm {~m} ^ { 2 }\) Luis makes a profit of \(\pounds 80\) on each small rug he sells that contains no faults but a profit of \(\pounds 60\) on any small rug he sells that contains faults. Luis sells \(n\) small rugs and expects to make a profit of at least \(\pounds 4000\)
  3. Calculate the minimum value of \(n\) Luis wishes to increase the productivity of his business and employs Rhiannon. Faults also occur randomly in Rhiannon's rugs and independently to faults made by Luis. Luis randomly selects 10 small rugs made by Rhiannon and finds 13 faults.
  4. Test, at the \(5 \%\) level of significance, whether or not there is evidence to support the suggestion that the rate at which faults occur is higher for Rhiannon than for Luis. State your hypotheses clearly.
Edexcel S2 2021 June Q3
15 marks Standard +0.3
  1. The continuous random variable \(Y\) has the following probability density function
$$f ( y ) = \begin{cases} \frac { 6 } { 25 } ( y - 1 ) & 1 \leqslant y < 2 \\ \frac { 3 } { 50 } \left( 4 y ^ { 2 } - y ^ { 3 } \right) & 2 \leqslant y < 4 \\ 0 & \text { otherwise } \end{cases}$$
  1. Sketch f(y)
  2. Find the mode of \(Y\)
  3. Use algebraic integration to calculate \(\mathrm { E } \left( Y ^ { 2 } \right)\) Given that \(\mathrm { E } ( Y ) = 2.696\)
  4. find \(\operatorname { Var } ( Y )\)
  5. Find the value of \(y\) for which \(\mathrm { P } ( Y \geqslant y ) = 0.9\) Give your answer to 3 significant figures.
Edexcel S2 2021 June Q4
10 marks Standard +0.8
  1. A bag contains a large number of balls, each with one of the numbers 1,2 or 5 written on it in the ratio \(2 : 3 : 4\) respectively.
A random sample of 3 balls is taken from the bag.
The random variable \(B\) represents the range of the numbers written on the balls in the sample.
  1. Find \(\mathrm { P } ( B = 4 )\)
  2. Find the sampling distribution of \(B\).
Edexcel S2 2021 June Q5
11 marks Standard +0.3
  1. A game uses two turntables, one red and one yellow. Each turntable has a point marked on the circumference that is lined up with an arrow at the start of the game. Jim spins both turntables and measures the distance, in metres, each point is from the arrow, around the circumference in an anticlockwise direction when the turntables stop spinning.
The continuous random variable \(Y\) represents the distance, in metres, the point is from the arrow for the yellow turntable. The cumulative distribution function of \(Y\) is given by \(\mathrm { F } ( y )\) where $$F ( y ) = \left\{ \begin{array} { c r } 0 & y < 0 \\ 1 - \left( \alpha + \beta y ^ { 2 } \right) & 0 \leqslant y \leqslant 5 \\ 1 & y > 5 \end{array} \right.$$
  1. Explain why (i) \(\alpha = 1\) $$\text { (ii) } \beta = - \frac { 1 } { 25 }$$
  2. Find the probability density function of \(Y\) The continuous random variable \(R\) represents the distance, in metres, the point is from the arrow for the red turntable. The distribution of \(R\) is modelled by a continuous uniform distribution over the interval \([ d , 3 d ]\) Given that \(\mathrm { P } \left( R > \frac { 11 } { 5 } \right) = \mathrm { P } \left( Y > \frac { 5 } { 3 } \right)\)
  3. find the value of \(d\) In the game each turntable is spun 3 times. The distance between the point and the arrow is determined for each spin. To win a prize, at least 5 of the distances the point is from the arrow when a turntable is spun must be less than \(\frac { 11 } { 5 } \mathrm {~m}\) Jo plays the game once.
  4. Calculate the probability of Jo winning a prize.
Edexcel S2 2021 June Q6
10 marks Standard +0.8
  1. The random variable \(Y \sim \mathrm {~B} ( 225 , p )\)
Using a normal approximation, the probability that \(Y\) is at least 188 is 0.1056 to 4 decimal places.
  1. Show that \(p\) satisfies \(145 p ^ { 2 } - 241 p + 100 = 0\) when the normal probability tables are used.
  2. Hence find the value of \(p\), justifying your answer.
Edexcel S2 2022 June Q1
10 marks Moderate -0.8
  1. The independent random variables \(W\) and \(X\) have the following distributions.
$$W \sim \operatorname { Po } ( 4 ) \quad X \sim \mathrm {~B} ( 3,0.8 )$$
  1. Write down the value of the variance of \(W\)
  2. Determine the mode of \(X\) Show your working clearly. One observation from each distribution is recorded as \(W _ { 1 }\) and \(X _ { 1 }\) respectively.
  3. Find \(\mathrm { P } \left( W _ { 1 } = 2 \right.\) and \(\left. X _ { 1 } = 2 \right)\)
  4. Find \(\mathrm { P } \left( X _ { 1 } < W _ { 1 } \right)\)
Edexcel S2 2022 June Q2
11 marks Standard +0.3
  1. The time, in minutes, spent waiting for a call to a call centre to be answered is modelled by the random variable \(T\) with probability density function
$$f ( t ) = \left\{ \begin{array} { l c } \frac { 1 } { 192 } \left( t ^ { 3 } - 48 t + 128 \right) & 0 \leqslant t \leqslant 4 \\ 0 & \text { otherwise } \end{array} \right.$$
  1. Use algebraic integration to find, in minutes and seconds, the mean waiting time.
  2. Show that \(\mathrm { P } ( 1 < T < 3 ) = \frac { 7 } { 16 }\) A supervisor randomly selects 256 calls to the call centre.
  3. Use a suitable approximation to find the probability that more than 125 of these calls take between 1 and 3 minutes to be answered.
Edexcel S2 2022 June Q3
10 marks Moderate -0.8
  1. A point is to be randomly plotted on the \(x\)-axis, where the units are measured in cm .
The random variable \(R\) represents the \(x\) coordinate of the point on the \(x\)-axis and \(R\) is uniformly distributed over the interval [-5,19] A negative value indicates that the point is to the left of the origin and a positive value indicates that the point is to the right of the origin.
  1. Find the exact probability that the point is plotted to the right of the origin.
  2. Find the exact probability that the point is plotted more than 3.5 cm away from the origin.
  3. Sketch the cumulative distribution function of \(R\) Three independent points with \(x\) coordinates \(R _ { 1 } , R _ { 2 }\) and \(R _ { 3 }\) are plotted on the \(x\)-axis.
  4. Find the exact probability that
    1. all three points are more than 10 cm from the origin
    2. the point furthest from the origin is more than 10 cm from the origin.
Edexcel S2 2022 June Q4
9 marks Standard +0.3
  1. Past evidence shows that \(7 \%\) of pears grown by a farmer are unfit for sale.
This season it is believed that the proportion of pears that are unfit for sale has decreased. To test this belief a random sample of \(n\) pears is taken. The random variable \(Y\) represents the number of pears in the sample that are unfit for sale.
  1. Find the smallest value of \(n\) such that \(Y = 0\) lies in the critical region for this test at a \(5 \%\) level of significance. In the past, \(8 \%\) of the pears grown by the farmer weigh more than 180 g . This season the farmer believes the proportion of pears weighing more than 180 g has changed. She takes a random sample of 75 pears and finds that 11 of them weigh more than 180 g .
  2. Test, using a suitable approximation, whether there is evidence of a change in the proportion of pears weighing more than 180 g .
    You should use a \(5 \%\) level of significance and state your hypotheses clearly.
Edexcel S2 2022 June Q5
10 marks Challenging +1.2
  1. The number of particles per millilitre in a solution is modelled by a Poisson distribution with mean 0.15
A randomly selected 50 millilitre sample of the solution is taken.
  1. Find the probability that
    1. exactly 10 particles are found,
    2. between 6 and 11 particles (inclusive) are found. Petra takes 12 independent samples of \(m\) millilitres of the solution.
      The probability that at least 2 of these samples contain no particles is 0.1184
  2. Using the Statistical Tables provided, find the value of \(m\)
Edexcel S2 2022 June Q6
13 marks Standard +0.3
  1. The continuous random variable \(X\) has probability density function
$$f ( x ) = \begin{cases} 0.1 x & 0 \leqslant x < 2 \\ k x ( 8 - x ) & 2 \leqslant x < 4 \\ a & 4 \leqslant x < 6 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) and \(a\) are constants.
It is known that \(\mathrm { P } ( X < 4 ) = \frac { 31 } { 45 }\)
  1. Find the exact value of \(k\)
    1. Find the exact value of \(a\)
    2. Find the exact value of \(\mathrm { P } ( 0 \leqslant X \leqslant 5.5 )\)
  3. Specify fully the cumulative distribution function of \(X\)
Edexcel S2 2022 June Q7
12 marks Standard +0.8
  1. A bag contains 10 counters each with exactly one number written on it.
There are 6 counters with the number 7 on them
There are 3 counters with the number 8 on them
There is 1 counter with the number 9 on it
A random sample of 3 counters is taken from the bag (without replacement).
These counters are then put back in the bag.
This process is then repeated until 20 samples have been taken.
The random variable \(Y\) represents the number of these 20 samples that contain the counter with the number 9 on it.
    1. Find the mean of \(Y\)
    2. Find the variance of \(Y\) A random sample of 3 counters is chosen from the bag (without replacement).
  2. List all possible samples where the median of the numbers on the 3 counters is 7
  3. Find the sampling distribution of the median of the numbers on the 3 counters.