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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.
Edexcel S2 2023 June Q1
11 marks Moderate -0.3
  1. In a large population \(40 \%\) of adults use online banking.
A random sample of 50 adults is taken.
The random variable \(X\) represents the number of adults in the sample that use online banking.
  1. Find
    1. \(\mathrm { P } ( X = 26 )\)
    2. \(\mathrm { P } ( X \geqslant 26 )\)
    3. the smallest value of \(k\) such that \(\mathrm { P } ( X \leqslant k ) > 0.4\) A random sample of 600 adults is taken.
    1. Find, using a normal approximation, the probability that no more than 222 of these 600 adults use online banking.
    2. Explain why a normal approximation is suitable in part (b)(i)
Edexcel S2 2023 June Q2
4 marks Easy -1.8
  1. (a) State one characteristic of a population that would make a census a practical alternative to sampling.
A leisure centre has 2500 members.
It asks a sample of 300 members for their opinions on the fees it charges for using the centre. For the sample,
(b) (i) identify a suitable sampling frame,
(ii) identify a sampling unit. The leisure centre has the following pieces of information.
\(A\) is the list of the different types of membership that can be paid for by members.
\(B\) is the mean of the membership fees paid by all 2500 members.
\(C\) is the number in the sample of 300 members who are satisfied with the fees they pay.
(c) State the piece of information that is a statistic. Give a reason for your answer.
Edexcel S2 2023 June Q3
9 marks Standard +0.3
  1. The continuous random variable \(X\) has probability density function given by
$$f ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { 48 } \left( x ^ { 2 } - 8 x + c \right) & 2 \leqslant x \leqslant 5 \\ 0 & \text { otherwise } \end{array} \right.$$
  1. Show that \(c = 31\)
  2. Find \(\mathrm { P } ( 2 < X < 3 )\)
  3. State whether the lower quartile of \(X\) is less than 3, equal to 3 or greater than 3 Give a reason for your answer. Kei does the following to work out the mode of \(X\) $$\begin{aligned} f ^ { \prime } ( x ) & = \frac { 1 } { 48 } ( 2 x - 8 ) \\ 0 & = \frac { 1 } { 48 } ( 2 x - 8 ) \\ x & = 4 \end{aligned}$$ Hence the mode of \(X\) is 4 Kei's answer for the mode is incorrect.
  4. Explain why Kei's method does not give the correct value for the mode.
  5. Find the mode of \(X\) Give a reason for your answer.
Edexcel S2 2023 June Q4
13 marks Moderate -0.8
  1. (a) Given \(n\) is large, state a condition for which the binomial distribution \(\mathrm { B } ( n , p )\) can be reasonably approximated by a Poisson distribution.
A manufacturer produces candles. Those candles that pass a quality inspection are suitable for sale. It is known that 2\% of the candles produced by the manufacturer are not suitable for sale. A random sample of 125 candles produced by the manufacturer is taken.
(b) Use a suitable approximation to find the probability that no more than 6 of the candles are not suitable for sale. The manufacturer also produces candle holders.
Charlie believes that 5\% of candle holders produced by the factory have minor defects.
The manufacturer claims that the true proportion is less than \(5 \%\)
To test the manufacturer's claim, a random sample of 30 candle holders is taken and none of them are found to contain minor defects.
(c) (i) Carry out a test of the manufacturer's claim using a \(5 \%\) level of significance. You should state your hypotheses clearly.
(ii) Give a reason why this is not an appropriate test. Ashley suggests changing the sample size to 50
(d) Comment on whether or not this change would make the test appropriate. Give a reason for your answer.
Edexcel S2 2023 June Q5
14 marks Standard +0.8
  1. A continuous random variable \(Y\) has cumulative distribution function given by
$$\mathrm { F } ( y ) = \left\{ \begin{array} { c r } 0 & y < 3 \\ \frac { 1 } { 16 } \left( y ^ { 2 } - 6 y + a \right) & 3 \leqslant y \leqslant 5 \\ \frac { 1 } { 12 } ( y + b ) & 5 < y \leqslant 9 \\ \frac { 1 } { 12 } \left( 100 y - 5 y ^ { 2 } + c \right) & 9 < y \leqslant 10 \\ 1 & y > 10 \end{array} \right.$$ where \(a\), \(b\) and \(c\) are constants.
  1. Find the value of \(a\) and the value of \(c\)
  2. Find the value of \(b\)
  3. Find \(\mathrm { P } ( 6 < Y \leqslant 9 )\) Show your working clearly.
  4. Specify the probability density function, f(y), for \(5 < y \leqslant 9\) Using the information $$\int _ { 3 } ^ { 5 } ( 6 y - 5 ) f ( y ) d y + \int _ { 9 } ^ { 10 } ( 6 y - 5 ) f ( y ) d y = 26.5$$
  5. find \(\mathrm { E } ( 6 Y - 5 )\) You should make your method clear.
Edexcel S2 2023 June Q6
12 marks Standard +0.3
  1. Akia selects at random a value from the continuous random variable \(W\), which is uniformly distributed over the interval \([ a , b ]\)
The probability that Akia selects a value greater than 17 is \(\frac { 1 } { 5 }\) The probability that Akia selects a value less than \(k\) is \(\frac { 53 } { 60 }\)
  1. Find the probability that Akia selects a value between 17 and \(k\) It is known that \(\operatorname { Var } ( W ) = 75\)
    1. Find the value of \(a\) and the value of \(b\)
    2. Find the value of \(k\)
  3. Find \(\mathrm { P } ( - 5 < W < 5 )\)
  4. Find \(\mathrm { E } \left( W ^ { 2 } \right)\)
Edexcel S2 2023 June Q7
12 marks Challenging +1.2
  1. A bakery sells muffins individually at an average rate of 8 muffins per hour.
    1. Find the probability that, in a randomly selected one-hour period, the bakery sells at least 4 but not more than 8 muffins.
    A sample of 5 non-overlapping half-hour periods is selected at random.
  2. Find the probability that the bakery sells fewer than 3 muffins in exactly 2 of these periods. Given that 4 muffins were sold in a one-hour period,
  3. find the probability that more muffins were sold in the first 15 minutes than in the last 45 minutes.
Edexcel S2 2024 June Q1
13 marks Standard +0.3
1 A garage sells tyres. The number of customers arriving at the garage to buy tyres in a 10-minute period is modelled by a Poisson distribution with mean 2
  1. Find the probability that
    1. fewer than 4 customers arrive to buy tyres in the next 10 minutes,
    2. more than 5 customers arrive to buy tyres in the next 10 minutes. The manager randomly selects 20 non-overlapping, 30-minute periods.
  2. Find the probability that there are between 4 and 7 (inclusive) customers arriving to buy tyres in exactly 15 of these 30-minute periods. The manager believes that placing an advert in the local paper will lead to a significant increase in the number of customers arriving at the garage.
    A week after the advert is placed, the manager randomly selects a 25 -minute period and finds that 10 customers arrive at the garage to buy tyres.
  3. Test, at the \(5 \%\) level of significance, whether or not there is evidence to support the manager's belief.
    State your hypotheses clearly.
  4. Explain why the Poisson distribution is unlikely to be valid for the number of tyres sold during a 10-minute period.
Edexcel S2 2024 June Q2
9 marks Standard +0.3
2 The continuous random variable \(H\) has cumulative distribution function given by $$\mathrm { F } ( h ) = \left\{ \begin{array} { l r } 0 & h \leqslant 0 \\ \frac { h ^ { 2 } } { 48 } & 0 < h \leqslant 4 \\ \frac { h } { 6 } - \frac { 1 } { 3 } & 4 < h \leqslant 5 \\ \frac { 3 } { 10 } h - \frac { h ^ { 2 } } { 75 } - \frac { 2 } { 3 } & 5 < h \leqslant d \\ 1 & h > d \end{array} \right.$$ where \(d\) is a constant.
  1. Show that \(2 d ^ { 2 } - 45 d + 250 = 0\)
  2. Find \(\mathrm { P } ( H < 1.5 \mid 1 < H < 4.5 )\)
  3. Find the probability density function \(\mathrm { f } ( h )\) You may leave the limits of \(h\) in terms of \(d\) where necessary.
Edexcel S2 2024 June Q3
15 marks Moderate -0.8
3 Jian owns a large group of shops. She decides to visit a random sample of the shops to check if the stocktaking system is being used incorrectly.
  1. Suggest a suitable sampling frame for Jian to use.
  2. Identify the sampling units.
  3. Give one advantage and one disadvantage of taking a sample rather than a census. Jian believes that the stocktaking system is being used incorrectly in \(40 \%\) of the shops.
    To investigate her belief, a random sample of 30 of the shops is taken.
  4. Using a 5\% level of significance, find the critical region for a two-tailed test of Jian’s belief.
    You should state the probability in each tail, which should each be as close as possible to 2.5\% The total number of shops, in the sample of 30, where the stocktaking system is being used incorrectly is 20
  5. Using the critical region from part (d), state what this suggests about Jian’s belief. Give a reason for your answer. Jian introduces a new, simpler, stocktaking system to all the shops.
    She takes a random sample of 150 shops and finds that in 47 of these shops the new stocktaking system is being used incorrectly.
  6. Using a suitable approximation, test, at the \(5 \%\) level of significance, whether or not there is evidence that the proportion of shops where the stocktaking system is being used incorrectly is now less than 0.4 You should state your hypotheses and show your working clearly.
Edexcel S2 2024 June Q4
10 marks Standard +0.3
4 A bag contains 50 counters, each with one of the numbers 4,7 or 10 written on it in the ratio \(2 : 3 : 5\) respectively. A random sample of 2 counters is taken from the bag. The numbers on the 2 counters are recorded as \(D _ { 1 }\) and \(D _ { 2 }\) The random variable \(M\) represents the mean of \(D _ { 1 }\) and \(D _ { 2 }\)
  1. Show that \(\mathrm { P } ( M = 4 ) = \frac { 9 } { 245 }\)
  2. Find the sampling distribution of \(M\) A random sample of \(n\) sets of 2 counters is taken. The random variable \(T\) represents the number of these \(n\) sets of 2 counters that have a mean of 4 Given that each set of 2 counters is replaced after it is drawn,
  3. calculate the minimum value of \(n\) such that \(\mathrm { P } ( T = 0 ) < 0.15\)
Edexcel S2 2024 June Q5
12 marks Standard +0.3
5 A receptionist receives incoming telephone calls and should connect them to the appropriate department. The probability of them being connected to the wrong department on the first attempt is 0.05 A random sample of 8 calls is taken.
  1. Find the probability that at least 2 of these calls are connected to the wrong department on the first attempt. The receptionist receives 1000 calls each day.
  2. Use a Poisson approximation to find the probability that exactly 45 callers are connected to the wrong department on the first attempt in a day. The total time, \(T\) seconds, taken for a call to be answered by a department has a continuous uniform distribution over the interval [10,50]
  3. Find \(\mathrm { P } ( T > 16 )\) The number of calls the receptionist receives in a one-minute interval is modelled by a Poisson distribution with mean 6 The receptionist receives a call from Jia and tries to connect it to the right department.
  4. Find the probability that in the next 40 seconds Jia's call is answered by the right department on the first attempt and the receptionist has received no other calls.
Edexcel S2 2024 June Q6
16 marks Standard +0.3
6 In this question solutions relying entirely on calculator technology are not acceptable.
The continuous random variable \(X\) has the following probability density function $$f ( x ) = \begin{cases} a + b x & - 1 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are constants.
  1. Show that \(4 a + 4 b = 1\) Given that \(\mathrm { E } \left( X ^ { 2 } \right) = \frac { 17 } { 5 }\)
    1. find an equation in terms of \(a\) only
    2. hence show that \(b = 0.1\)
  3. Sketch the probability density function \(\mathrm { f } ( x )\) of \(X\)
  4. Find the value of \(k\) for which \(\mathrm { P } ( X \geqslant k ) = 0.8\)
Edexcel S2 2016 October Q1
10 marks Moderate -0.3
  1. A mobile phone company claims that each year \(5 \%\) of its customers have their mobile phone stolen. An insurance company claims this percentage is higher. A random sample of 30 of the mobile phone company's customers is taken and 4 of them have had their mobile phone stolen during the last year.
    1. Test the insurance company's claim at the \(10 \%\) level of significance. State your hypotheses clearly.
    A new random sample of 90 customers is taken. A test is carried out using these 90 customers, to see if the percentage of customers who have had a mobile phone stolen in the last year is more than 5\%
  2. Using a suitable approximation and a \(10 \%\) level of significance, find the critical region for this test.
Edexcel S2 2016 October Q2
14 marks Moderate -0.3
  1. The lifetime of a particular battery, \(T\) hours, is modelled using the cumulative distribution function
$$\mathrm { F } ( t ) = \left\{ \begin{array} { l r } 0 & t < 8 \\ \frac { 1 } { 96 } \left( 74 t - \frac { 5 } { 2 } t ^ { 2 } + k \right) & 8 \leqslant t \leqslant 12 \\ 1 & t > 12 \end{array} \right.$$
  1. Show that \(k = - 432\)
  2. Find the probability density function of \(T\), for all values of \(t\).
  3. Write down the mode of \(T\).
  4. Find the median of \(T\).
  5. Find the probability that a randomly selected battery has a lifetime of less than 9 hours. A battery is selected at random. Given that its lifetime is at least 9 hours,
  6. find the probability that its lifetime is no more than 11 hours.
Edexcel S2 2016 October Q3
10 marks Moderate -0.3
  1. A large number of students sat an examination. All of the students answered the first question. The first question was answered correctly by \(40 \%\) of the students.
In a random sample of 20 students who sat the examination, \(X\) denotes the number of students who answered the first question correctly.
  1. Write down the distribution of the random variable \(X\)
  2. Find \(\mathrm { P } ( 4 \leqslant X < 9 )\) Students gain 7 points if they answer the first question correctly and they lose 3 points if they do not answer it correctly.
  3. Find the probability that the total number of points scored on the first question by the 20 students is more than 0
  4. Calculate the variance of the total number of points scored on the first question by a random sample of 20 students.