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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.
Edexcel S2 2016 October Q4
10 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4ecee051-3a6f-4c12-8c53-926e8c3e241f-14_451_976_233_484} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A continuous random variable \(X\) has the probability density function \(\mathrm { f } ( x )\) shown in Figure 1 $$\mathrm { f } ( x ) = \begin{cases} m x & 0 \leqslant x \leqslant 5 \\ k & 5 < x \leqslant 10.5 \\ 0 & \text { otherwise } \end{cases}$$ where \(m\) and \(k\) are constants.
    1. Show that \(k = \frac { 1 } { 8 }\)
    2. Find the value of \(m\)
  2. Find \(\mathrm { E } ( X )\)
  3. Find the interquartile range of \(X\)
Edexcel S2 2016 October Q5
11 marks Standard +0.3
  1. A string of length 40 cm is cut into 2 pieces at a random point. The continuous random variable \(L\) represents the length of the longer piece of string.
    1. Write down the distribution of \(L\)
    2. Find the probability that the length of the longer piece of string is 28 cm to the nearest cm
    Each piece of string is used to form the perimeter of a square.
  2. Calculate the probability that the area of the larger square is less than \(64 \mathrm {~cm} ^ { 2 }\)
  3. Calculate the probability that the difference in area between the two squares is greater than \(81 \mathrm {~cm} ^ { 2 }\)
Edexcel S2 2016 October Q6
12 marks Standard +0.3
  1. According to an electric company, power failures occur randomly at a rate of \(\lambda\) every 10 weeks, \(1 < \lambda < 10\)
    1. Write down an expression in terms of \(\lambda\) for the probability that there are fewer than 2 power failures in a randomly selected 10 week period.
    2. Write down an expression in terms of \(\lambda\) for the probability that there is exactly 1 power failure in a randomly selected 5 week period.
    Over a 100 week period, the probability, using a normal approximation, that fewer than 15 power failures occur is 0.0179 (to 3 significant figures).
    1. Justify the use of a normal approximation.
    2. Find the value of \(\lambda\). Show each stage of your working clearly.
Edexcel S2 2016 October Q7
8 marks Standard +0.3
  1. An ice cream shop sells a large number of 1 scoop, 2 scoop and 3 scoop ice cream cones to its customers in the ratio \(5 : 2 : 1\)
A random sample of 2 customers at the ice cream shop is taken. Each customer orders a 1 scoop or a 2 scoop or a 3 scoop ice cream cone. Let \(S\) represent the total number of ice cream scoops ordered by these 2 customers.
  1. Find the sampling distribution of \(S\) A random sample of \(n\) customers at the ice cream shop is taken. Each customer orders a 1 scoop or a 2 scoop or a 3 scoop ice cream cone. The probability that more than \(n\) scoops of ice cream are ordered by these customers is greater than 0.99
  2. Find the smallest possible value of \(n\)
    \includegraphics[max width=\textwidth, alt={}]{4ecee051-3a6f-4c12-8c53-926e8c3e241f-28_2632_1828_121_121}
Edexcel S2 2017 October Q1
9 marks Standard +0.3
  1. A shop sells rods of nominal length 200 cm . The rods are bought from a manufacturer who uses a machine to cut rods of length \(L \mathrm {~cm}\), where \(L \sim \mathrm {~N} \left( \mu , 0.2 ^ { 2 } \right)\)
The value of \(\mu\) is such that there is only a \(5 \%\) chance that a rod, selected at random from those supplied to the shop, will have length less than 200 cm .
  1. Find the value of \(\mu\) to one decimal place. A customer buys a random sample of 8 of these rods.
  2. Find the probability that at least 3 of these rods will have length less than 200 cm . Another customer buys a random sample of 60 of these rods.
  3. Using a suitable approximation, find the probability that more than 5 of these rods will have length less than 200 cm .
Edexcel S2 2017 October Q2
18 marks Standard +0.3
2. The weekly sales, \(S\), in thousands of pounds, of a small business has probability density function $$\mathrm { f } ( s ) = \left\{ \begin{array} { c c } k ( s - 2 ) ( 10 - s ) & 2 < s < 10 \\ 0 & \text { otherwise } \end{array} \right.$$
  1. Use algebraic integration to show that \(k = \frac { 3 } { 256 }\)
  2. Write down the value of \(\mathrm { E } ( S )\)
  3. Use algebraic integration to find the standard deviation of the weekly sales. A week is selected at random.
  4. Showing your working, find the probability that this week's sales exceed \(\pounds 7100\) Give your answer to one decimal place. A quarter is defined as 12 consecutive weeks. The discrete random variable \(X\) is the number of weeks in a quarter in which the weekly sales exceed £7100 The manager earns a bonus at the following rates:
    \(\boldsymbol { X }\)Bonus Earned
    \(X \leqslant 5\)\(\pounds 0\)
    \(X = 6\)\(\pounds 1000\)
    \(X \geqslant 7\)\(\pounds 5000\)
  5. Using your answer to part (d), calculate the manager's expected bonus per quarter.
Edexcel S2 2017 October Q3
14 marks Standard +0.3
3. In a shop, the weekly demand for Birdscope cameras is modelled by a Poisson distribution with mean 8 The shop has 9 Birdscope cameras in stock at the start of each week. A week is selected at random.
  1. Find the probability that the demand for Birdscope cameras cannot be met in this particular week. In a year, there are 50 weeks in which Birdscope cameras can be sold.
  2. Find the expected number of weeks in the year that the shop will not be able to meet the demand for Birdscope cameras.
  3. Find the number of Birdscope cameras the shop should stock at the beginning of each week if it wants the estimated number of weeks in the year in which demand cannot be met to be less than 2 The shop increases its stock and reduces the price of Birdscope cameras in order to increase demand. A random sample of 10 weeks is selected and it is found that, in the 10 weeks, a total of 95 Birdscope cameras were sold. Given that there were no weeks when the shop was unable to meet the demand for Birdscope cameras,
  4. use a suitable approximation to test whether or not the demand for Birdscope cameras has increased following the price reduction. You should state your hypotheses clearly and use a 5\% level of significance.
Edexcel S2 2017 October Q4
14 marks Moderate -0.8
4. In a computer game, a ship appears randomly on a rectangular screen. The continuous random variable \(X \mathrm {~cm}\) is the distance of the centre of the ship from the bottom of the screen. The random variable \(X\) is uniformly distributed over the interval \([ 0 , \alpha ]\) where \(\alpha \mathrm { cm }\) is the height of the screen. Given that \(\mathrm { P } ( X > 6 ) = 0.6\)
  1. find the value of \(\alpha\)
  2. find \(\mathrm { P } ( 4 < X < 10 )\) The continuous random variable \(Y\) cm is the distance of the centre of the ship from the left-hand side of the screen. The random variable \(Y\) is uniformly distributed over the interval [ 0,20 ] where 20 cm is the width of the screen.
  3. Find the mean and the standard deviation of \(Y\).
  4. Find \(\mathrm { P } ( | Y - 4 | < 2 )\)
  5. Given that \(X\) and \(Y\) are independent, find the probability that the centre of the ship appears
    1. in a square of side 4 cm which is at the centre of the screen,
    2. within 5 cm of a side or the top or the bottom of the screen.
Edexcel S2 2017 October Q5
10 marks Standard +0.3
5. The continuous random variable \(Y\) has cumulative distribution function \(\mathrm { F } ( y )\) given by $$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 3 \\ k \left( y ^ { 2 } - 2 y - 3 \right) & 3 \leqslant y \leqslant \alpha \\ 4 k ( 2 y - 7 ) & \alpha < y \leqslant 6 \\ 1 & y > 6 \end{array} \right.$$ where \(k\) and \(\alpha\) are constants.
  1. Find \(\mathrm { P } ( 4.5 < Y \leqslant 5.5 )\)
  2. Find the probability density function \(\mathrm { f } ( \mathrm { y } )\)