Questions — Edexcel S2 (494 questions)

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Edexcel S2 2018 June Q6
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
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
  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
  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
  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
  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
  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
  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
  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
  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
  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
  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
  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
  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
  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
  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
  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
  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
  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
  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
  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
  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
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
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
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.