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Edexcel S2 2019 January Q7
14 marks Standard +0.3
  1. The continuous random variable \(X\) has probability density function
$$f ( x ) = \begin{cases} c ( x + 3 ) & - 3 \leqslant x < 0 \\ \frac { 5 } { 36 } ( 3 - x ) & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$ where \(c\) is a positive constant.
  1. Show that \(c = \frac { 1 } { 12 }\)
    1. Sketch the probability density function.
    2. Explain why the mode of \(X = 0\)
  3. Find the cumulative distribution function of \(X\), for all values of \(x\)
  4. Find, to 3 significant figures, the value of \(d\) such that \(\mathrm { P } ( X > d \mid X > 0 ) = \frac { 2 } { 5 }\)
    Leave blank
    Q7

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Edexcel S2 2021 January Q1
14 marks Standard +0.3
  1. Jim farms oysters in a particular lake. He knows from past experience that \(5 \%\) of young oysters do not survive to be harvested.
In a random sample of 30 young oysters, the random variable \(X\) represents the number that do not survive to be harvested.
  1. Write down a suitable model for the distribution of \(X\).
  2. State an assumption that has been made for the model in part (a).
  3. Find the probability that
    1. exactly 24 young oysters do survive to be harvested,
    2. at least 3 young oysters do not survive to be harvested. A second random sample, of 200 young oysters, is taken. The probability that at least \(n\) of these young oysters do not survive to be harvested is more than 0.8
  4. Using a suitable approximation, find the maximum value of \(n\). Jim believes that the level of salt in the lake water has changed and it has altered the survival rate of his oysters. He takes a random sample of 25 young oysters and places them in the lake.
    When Jim harvests the oysters, he finds that 21 do survive to be harvested.
  5. Use a suitable test, at the \(5 \%\) level of significance, to assess whether or not there is evidence that the proportion of oysters not surviving to be harvested is more than \(5 \%\). State your hypotheses clearly.
Edexcel S2 2021 January Q2
10 marks Standard +0.3
2. The distance, in metres, a novice tightrope artist, walking on a wire, walks before falling is modelled by the random variable \(W\) with cumulative distribution function $$\mathrm { F } ( w ) = \left\{ \begin{array} { c c } 0 & w < 0 \\ \frac { 1 } { 3 } \left( w - \frac { w ^ { 4 } } { 256 } \right) & 0 \leqslant w \leqslant 4 \\ 1 & w > 4 \end{array} \right.$$
  1. Find the probability that a novice tightrope artist, walking on the wire, walks at least 3.5 metres before falling. A random sample of 30 novice tightrope artists is taken.
  2. Find the probability that more than 1 of these novice tightrope artists, walking on the wire, walks at least 3.5 metres before falling. Given \(\mathrm { E } ( W ) = 1.6\)
  3. use algebraic integration to find \(\operatorname { Var } ( W )\) DO NOT WRITEIN THIS AREA
Edexcel S2 2021 January Q3
17 marks Standard +0.3
3. The number of water fleas, in 100 ml of pond water, has a Poisson distribution with mean 7
  1. Find the probability that a sample of 100 ml of the pond water does not contain exactly 4 water fleas. Aja collects 5 separate samples, each of 100 ml , of the pond water.
  2. Find the probability that exactly 1 of these samples contains exactly 4 water fleas. Using a normal approximation, the probability that more than 3 water fleas will be found in a random sample of \(n \mathrm { ml }\) of the pond water is 0.9394 correct to 4 significant figures.
    1. Show that \(n - 1.55 \sqrt { \frac { n } { 0.07 } } - 50 = 0\)
    2. Hence find the value of \(n\) After the pond has been cleaned, the number of water fleas in a 100 ml random sample of the pond water is 15
  4. Using a suitable test, at the \(1 \%\) level of significance, assess whether or not there is evidence that the number of water fleas per 100 ml of the pond water has increased. State your hypotheses clearly. \includegraphics[max width=\textwidth, alt={}, center]{f63c39df-cfc9-4a6b-838d-67613710b0ce-11_2255_50_314_34}
    VIXV SIHIANI III IM IONOOVIAV SIHI NI JYHAM ION OOVI4V SIHI NI JLIYM ION OO
Edexcel S2 2021 January Q4
10 marks Standard +0.3
4. A continuous random variable \(X\) has probability density function $$\mathrm { f } ( x ) = \left\{ \begin{array} { c c } k ( a - x ) ^ { 2 } & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{array} \right.$$ where \(k\) and \(a\) are constants.
  1. Show that \(k a ^ { 3 } = 3\) Given that \(\mathrm { E } ( X ) = 1.5\)
  2. use algebraic integration to show that \(a = 6\)
  3. Verify that the median of \(X\) is 1.2 to one decimal place. \includegraphics[max width=\textwidth, alt={}, center]{f63c39df-cfc9-4a6b-838d-67613710b0ce-15_2255_50_314_34}
    VIXV SIHIANI III IM IONOOVIAV SIHI NI JYHAM ION OOVI4V SIHI NI JLIYM ION OO
Edexcel S2 2021 January Q5
14 marks Standard +0.3
5. A piece of wood \(A B\) is 3 metres long. The wood is cut at random at a point \(C\) and the random variable \(W\) represents the length of the piece of wood \(A C\).
  1. Find the probability that the length of the piece of wood \(A C\) is more than 1.8 metres. The two pieces of wood \(A C\) and \(C B\) form the two shortest sides of a right-angled triangle. The random variable \(X\) represents the length of the longest side of the right-angled triangle.
  2. Show that \(X ^ { 2 } = 2 W ^ { 2 } - 6 W + 9\) [0pt] [You may assume for random variables \(S , T\) and \(U\) and for constants \(a\) and \(b\) that if \(S = a T + b U\) then \(\mathrm { E } ( S ) = a \mathrm { E } ( T ) + b \mathrm { E } ( U ) ]\)
  3. Find \(\mathrm { E } \left( X ^ { 2 } \right)\)
  4. Find \(\mathrm { P } \left( X ^ { 2 } > 5 \right)\)
Edexcel S2 2021 January Q6
10 marks Moderate -0.8
6. The owner of a very large youth club has designed a new method for allocating people to teams. Before introducing the method he decided to find out how the members of the youth club might react.
  1. Explain why the owner decided to take a random sample of the youth club members rather than ask all the youth club members.
  2. Suggest a suitable sampling frame.
  3. Identify the sampling units. The new method uses a bag containing a large number of balls. Each ball is numbered either 20, 50 or 70
    When a ball is selected at random, the random variable \(X\) represents the number on the ball where $$\mathrm { P } ( X = 20 ) = p \quad \mathrm { P } ( X = 50 ) = q \quad \mathrm { P } ( X = 70 ) = r$$ A youth club member takes a ball from the bag, records its number and replaces it in the bag. He then takes a second ball from the bag, records its number and replaces it in the bag. The random variable \(M\) is the mean of the 2 numbers recorded. Given that $$\mathrm { P } ( M = 20 ) = \frac { 25 } { 64 } \quad \mathrm { P } ( M = 60 ) = \frac { 1 } { 16 } \quad \text { and } \quad q > r$$
  4. show that \(\mathrm { P } ( M = 50 ) = \frac { 1 } { 16 }\)
    VIHV SIHII NI I IIIM I ON OCVIAV SIHI NI JYHAM ION OOVI4V SIHI NI JLIYM ION OO
    \includegraphics[max width=\textwidth, alt={}, center]{f63c39df-cfc9-4a6b-838d-67613710b0ce-24_111_65_2525_1880} \includegraphics[max width=\textwidth, alt={}, center]{f63c39df-cfc9-4a6b-838d-67613710b0ce-24_140_233_2625_1733}
Edexcel S2 2022 January Q1
11 marks Standard +0.3
1 A local pottery makes cups. The number of faulty cups made by the pottery in a week follows a Poisson distribution with a mean of 6 In a randomly chosen week, the probability that there will be at least \(x\) faulty cups made is 0.1528
  1. Find the value of \(x\)
  2. Use a normal approximation to find the probability that in 6 randomly chosen weeks the total number of faulty cups made is fewer than 32 A week is called a "poor week" if at least \(x\) faulty cups are made, where \(x\) is the value found in part (a).
  3. Find the probability that in 50 randomly chosen weeks, more than 1 is a "poor week".
Edexcel S2 2022 January Q2
8 marks Moderate -0.8
2 The continuous random variable \(X\) has cumulative distribution function given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < - k \\ \frac { x + k } { 4 k } & - k \leqslant x \leqslant 3 k \\ 1 & x > 3 k \end{array} \right.$$ where \(k\) is a positive constant.
  1. Specify fully, in terms of \(k\), the probability density function of \(X\)
  2. Write down, in terms of \(k\), the value of \(\mathrm { E } ( X )\)
  3. Show that \(\operatorname { Var } ( X ) = \frac { 4 } { 3 } k ^ { 2 }\)
  4. Find, in terms of \(k\), the value of \(\mathrm { E } \left( 3 X ^ { 2 } \right)\)
Edexcel S2 2022 January Q3
9 marks Standard +0.3
3 A photocopier in a school is known to break down at random at a mean rate of 8 times per week.
  1. Give a reason why a Poisson distribution could be used to model the number of breakdowns. The headteacher of the school replaces the photocopier with a refurbished one and wants to find out if the rate of breakdowns has increased or decreased.
  2. Write down suitable null and alternative hypotheses that the headteacher should use. The refurbished photocopier was monitored for the first week after it was installed.
  3. Using a \(5 \%\) level of significance, find the critical region to test whether the rate of breakdowns has now changed.
  4. Find the actual significance level of a test based on the critical region from part (c). During the first week after it was installed there were 4 breakdowns.
  5. Comment on this finding in the light of the critical region found in part (c).
Edexcel S2 2022 January Q4
15 marks Standard +0.3
4 The continuous random variable \(X\) has a probability density function given by $$\mathrm { f } ( x ) = \begin{cases} \frac { 1 } { 2 } k ( x - 1 ) & 1 \leqslant x \leqslant 3 \\ k & 3 < x \leqslant 6 \\ \frac { 1 } { 4 } k ( 10 - x ) & 6 < x \leqslant 10 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a positive constant.
  1. Sketch \(\mathrm { f } ( x )\) for all values of \(x\)
  2. Show that \(k = \frac { 1 } { 6 }\)
  3. Specify fully the cumulative distribution function \(\mathrm { F } ( x )\) of \(X\) Given that \(\mathrm { E } ( X ) = \frac { 61 } { 12 }\)
  4. find \(\mathrm { P } ( X > \mathrm { E } ( X ) )\)
  5. Describe the skewness of the distribution, giving a reason for your answer.
Edexcel S2 2022 January Q5
14 marks Moderate -0.3
5 Applicants for a pilot training programme with a passenger airline are screened for colour blindness. Past records show that the proportion of applicants identified as colour blind is 0.045
  1. Write down a suitable model for the distribution of the number of applicants identified as colour blind from a total of \(n\) applicants.
  2. State one assumption necessary for this distribution to be a suitable model of this situation.
  3. Using a suitable approximation, find the probability that exactly 5 out of 120 applicants are identified as colour blind.
  4. Explain why the approximation that you used in part (c) is appropriate. Jaymini claims that 75\% of all applicants for this training programme go on to become pilots. From a random sample of 96 applicants for this training programme 67 go on to become pilots.
  5. Using a suitable approximation, test Jaymini's claim at the \(5 \%\) level of significance. State your hypotheses clearly.
Edexcel S2 2022 January Q6
10 marks Standard +0.3
6
  1. Explain what you understand by the sampling distribution of a statistic. At Sam's cafe a standard breakfast consists of 6 breakfast items. Customers can then choose to upgrade to a medium breakfast by adding 1 extra breakfast item or they can upgrade to a large breakfast by adding 2 extra breakfast items. Standard, medium and large breakfasts are sold in the ratio \(6 : 3 : 2\) respectively. A random sample of 2 customers is taken from customers who have bought a breakfast from Sam's cafe on a particular day.
  2. Find the sampling distribution for the total number, \(T\), of breakfast items bought by these 2 customers. Show your working clearly.
  3. Find \(\mathrm { E } ( T )\)
Edexcel S2 2022 January Q7
8 marks Standard +0.8
7 The sides of a square are each of length \(L \mathrm {~cm}\) and its area is \(A \mathrm {~cm} ^ { 2 }\) Given that \(A\) is uniformly distributed on the interval [10,30]
  1. find \(\mathrm { P } ( L \geqslant 4.5 )\)
  2. find \(\operatorname { Var } ( L )\)
    \includegraphics[max width=\textwidth, alt={}]{a009b02e-4cd3-497b-a141-4630c653e20b-28_2655_1947_114_116}
Edexcel S2 2023 January Q1
11 marks Moderate -0.3
  1. A shop sells shoes at a mean rate of 4 pairs of shoes per hour on a weekday.
    1. Suggest a suitable distribution for modelling the number of sales of pairs of shoes made per hour on a weekday.
    2. State one assumption necessary for this distribution to be a suitable model of this situation.
    3. Find the probability that on a weekday the shop sells
      1. more than 4 pairs of shoes in a one-hour period,
      2. more than 4 pairs of shoes in each of 3 consecutive one-hour periods.
    The area manager visits the shop on a weekday, the day after an advert for the shop appears in a local paper. In a one-hour period during the manager's visit, the shop sells 7 pairs of shoes. This leads the manager to believe that the advert has increased the shop's sales of pairs of shoes.
  2. Stating your hypotheses clearly, test at the \(5 \%\) level of significance whether or not there is evidence of an increase in sales of pairs of shoes following the appearance of the advert.
Edexcel S2 2023 January Q2
11 marks Moderate -0.8
  1. A bag contains a large number of coins. It only contains 20 p and 50 p coins. A random sample of 3 coins is taken from the bag.
    1. List all the possible combinations of 3 coins that might be taken.
    Let \(\bar { X }\) represent the mean value of the 3 coins taken.
    Part of the sampling distribution of \(\bar { X }\) is given below.
    \(\bar { x }\)20\(a\)\(b\)50
    \(\mathrm { P } ( \bar { X } = \bar { x } )\)\(\frac { 4913 } { 8000 }\)\(c\)\(d\)\(\frac { 27 } { 8000 }\)
  2. Write down the value of \(a\) and the value of \(b\) The probability of taking a 20p coin at random from the bag is \(p\) The probability of taking a 50p coin at random from the bag is \(q\)
  3. Find the value of \(p\) and the value of \(q\)
  4. Hence, find the value of \(c\) and the value of \(d\) Let \(M\) represent the mode of the 3 coins taken at random from the bag.
  5. Find the sampling distribution of \(M\)
Edexcel S2 2023 January Q3
11 marks Standard +0.3
  1. Superbounce is a manufacturer of tennis balls.
It knows from past records that 10\% of its tennis balls fail a bounce test.
  1. Find the probability that from a random sample of 10 of these tennis balls
    1. at least 4 fail the bounce test
    2. more than 1 but fewer than 5 fail the bounce test. The managing director makes changes to the production process and claims that these changes will reduce the probability of its tennis balls failing the bounce test. After the changes were made a random sample of 50 of the tennis balls were tested and it was found that 2 failed the bounce test.
  2. Test, at the \(5 \%\) significance level, whether or not this result supports the managing director's claim. In a second random sample of \(n\) tennis balls it was found that none failed the bounce test. As a result of this sample, the managing director's claim is supported at the 1\% significance level.
  3. Find the smallest possible value of \(n\)
Edexcel S2 2023 January Q4
10 marks Standard +0.3
  1. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\), shown in the diagram, where \(k\) is a constant. \includegraphics[max width=\textwidth, alt={}, center]{f4fa6add-5860-4c88-bb70-f3edd9b22211-12_511_1096_351_351}
    1. Find \(\mathrm { P } ( X < 10 k )\)
    2. Show that \(k = \frac { 1 } { \pi }\)
    3. Find, in terms of \(\pi\), the values of
      1. \(\mathrm { E } ( X )\)
      2. \(\operatorname { Var } ( X )\)
    Circles are drawn with area \(A\), where $$A = \pi \left( X + \frac { 2 } { \pi } \right) ^ { 2 }$$
  2. Find \(\mathrm { E } ( A )\)
Edexcel S2 2023 January Q5
14 marks Standard +0.3
  1. A company produces steel cable.
Defects in the steel cable produced by this company occur at random, at a constant rate of 1 defect per 16 metres. On one day the company produces a piece of steel cable 80 metres long.
  1. Find the probability that there are at most 5 defects in this piece of steel cable. The company produces a piece of steel cable 80 metres long on each of the next 4 days.
  2. Find the probability that fewer than 2 of these 4 pieces of steel cable contain at most 5 defects. The following week the company produces a piece of steel cable \(x\) metres long.
    Using a normal approximation, the probability that this piece of steel cable has fewer than 26 defects is 0.5398
  3. Find the value of \(x\)
Edexcel S2 2023 January Q6
18 marks Standard +0.8
  1. The continuous random variable \(X\) has cumulative distribution function
$$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < 0 \\ a x + b x ^ { 2 } & 0 \leqslant x \leqslant k \\ 1 & x > k \end{array} \right.$$ where \(a , b\) and \(k\) are positive constants.
  1. Show that \(a k = 1 - b k ^ { 2 }\) Using part (a) and given that \(\mathrm { E } ( X ) = \frac { 6 } { 5 }\)
  2. show that \(5 b k ^ { 3 } = 36 - 15 k\) Using part (a) and given that \(\mathrm { E } ( X ) = \frac { 6 } { 5 }\) and \(\operatorname { Var } ( X ) = \frac { 22 } { 75 }\)
  3. show that \(5 b k ^ { 4 } = 52 - 10 k ^ { 2 }\) Given that \(k < 3\)
  4. find the value of \(k\)
  5. Hence find the value of \(a\) and the value of \(b\)
Edexcel S2 2024 January Q1
16 marks Standard +0.3
  1. The manager of a supermarket is investigating the number of complaints per day received from customers.
A random sample of 180 days is taken and the results are shown in the table below.
Number of complaints per day0123456\(\geqslant 7\)
Frequency122837382917190
  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 complaints per day. The manager uses a Poisson distribution with mean 3 to model the number of complaints per day.
  3. For a randomly selected day find, using the manager's model, the probability that there are
    1. at least 3 complaints,
    2. more than 4 complaints but less than 8 complaints. A week consists of 7 consecutive days.
  4. Using the manager's model and a suitable approximation, show that the probability that there are less than 19 complaints in a randomly selected week is 0.29 to 2 decimal places.
    Show your working clearly.
    (Solutions relying on calculator technology are not acceptable.) A period of 13 weeks is selected at random.
  5. Find the probability that in this period there are exactly 5 weeks that have less than 19 complaints.
    Show your working clearly.
Edexcel S2 2024 January Q2
8 marks Standard +0.3
  1. The length of pregnancy for a randomly selected pregnant sheep is \(D\) days where
$$D \sim \mathrm {~N} \left( 112.4 , \sigma ^ { 2 } \right)$$ Given that 5\% of pregnant sheep have a length of pregnancy of less than 108 days,
  1. find the value of \(\sigma\) Qiang selects 25 pregnant sheep at random from a large flock.
  2. Find the probability that more than 3 of these pregnant sheep have a length of pregnancy of less than 108 days. Charlie takes 200 random samples of 25 pregnant sheep.
  3. Use a Poisson approximation to estimate the probability that at least 2 of the samples have more than 3 pregnant sheep with a length of pregnancy of less than 108 days.
Edexcel S2 2024 January Q3
12 marks Standard +0.3
  1. Rowan believes that \(35 \%\) of type \(A\) vacuum tubes shatter when exposed to alternating high and low temperatures.
Rowan takes a random sample of 15 of these type \(A\) vacuum tubes and uses a two-tailed test, at the \(5 \%\) level of significance, to test his belief.
  1. Give two assumptions, in context, that Rowan needs to make for a binomial distribution to be a suitable model for the number of these type \(A\) vacuum tubes that shatter when exposed to alternating high and low temperatures.
  2. Using a binomial distribution, find the critical region for the test. You should state the probability of rejection in each tail, which should be as close as possible to 0.025
  3. Find the actual level of significance of the test based on your critical region from part (b) Rowan records that in the latest batch of 15 type \(A\) vacuum tubes exposed to alternating high and low temperatures, 4 of them shattered.
  4. With reference to part (b), comment on Rowan’s belief. Give a reason for your answer. Rowan changes to type \(B\) vacuum tubes. He takes a random sample of 40 type \(B\) vacuum tubes and finds that 8 of them shatter when exposed to alternating high and low temperatures.
  5. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the proportion of type \(B\) vacuum tubes that shatter when exposed to alternating high and low temperatures is lower than \(35 \%\) You should state your hypotheses clearly.
Edexcel S2 2024 January Q4
12 marks Standard +0.3
  1. The continuous random variable \(G\) has probability density function \(\mathrm { f } ( \mathrm { g } )\) given by
$$f ( g ) = \begin{cases} \frac { 1 } { 15 } ( g + 3 ) & - 1 < g \leqslant 2 \\ \frac { 3 } { 20 } & 2 < g \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$
  1. Sketch the graph of \(\mathrm { f } ( \mathrm { g } )\)
  2. Find \(\mathrm { P } ( ( 1 \leqslant 2 G \leqslant 6 ) \mid G \leqslant 2 )\) The continuous random variable \(H\) is such that \(\mathrm { E } ( H ) = 12\) and \(\operatorname { Var } ( H ) = 2.4\)
  3. Find \(\mathrm { E } \left( 2 H ^ { 2 } + 3 G + 3 \right)\) Show your working clearly.
    (Solutions relying on calculator technology are not acceptable.)
Edexcel S2 2024 January Q5
10 marks Standard +0.8
  1. The random variable \(W\) has a continuous uniform distribution over the interval \([ - 6 , a ]\) where \(a\) is a constant.
Given that \(\operatorname { Var } ( W ) = 27\)
  1. show that \(a = 12\) Given that \(\mathrm { P } ( W > b ) = \frac { 3 } { 5 }\)
    1. find the value of \(b\)
    2. find \(\mathrm { P } \left( - 12 < W < \frac { b } { 2 } \right)\) A piece of wood \(A B\) has length 160 cm . The wood is cut at random into 2 pieces. Each of the pieces is then cut in half. The four pieces are used to form the sides of a rectangle.
  3. Calculate the probability that the area of the rectangle is greater than \(975 \mathrm {~cm} ^ { 2 }\)