Questions — Edexcel S2 (494 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Edexcel S2 2017 January Q6
  1. A seed producer claims that \(96 \%\) of its bean seeds germinate.
To test the producer's claim, a random sample of 75 bean seeds was planted and 66 of these seeds germinated. Use a suitable approximation to test, at the \(1 \%\) level of significance, whether or not the producer is overstating the probability of its bean seeds germinating. State your hypotheses clearly.
Edexcel S2 2017 January Q7
7. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by $$f ( x ) = \begin{cases} \frac { 1 } { 20 } x ^ { 3 } & 0 \leqslant x \leqslant 2
\frac { 1 } { 10 } ( 6 - x ) & 2 < x \leqslant 6
0 & \text { otherwise } \end{cases}$$
  1. Sketch the graph of \(\mathrm { f } ( x )\) for all values of \(x\).
  2. Write down the mode of \(X\).
  3. Show that \(\mathrm { P } ( X > 2 ) = 0.8\)
  4. Define fully the cumulative distribution function \(\mathrm { F } ( x )\). Given that \(\mathrm { P } ( X < a \mid X > 2 ) = \frac { 5 } { 8 }\)
  5. find the value of \(\mathrm { F } ( a )\).
  6. Hence, or otherwise, find the value of \(a\). Give your answer to 3 significant figures.
Edexcel S2 2018 January Q1
  1. A continuous random variable \(X\) has cumulative distribution function
$$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < 1
\frac { 1 } { 16 } ( x - 1 ) ^ { 2 } & 1 \leqslant x \leqslant 5
1 & x > 5 \end{array} \right.$$
  1. Find \(\mathrm { P } ( X > 4 )\)
  2. Find \(\mathrm { P } ( X > 3 \mid 2 < X < 4 )\)
  3. Find the exact value of \(\mathrm { E } ( X )\)
Edexcel S2 2018 January Q2
2. A farmer sells boxes of eggs. The eggs are sold in boxes of 6 eggs and boxes of 12 eggs in the ratio \(n : 1\) A random sample of three boxes is taken.
The number of eggs in the first box is denoted by \(X _ { 1 }\)
The number of eggs in the second box is denoted by \(X _ { 2 }\)
The number of eggs in the third box is denoted by \(X _ { 3 }\)
The random variable \(T = X _ { 1 } + X _ { 2 } + X _ { 3 }\)
Given that \(\mathrm { P } ( T = 18 ) = 0.729\)
  1. show that \(n = 9\)
  2. find the sampling distribution of \(T\) The random variable \(R\) is the range of \(X _ { 1 } , X _ { 2 } , X _ { 3 }\)
  3. Using your answer to part (b), or otherwise, find the sampling distribution of \(R\)
Edexcel S2 2018 January Q3
  1. Albert uses scales in his kitchen to weigh some fruit.
The random variable \(D\) represents, in grams, the weight of the fruit given by the scales minus the true weight of the fruit. The random variable \(D\) is uniformly distributed over the interval \([ - 2.5,2.5 ]\)
  1. Specify the probability density function of \(D\)
  2. Find the standard deviation of \(D\) Albert weighs a banana on the scales.
  3. Write down the probability that the weight given by the scales equals the true weight of the banana.
  4. Find the probability that the weight given by the scales is within 1 gram of the banana's true weight. Albert weighs 10 bananas on the scales, one at a time.
  5. Find the probability that the weight given by the scales is within 1 gram of the true weight for at least 6 of the bananas.
Edexcel S2 2018 January Q4
4. A sweet shop produces different coloured sweets and sells them in bags. The proportion of green sweets produced is \(p\) Each bag is filled with a random sample of \(n\) sweets. The mean number of green sweets in a bag is 4.2 and the variance is 3.57
  1. Find the value of \(n\) and the value of \(p\) The proportion of red sweets produced by the shop is 0.35
  2. Find the probability that, in a random sample of 25 sweets, the number of red sweets exceeds the expected number of red sweets. The shop claims that \(10 \%\) of its customers buy more than two bags of sweets. A random sample of 40 customers is taken and 1 customer buys more than two bags of sweets.
  3. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the proportion of customers who buy more than two bags of sweets is less than the shop's claim. State your hypotheses clearly.
Edexcel S2 2018 January Q5
5. A delivery company loses packages randomly at a mean rate of 10 per month. The probability that the delivery company loses more than 12 packages in a randomly selected month is \(p\)
  1. Find the value of \(p\) The probability that the delivery company loses more than \(k\) packages in a randomly selected month is at least \(2 p\)
  2. Find the largest possible value of \(k\) In a randomly selected month,
  3. find the probability that exactly 4 packages were lost in each half of the month. In a randomly selected two-month period, 21 packages were lost.
  4. Find the probability that at least 10 packages were lost in each of these two months.
  5. Using a suitable approximation, find the probability that more than 27 packages are lost during a randomly selected 4-month period.
Edexcel S2 2018 January Q6
  1. In a local council, \(60 \%\) of households recycle at least half of their waste. A random sample of 80 households is taken.
The random variable \(X\) represents the number of households in the sample that recycle at least half of their waste.
  1. Using a suitable approximation, find the smallest number of households, \(n\), such that $$\mathrm { P } ( X \geqslant n ) < 0.05$$ The number of bags recycled per family per week was known to follow a Poisson distribution with mean 1.5 Following a recycling campaign, the council believes the mean number of bags recycled per family per week has increased. To test this belief, 6 families are selected at random and the total number of bags they recycle the following week is recorded. The council wishes to test, at the 5\% level of significance, whether or not there is evidence that the mean number of bags recycled per family per week has increased.
  2. Find the critical region for the total number of bags recycled by the 6 families.
Edexcel S2 2018 January Q7
  1. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by
$$f ( x ) = \begin{cases} \frac { 1 } { 16 } x ^ { 2 } & 1 \leqslant x < 3
k ( 4 - x ) & 3 \leqslant x \leqslant 4
0 & \text { otherwise } \end{cases}$$
  1. Show that \(k = \frac { 11 } { 12 }\)
  2. Sketch \(\mathrm { f } ( x )\) for \(1 \leqslant x \leqslant 4\)
  3. Write down the mode of \(X\) Given that \(\mathrm { E } ( X ) = \frac { 25 } { 9 }\)
  4. use algebraic integration to find \(\operatorname { Var } ( X )\), giving your answer to 3 significant figures. The cumulative distribution function of \(X\) is given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < 1
    \frac { 1 } { 48 } \left( x ^ { 3 } + c \right) & 1 \leqslant x < 3
    \frac { 11 } { 12 } \left( 4 x - \frac { 1 } { 2 } x ^ { 2 } + d \right) & 3 \leqslant x \leqslant 4
    1 & x > 4 \end{array} \right.$$
    1. Find the exact value of \(C\)
    2. Find the exact value of \(d\)
  6. Calculate, to 3 significant figures, the upper quartile of \(X\)
    \includegraphics[max width=\textwidth, alt={}]{a814156d-6945-4601-9cae-d28d8ae0db1e-28_2632_1826_121_121}
Edexcel S2 2019 January Q1
  1. A bus company sells tickets for a journey from London to Oxford every Saturday. Past records show that \(5 \%\) of people who buy a ticket do not turn up for the journey.
The bus has seats for 48 people.
Each week the bus company sells tickets to exactly 50 people for the journey.
The random variable \(X\) represents the number of these people who do not turn up for the journey.
  1. State one assumption required to model \(X\) as a binomial distribution. For this week's journey find,
    1. the probability that all 50 people turn up for the journey,
    2. \(\mathrm { P } ( X = 1 )\) The bus company receives \(\pounds 20\) for each ticket sold and all 50 tickets are sold. It must pay out \(\pounds 60\) to each person who buys a ticket and turns up for the journey but does not have a seat.
  3. Find the bus company's expected total earnings per journey.
Edexcel S2 2019 January Q2
  1. During morning hours, employees arrive randomly at an office drinks dispenser at a rate of 2 every 10 minutes.
The number of employees arriving at the drinks dispenser is assumed to follow a Poisson distribution.
  1. Find the probability that fewer than 5 employees arrive at the drinks dispenser during a 10-minute period one morning. During a 30 -minute period one morning, the probability that \(n\) employees arrive at the drinks dispenser is the same as the probability that \(n + 1\) employees arrive at the drinks dispenser.
  2. Find the value of \(n\) During a 45-minute period one morning, the probability that between \(c\) and 12, inclusive, employees arrive at the drinks dispenser is 0.8546
  3. Find the value of \(C\)
  4. Find the probability that exactly 2 employees arrive at the drinks dispenser in exactly 4 of the 6 non-overlapping 10-minute intervals between 10 am and 11am one morning.
Edexcel S2 2019 January Q3
3. Figure 1 shows an accurate graph of the cumulative distribution function, \(\mathrm { F } ( x )\), for the continuous random variable \(X\) \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{17296edc-9ab4-4f81-ae68-c76190986fd1-08_535_1152_354_342} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure}
  1. Find \(\mathrm { P } ( 3 < X < 7 )\) The probability density function of \(X\) is given by $$\mathrm { f } ( x ) = \begin{cases} a & 2 \leqslant x < 4
    b & 4 \leqslant x < 6
    c & 6 \leqslant x \leqslant 8
    0 & \text { otherwise } \end{cases}$$ where \(a\), \(b\) and \(c\) are constants.
  2. Find the value of \(a\), the value of \(b\) and the value of \(c\)
  3. Find \(\mathrm { E } ( X )\)
Edexcel S2 2019 January Q4
  1. At a shop, past figures show that \(35 \%\) of customers pay by credit card. Following the shop’s decision to no longer charge a fee for using a credit card, a random sample of 20 customers is taken and 11 are found to have paid by credit card.
Hadi believes that the proportion of customers paying by credit card is now greater than 35\%
  1. Test Hadi's belief at the \(5 \%\) level of significance. State your hypotheses clearly. For a random sample of 20 customers,
  2. show that 11 lies less than 2 standard deviations above the mean number of customers paying by credit card.
    You may assume that \(35 \%\) is the true proportion of customers who pay by credit card.
Edexcel S2 2019 January Q5
  1. The continuous random variable \(X\) is uniformly distributed over the interval \([ a , b ]\) where \(0 < a < b\)
Given that \(\mathrm { P } ( X < b - 2 a ) = \frac { 1 } { 3 }\)
    1. show that \(\mathrm { E } ( X ) = \frac { 5 a } { 2 }\)
    2. find \(\mathrm { P } ( X > b - 4 a )\) The continuous random variable \(Y\) is uniformly distributed over the interval [3, c] where \(c > 3\) Given that \(\operatorname { Var } ( Y ) = 3 c - 9\), find
    1. the value of \(c\)
    2. \(\mathrm { P } ( 2 Y - 7 < 20 - Y )\)
    3. \(\mathrm { E } \left( Y ^ { 2 } \right)\)
Edexcel S2 2019 January Q6
  1. (i) (a) State the conditions under which the Poisson distribution may be used as an approximation to the binomial distribution.
A factory produces tyres for bicycles and \(0.25 \%\) of the tyres produced are defective. A company orders 3000 tyres from the factory.
(b) Find, using a Poisson approximation, the probability that there are more than 7 defective tyres in the company’s order.
(ii) At the company \(40 \%\) of employees are known to cycle to work. A random sample of 150 employees is taken. The random variable \(C\) represents the number of employees in the sample who cycle to work.
(a) Describe a suitable sampling frame that can be used to take this sample.
(b) Explain what you understand by the sampling distribution of \(C\) Louis uses a normal approximation to calculate the probability that at most \(\alpha\) employees in the sample cycle to work. He forgets to use a continuity correction and obtains the incorrect probability 0.0668 Find, showing all stages of your working,
(c) the value of \(\alpha\)
(d) the correct probability.
Edexcel S2 2019 January Q7
  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

    \hline &
    \hline \end{tabular}
Edexcel S2 2021 January Q1
  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
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
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
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
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
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
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
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
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).