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Edexcel S2 2017 October Q6
10 marks Standard +0.8
6. A fair 6 -sided die is thrown \(n\) times. The number of sixes, \(X\), is recorded. Using a normal approximation, \(\mathrm { P } ( X < 50 ) = 0.0082\) correct to 4 decimal places. Find the value of \(n\).
(10)
END
Edexcel S2 2018 October Q1
7 marks Standard +0.3
  1. Each day a restaurant opens between 11 am and 11 pm . During its opening hours, the restaurant receives calls for reservations at an average rate of 6 per hour.
    1. Find the probability that the restaurant receives exactly 1 call for a reservation between 6 pm and 7 pm .
    The restaurant distributes leaflets to local residents to try and increase the number of calls for reservations. After distributing the leaflets, it records the number of calls for reservations it receives over a 90 minute period. Given that it receives 14 calls for reservations during the 90 minute period,
  2. test, at the \(5 \%\) level of significance, whether the rate of calls for reservations has increased from 6 per hour. State your hypotheses clearly.
Edexcel S2 2018 October Q2
13 marks Standard +0.3
  1. At a cafe, customers ordering hot drinks order either tea or coffee.
Of all customers ordering hot drinks, \(80 \%\) order tea and \(20 \%\) order coffee. Of those who order tea, \(35 \%\) take sugar and of those who order coffee \(60 \%\) take sugar.
  1. A random sample of 12 customers ordering hot drinks is selected. Find the probability that fewer than 3 of these customers order coffee.
    1. A randomly selected customer who orders a hot drink is chosen. Show that the probability that the customer takes sugar is 0.4
    2. Write down the distribution for the number of customers who take sugar from a random sample of \(n\) customers ordering hot drinks.
  3. A random sample of 10 customers ordering hot drinks is selected.
    1. Find the probability that exactly 4 of these 10 customers take sugar.
    2. Given that at least 3 of these 10 customers take sugar, find the probability that no more than 6 of these 10 customers take sugar.
  4. In a random sample of 150 customers ordering hot drinks, find, using a suitable approximation, the probability that at least half of them take sugar.
Edexcel S2 2018 October Q3
14 marks Standard +0.3
3. The function \(\mathrm { f } ( x )\) is defined as $$f ( x ) = \begin{cases} \frac { 1 } { 9 } ( x + 5 ) ( 3 - x ) & 1 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$ Albert believes that \(\mathrm { f } ( x )\) is a valid probability density function.
  1. Sketch \(\mathrm { f } ( x )\) and comment on Albert's belief. The continuous random variable \(Y\) has probability density function given by $$g ( y ) = \begin{cases} k y \left( 12 - y ^ { 2 } \right) & 1 \leqslant y \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a positive constant.
  2. Use calculus to find the mode of \(Y\)
  3. Use algebraic integration to find the value of \(k\)
  4. Find the median of \(Y\) giving your answer to 3 significant figures.
  5. Describe the skewness of the distribution of \(Y\) giving a reason for your answer.
Edexcel S2 2018 October Q4
9 marks Standard +0.3
4. A bag contains a large number of marbles, each of which is blue or red. A random sample of 3 marbles is taken from the bag. The random variable \(D\) represents the number of blue marbles taken minus the number of red marbles taken. Given that 20\% of the marbles in the bag are blue,
  1. show that \(\mathrm { P } ( D = - 1 ) = 0.384\)
  2. find the sampling distribution of \(D\)
  3. write down the mode of \(D\) Takashi claims that the true proportion of blue marbles is greater than 20\% and tests his claim by selecting a random sample of 12 marbles from the bag.
  4. Find the critical region for this test at the 10\% level of significance.
  5. State the actual significance level of this test. \includegraphics[max width=\textwidth, alt={}, center]{d2f40cdb-917a-4377-88f4-396766a299e2-15_2255_47_314_37}
Edexcel S2 2018 October Q5
11 marks Standard +0.8
5. The random variable \(X\) has cumulative distribution function given by $$F ( x ) = \left\{ \begin{array} { l r } 0 & x < 0 \\ \frac { 1 } { 100 } \left( a x ^ { 3 } + b x ^ { 2 } + 15 x \right) & 0 \leqslant x \leqslant 5 \\ 1 & x > 5 \end{array} \right.$$ Given that \(\mathrm { E } \left( X ^ { 2 } \right) = 6.25\)
  1. show that \(6 a + b = 0\)
  2. find the value of \(a\) and the value of \(b\)
  3. find \(\mathrm { P } ( 3 \leqslant X \leqslant 7 )\)
Edexcel S2 2018 October Q6
9 marks Standard +0.3
  1. One side of a square is measured to the nearest centimetre and this measurement is multiplied by 4 to estimate the perimeter of the square. The random variable, \(W \mathrm {~cm}\), represents the estimated perimeter of the square minus the true perimeter of the square. \(W\) is uniformly distributed over the interval \([ a , b ]\)
    1. Explain why \(a = - 2\) and \(b = 2\)
    The standard deviation of \(W\) is \(\sigma\)
    1. Find \(\sigma\)
    2. Find the probability that the estimated perimeter of the square is within \(\sigma\) of the true perimeter of the square. One side of each of 100 squares are now measured. Using a suitable approximation,
  3. find the probability that \(W\) is greater than 1.9 for at least 5 of these squares.
    VIAN SIHI NI IIIHM ION OCVIUV SIHILNI JMAMALONOOVI4V SIHI NI JIIYM ION OC
Edexcel S2 2018 October Q7
12 marks Standard +0.8
7. Members of a conservation group record the number of sightings of a rare animal. The number of sightings follows a Poisson distribution with a rate of 1 every 2 months.
  1. Find the smallest value of \(n\) such that the probability that there are at least \(n\) sightings in 2 months is less than 0.05
  2. Find the smallest number of months, \(m\), such that the probability of no sightings in \(m\) months is less than 0.05
  3. Find the probability that there is at least 1 sighting per month in each of 3 consecutive months.
  4. Find the probability that the number of sightings in an 8 month period is equal to the expected number of sightings for that period.
  5. Given that there were 4 sightings in a 4 month period, find the probability that there were more sightings in the last 2 months than in the first 2 months.
Edexcel S2 2020 October Q1
8 marks Standard +0.3
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3a781851-e2cc-4379-8b8c-abb3060a6019-02_572_497_299_726} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the probability density function \(\mathrm { f } ( x )\) of the random variable \(X\). For \(1 \leqslant x \leqslant 2 , \mathrm { f } ( x )\) is represented by a curve with equation \(\mathrm { f } ( x ) = k \left( \frac { 1 } { 2 } x ^ { 3 } - 3 x ^ { 2 } + a x + 1 \right)\) where \(k\) and \(a\) are constants. For all other values of \(x , \mathrm { f } ( x ) = 0\)
  1. Use algebraic integration to show that \(k ( 12 a - 33 ) = 8\) Given that \(a = 5\)
  2. calculate the mode of \(X\).
    VI4V SIHI NI JIIIM ION OCVIAN SIHI NI IHMM I ON OOVAYV SIHI NI JIIIM ION OO
Edexcel S2 2020 October Q2
12 marks Moderate -0.3
  1. In the summer Kylie catches a local steam train to work each day. The published arrival time for the train is 10 am.
The random variable \(W\) is the train's actual arrival time minus the published arrival time, in minutes. When the value of \(W\) is positive, the train is late. The cumulative distribution function \(\mathrm { F } ( w )\) is shown in the sketch below. \includegraphics[max width=\textwidth, alt={}, center]{3a781851-e2cc-4379-8b8c-abb3060a6019-06_583_1235_589_349}
  1. Specify fully the probability density function \(\mathrm { f } ( w )\) of \(W\).
  2. Write down the value of \(\mathrm { E } ( \mathrm { W } )\)
  3. Calculate \(\alpha\) such that \(\mathrm { P } ( \alpha \leqslant W \leqslant 1.6 ) = 0.35\) A day is selected at random.
  4. Calculate the probability that on this day the train arrives between 1.2 minutes late and 2.4 minutes late. Given that on this day the train was between 1.2 minutes late and 2.4 minutes late,
  5. calculate the probability that it was more than 2 minutes late. A random sample of 40 days is taken.
  6. Calculate the probability that for at least 10 of these days the train is between 1.2 minutes late and 2.4 minutes late. DO NOT WRITEIN THIS AREA
Edexcel S2 2020 October Q3
15 marks Moderate -0.3
3. A manufacturer produces plates. The proportion of plates that are flawed is \(45 \%\), with flawed plates occurring independently. A random sample of 10 of these plates is selected.
  1. Find the probability that the sample contains
    1. fewer than 2 flawed plates,
    2. at least 6 flawed plates.
      (4) George believes that the proportion of flawed plates is not \(45 \%\). To assess his belief George takes a random sample of 120 plates. The random variable \(F\) represents the number of flawed plates found in the sample.
  2. Using a normal approximation, find the maximum number of plates, \(c\), and the minimum number of plates, \(d\), such that $$\mathrm { P } ( F \leqslant c ) \leqslant 0.05 \text { and } \mathrm { P } ( F \geqslant d ) \leqslant 0.05$$ where \(F \sim \mathrm {~B} ( 120,0.45 )\) The manufacturer claims that, after a change to the production process, the proportion of flawed plates has decreased. A random sample of 30 plates, taken after the change to the production process, contains 8 flawed plates.
  3. Use a suitable hypothesis test, at the \(5 \%\) level of significance, to assess the manufacturer's claim. State your hypotheses clearly. \includegraphics[max width=\textwidth, alt={}, center]{3a781851-e2cc-4379-8b8c-abb3060a6019-11_2255_50_314_34}
Edexcel S2 2020 October Q4
16 marks Moderate -0.8
4. In a peat bog, Common Spotted-orchids occur at a mean rate of 4.5 per \(\mathrm { m } ^ { 2 }\)
  1. Give an assumption, not already stated, that is required for the number of Common Spotted-orchids per \(\mathrm { m } ^ { 2 }\) of the peat bog to follow a Poisson distribution.
    (1) Given that the number of Common Spotted-orchids in \(1 \mathrm {~m} ^ { 2 }\) of the peat bog can be modelled by a Poisson distribution,
  2. find the probability that in a randomly selected \(1 \mathrm {~m} ^ { 2 }\) of the peat bog
    1. there are exactly 6 Common Spotted-orchids,
    2. there are fewer than 10 but more than 4 Common Spotted-orchids.
      (4) Juan believes that by introducing a new management scheme the number of Common Spotted-orchids in the peat bog will increase. After three years under the new management scheme, a randomly selected \(2 \mathrm {~m} ^ { 2 }\) of the peat bog contains 11 Common Spotted-orchids.
  3. Using a \(5 \%\) significance level assess Juan’s belief. State your hypotheses clearly. Assuming that in the peat bog, Common Spotted-orchids still occur at a mean rate of 4.5 per \(\mathrm { m } ^ { 2 }\)
  4. use a normal approximation to find the probability that in a randomly selected \(20 \mathrm {~m} ^ { 2 }\) of the peat bog there are fewer than 70 Common Spotted-orchids. Following a period of dry weather, the probability that there are fewer than 70 Common Spotted-orchids in a randomly selected \(20 \mathrm {~m} ^ { 2 }\) of the peat bog is 0.012 A random sample of 200 non-overlapping \(20 \mathrm {~m} ^ { 2 }\) areas of the peat bog is taken.
  5. Using a suitable approximation, calculate the probability that at most 1 of these areas contains fewer than 70 Common Spotted-orchids. \includegraphics[max width=\textwidth, alt={}, center]{3a781851-e2cc-4379-8b8c-abb3060a6019-15_2255_50_314_34}
Edexcel S2 2020 October Q5
13 marks Standard +0.3
5. The waiting time, \(T\) minutes, of a customer to be served in a local post office has probability density function $$\mathrm { f } ( t ) = \begin{cases} \frac { 1 } { 50 } ( 18 - 2 t ) & 0 \leqslant t \leqslant 3 \\ \frac { 1 } { 20 } & 3 < t \leqslant 5 \\ 0 & \text { otherwise } \end{cases}$$ Given that the mean number of minutes a customer waits to be served is 1.66
  1. use algebraic integration to find \(\operatorname { Var } ( T )\), giving your answer to 3 significant figures.
  2. Find the cumulative distribution function \(\mathrm { F } ( t )\) for all values of \(t\).
  3. Calculate the probability that a randomly chosen customer's waiting time will be more than 2 minutes.
  4. Calculate \(\mathrm { P } ( [ \mathrm { E } ( T ) - 2 ] < T < [ \mathrm { E } ( T ) + 2 ] )\)
    VIXV SIHIANI III IM IONOOVIAV SIHI NI JYHAM ION OOVI4V SIHI NI JLIYM ION OO
Edexcel S2 2020 October Q6
11 marks Standard +0.8
6. (a) Explain what you understand by the sampling distribution of a statistic. A factory produces beads in bags for craft shops. A small bag contains 40 beads, a medium bag contains 80 beads and a large bag contains 150 beads. The factory produces small, medium and large bags in the ratio 5:3:2 respectively. A random sample of 3 bags is taken from the factory.
(b) Find the sampling distribution for the range of the number of beads in the 3 bags in the sample. A random sample of \(n\) sets of 3 bags is taken. The random variable \(Y\) represents the number of these \(n\) sets of 3 bags that have a range of 70
(c) Calculate the minimum value of \(n\) such that \(\mathrm { P } ( Y = 0 ) < 0.2\)
Edexcel S2 2021 October Q1
14 marks Standard +0.3
  1. A research project into food purchases found that \(35 \%\) of people who buy eggs do not buy free range eggs.
A random sample of 30 people who bought eggs is taken. The random variable \(F\) denotes the number of people who do not buy free range eggs.
  1. Find \(\mathrm { P } ( F \geqslant 12 )\)
  2. Find \(\mathrm { P } ( 8 \leqslant F < 15 )\) A farm shop gives 3 loyalty points with every purchase of free range eggs. With every purchase of eggs that are not free range the farm shop gives 1 loyalty point. A random sample of 30 customers who buy eggs from the farm shop is taken.
  3. Find the probability that the total number of points given to these customers is less than 70 The manager of the farm shop believes that the proportion of customers who buy eggs but do not buy free range eggs is more than \(35 \%\) In a survey of 200 customers who buy eggs, 86 do not buy free range eggs. Using a suitable test and a normal approximation,
  4. determine, at the \(5 \%\) level of significance, whether there is evidence to support the manager's belief. State your hypotheses clearly.
Edexcel S2 2021 October Q2
11 marks Standard +0.8
2. (i) The continuous random variable \(X\) is uniformly distributed over the interval \([ a , b ]\) Given that \(\mathrm { P } ( 8 < X < 14 ) = \frac { 1 } { 5 }\) and \(\mathrm { E } ( X ) = 11\)
  1. write down \(\mathrm { P } ( X > 14 )\)
  2. find \(\mathrm { P } ( 6 X > a + b )\) (ii) Susie makes a strip of pasta 45 cm long. She then cuts the strip of pasta, at a randomly chosen point, into two pieces. The random variable \(S\) is the length of the shortest piece of pasta.
  3. Write down the distribution of \(S\)
  4. Calculate the probability that the shortest piece of pasta is less than 12 cm long. Susie makes 20 strips of pasta, all 45 cm long, and separately cuts each strip of pasta, at a randomly chosen point, into two pieces.
  5. Calculate the probability that exactly 6 of the pieces of pasta are less than 12 cm long.
Edexcel S2 2021 October Q3
10 marks Standard +0.8
3. A continuous random variable \(X\) has cumulative distribution function $$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < 0 \\ 4 a x ^ { 2 } & 0 \leqslant x \leqslant 1 \\ a \left( b x ^ { 3 } - x ^ { 4 } + 1 \right) & 1 < x \leqslant 3 \\ 1 & x > 3 \end{array} \right.$$ where \(a\) and \(b\) are positive constants.
  1. Show that \(b = 4\)
  2. Find the exact value of \(a\)
  3. Find \(\mathrm { P } ( X > 2.25 )\)
  4. Showing your working clearly,
    1. sketch the probability density function of \(X\)
    2. calculate the mode of \(X\)
Edexcel S2 2021 October Q4
15 marks Challenging +1.2
  1. The number of cars entering a safari park per 10 -minute period can be modelled by a Poisson distribution with mean 6
    1. Find the probability that in a given 10 -minute period exactly 8 cars will enter the safari park.
    2. Find the smallest value of \(n\) such that the probability that at least \(n\) cars enter the safari park in 10 minutes is less than 0.05
    The probability that no cars enter the safari park in \(m\) minutes, where \(m\) is an integer, is less than 0.05
  2. Find the smallest value of \(m\) A car enters the safari park.
  3. Find the probability that there is less than 5 minutes before the next car enters the safari park. Given that exactly 15 cars entered the safari park in a 30-minute period,
  4. find the probability that exactly 1 car entered the safari park in the first 5 minutes of the 30-minute period. Aston claims that the mean number of cars entering the safari park per 10-minute period is more than 6 He selects a 15-minute period at random in order to test whether there is evidence to support his claim.
  5. Determine the critical region for the test at the \(5 \%\) level of significance.
Edexcel S2 2021 October Q5
8 marks Standard +0.3
  1. A bag contains a large number of counters.
40\% of the counters are numbered 1 \(35 \%\) of the counters are numbered 2 \(25 \%\) of the counters are numbered 3 In a game Alif draws two counters at random from the bag. His score is 4 times the number on the first counter minus 2 times the number on the second counter.
  1. Show that Alif gets a score of 8 with probability 0.0875
  2. Find the sampling distribution of Alif's score.
  3. Calculate Alif's expected score.
Edexcel S2 2021 October Q6
17 marks Standard +0.3
6. The continuous random variable \(Y\) has probability density function \(\mathrm { f } ( y )\) given by $$f ( y ) = \begin{cases} \frac { 1 } { 14 } ( y + 2 ) & - 1 < y \leqslant 1 \\ \frac { 3 } { 14 } & 1 < y \leqslant 3 \\ \frac { 1 } { 14 } ( 6 - y ) & 3 < y \leqslant 5 \\ 0 & \text { otherwise } \end{cases}$$
  1. Sketch the probability density function \(\mathrm { f } ( \mathrm { y } )\) Given that \(\mathrm { E } \left( Y ^ { 2 } \right) = \frac { 131 } { 21 }\)
  2. find \(\operatorname { Var } ( 2 Y - 3 )\) The cumulative distribution function of \(Y\) is \(\mathrm { F } ( y )\)
  3. Show that \(\mathrm { F } ( y ) = \frac { 1 } { 14 } \left( \frac { y ^ { 2 } } { 2 } + 2 y + \frac { 3 } { 2 } \right)\) for \(- 1 < y \leqslant 1\)
  4. Find \(\mathrm { F } ( y )\) for all values of \(y\)
  5. Find the exact value of the 30th percentile of \(Y\)
  6. Find \(\mathrm { P } ( 4 Y \leqslant 5 \mid Y \leqslant 3 )\)
Edexcel S2 2022 October Q1
11 marks Standard +0.3
  1. Bhavna produces rolls of cloth. She knows that faults occur randomly in her cloth at a mean rate of 1.5 every 15 metres.
    1. Find the probability that in 15 metres of her cloth there are
      1. less than 3 faults,
      2. at least 6 faults.
    Each roll contains 100 metres of Bhavna's cloth.
    She selects 15 rolls at random.
  2. Find the probability that exactly 10 of these rolls each have fewer than 13 faults. Bhavna decides to sell her cloth in pieces.
    Each piece of her cloth is 4 metres long.
    The cost to make each piece is \(\pounds 5.00\) She sells each piece of her cloth that contains no faults for \(\pounds 7.40\) She sells each piece of her cloth that contains faults for \(\pounds 2.00\)
  3. Find the expected profit that Bhavna will make on each piece of her cloth that she sells.
Edexcel S2 2022 October Q2
15 marks Moderate -0.3
  1. A random variable \(X\) has probability density function given by
$$f ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { 4 } & - \frac { 1 } { 2 } \leqslant x < \frac { 1 } { 2 } \\ 2 x - \frac { 3 } { 4 } & \frac { 1 } { 2 } \leqslant x \leqslant k \\ 0 & \text { otherwise } \end{array} \right.$$ where \(k\) is a positive constant.
  1. Sketch the graph of \(\mathrm { f } ( x )\)
  2. By forming and solving an equation in \(k\), show that \(k = 1.25\)
  3. Use calculus to find \(\mathrm { E } ( X )\)
  4. Calculate the interquartile range of \(X\)
Edexcel S2 2022 October Q3
10 marks Standard +0.3
  1. A company produces packets of sunflower seeds. Each packet contains 40 seeds. The company claims that, on average, only 35\% of its sunflower seeds do not germinate.
A packet is selected at random.
  1. Using a \(5 \%\) level of significance, find an appropriate critical region for a two-tailed test that the proportion of sunflower seeds that do not germinate is 0.35 You should state your hypotheses clearly and state the probability, which should be as close as possible to \(2.5 \%\), for each tail of your critical region.
  2. Write down the actual significance level of this test. Past records suggest that \(2.8 \%\) of the company's sunflower seeds grow to a height of more than 3 metres.
    A random sample of 250 of the company's sunflower seeds is taken and 11 of them grow to a height of more than 3 metres.
  3. Using a suitable approximation test, at the \(5 \%\) level of significance, whether or not there is evidence that the proportion of sunflower seeds that grow to a height of more than 3 metres is now greater than \(2.8 \%\) State your hypotheses clearly.
Edexcel S2 2022 October Q4
9 marks Standard +0.3
  1. The probability that a person completes a particular task in less than 15 minutes is 0.4 Jeffrey selects 20 people at random and asks them to complete the task. The random variable, \(X\), represents the number of people who complete the task in less than 15 minutes.
    1. Find \(\mathrm { P } ( 5 \leqslant X < 8 )\)
    Mia takes a random sample of 140 people.
    Using a normal approximation, the probability that fewer than \(n\) of these 140 people complete the task in less than 15 minutes is 0.0239 to 4 decimal places.
  2. Find the value of \(n\) Show your working clearly.
Edexcel S2 2022 October Q5
9 marks Standard +0.3
  1. The continuous random variable \(X\) has cumulative distribution function given by
$$\mathrm { F } ( x ) = \left\{ \begin{array} { c r } 0 & x < 3 \\ \frac { 1 } { 6 } ( x - 3 ) ^ { 2 } & 3 \leqslant x < 4 \\ \frac { x } { 3 } - \frac { 7 } { 6 } & 4 \leqslant x < c \\ 1 - \frac { 1 } { 6 } ( d - x ) ^ { 2 } & c \leqslant x < 7 \\ 1 & x \geqslant 7 \end{array} \right.$$ where \(c\) and \(d\) are constants.
  1. Show that \(c = 6\)
  2. Find \(\mathrm { P } ( X > 3.5 )\)
  3. Find \(\mathrm { P } ( X > 4.5 \mid 3.5 < X < 5.5 )\)