Questions S2 (1597 questions)

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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 \(\mathrm { f } ( \mathrm { y } )\)
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.