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Edexcel S2 2016 January Q2
10 marks Moderate -0.3
  1. The continuous random variable \(X\) is uniformly distributed over the interval \([ a , b ]\) Given that \(\mathrm { P } ( 3 < X < 5 ) = \frac { 1 } { 8 }\) and \(\mathrm { E } ( X ) = 4\)
    1. find the value of \(a\) and the value of \(b\)
    2. find the value of the constant, \(c\), such that \(\mathrm { E } ( c X - 2 ) = 0\)
    3. find the exact value of \(\mathrm { E } \left( X ^ { 2 } \right)\)
    4. find \(\mathrm { P } ( 2 X - b > a )\)
    5. Left-handed people make up \(10 \%\) of a population. A random sample of 60 people is taken from this population. The discrete random variable \(Y\) represents the number of left-handed people in the sample.
      1. Write down an expression for the exact value of \(\mathrm { P } ( Y \leqslant 1 )\)
      2. Evaluate your expression, giving your answer to 3 significant figures.
    7. Using a Poisson approximation, estimate \(\mathrm { P } ( Y \leqslant 1 )\)
    8. Using a normal approximation, estimate \(\mathrm { P } ( Y \leqslant 1 )\)
    9. Give a reason why the Poisson approximation is a more suitable estimate of \(\mathrm { P } ( Y \leqslant 1 )\)
Edexcel S2 2016 January Q4
12 marks Standard +0.3
4. A continuous random variable \(X\) has cumulative distribution function $$F ( x ) = \left\{ \begin{array} { l r } 0 & x < 0 \\ \frac { 1 } { 4 } x & 0 \leqslant x \leqslant 1 \\ \frac { 1 } { 20 } x ^ { 4 } + \frac { 1 } { 5 } & 1 < x \leqslant d \\ 1 & x > d \end{array} \right.$$
  1. Show that \(d = 2\)
  2. Find \(\mathrm { P } ( X < 1.5 )\)
  3. Write down the value of the lower quartile of \(X\)
  4. Find the median of \(X\)
  5. Find, to 3 significant figures, the value of \(k\) such that \(\mathrm { P } ( X > 1.9 ) = \mathrm { P } ( X < k )\)
Edexcel S2 2016 January Q5
10 marks Standard +0.3
5. The number of eruptions of a volcano in a 10 year period is modelled by a Poisson distribution with mean 1
  1. Find the probability that this volcano erupts at least once in each of 2 randomly selected 10 year periods.
  2. Find the probability that this volcano does not erupt in a randomly selected 20 year period. The probability that this volcano erupts exactly 4 times in a randomly selected \(w\) year period is 0.0443 to 3 significant figures.
  3. Use the tables to find the value of \(w\) A scientist claims that the mean number of eruptions of this volcano in a 10 year period is more than 1 She selects a 100 year period at random in order to test her claim.
  4. State the null hypothesis for this test.
  5. Determine the critical region for the test at the \(5 \%\) level of significance.
Edexcel S2 2016 January Q6
15 marks Standard +0.3
6. A continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} a x ^ { 2 } + b x & 1 \leqslant x \leqslant 7 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are constants.
  1. Show that \(114 a + 24 b = 1\) Given that \(a = \frac { 1 } { 90 }\)
  2. use algebraic integration to find \(\mathrm { E } ( X )\)
  3. find the cumulative distribution function of \(X\), specifying it for all values of \(x\)
  4. find \(\mathrm { P } ( X > \mathrm { E } ( X ) )\)
  5. use your answer to part (d) to describe the skewness of the distribution.
Edexcel S2 2016 January Q7
12 marks Standard +0.3
  1. A fisherman is known to catch fish at a mean rate of 4 per hour. The number of fish caught by the fisherman in an hour follows a Poisson distribution.
The fisherman takes 5 fishing trips each lasting 1 hour.
  1. Find the probability that this fisherman catches at least 6 fish on exactly 3 of these trips. The fisherman buys some new equipment and wants to test whether or not there is a change in the mean number of fish caught per hour. Given that the fisherman caught 14 fish in a 2 hour period using the new equipment,
  2. carry out the test at the \(5 \%\) level of significance. State your hypotheses clearly.
    (6) by the fisherman in an hour follows a Poisson distribution.
    The fisherman takes 5 fishing trips each lasting 1 hour.
  3. Find the probability that this fisherman catches at least 6 fish on exactly 3 of these trips.
    7.
    \includegraphics[max width=\textwidth, alt={}]{00d4f7c7-6ad5-43c0-8512-16c83cde107a-13_2632_1826_121_123}
Edexcel S2 2017 January Q1
7 marks Easy -1.2
  1. The continuous random variable \(W\) has the normal distribution \(\mathrm { N } \left( 32,4 { } ^ { 2 } \right)\)
    1. Write down the value of \(\mathrm { P } ( W = 36 )\)
    The discrete random variable \(X\) has the binomial distribution \(\mathrm { B } ( 20,0.45 )\)
  2. Find \(\mathrm { P } ( X = 8 )\)
  3. Find the probability that \(X\) lies within one standard deviation of its mean.
Edexcel S2 2017 January Q2
7 marks Moderate -0.8
2. The continuous random variable \(X\) is uniformly distributed over the interval \([ \alpha , \beta ]\) where \(\beta > \alpha\) Given that \(\mathrm { E } ( X ) = 8\)
  1. write down an equation involving \(\alpha\) and \(\beta\) Given also that \(\mathrm { P } ( X \leqslant 13 ) = 0.7\)
  2. find the value of \(\alpha\) and the value of \(\beta\)
  3. find \(\operatorname { Var } ( X )\)
  4. find \(\mathrm { P } ( 5 \leqslant X \leqslant 35 )\)
Edexcel S2 2017 January Q3
16 marks Standard +0.3
3. (a) State the condition under which the normal distribution may be used as an approximation to the Poisson distribution. The number of reported first aid incidents per week at an airport terminal has a Poisson distribution with mean 3.5
(b) Find the modal number of reported first aid incidents in a randomly selected week. Justify your answer. The random variable \(X\) represents the number of reported first aid incidents at this airport terminal in the next 2 weeks.
(c) Find \(\mathrm { P } ( X > 5 )\) (d) Given that there were exactly 6 reported first aid incidents in a 2 week period, find the probability that exactly 4 were reported in the first week.
(e) Using a suitable approximation, find the probability that in the next 40 weeks there will be at least 120 reported first aid incidents.
Edexcel S2 2017 January Q4
10 marks Standard +0.3
  1. The time, in thousands of hours, that a certain electrical component will last is modelled by the random variable \(X\), with probability density function
$$f ( x ) = \begin{cases} \frac { 3 } { 64 } x ^ { 2 } ( 4 - x ) & 0 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$ Using this model, find, by algebraic integration,
  1. the mean number of hours that a component will last,
  2. the standard deviation of \(X\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{ce1f9aa7-cf16-4293-98b1-157eed35b761-06_478_974_1069_479} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows a sketch of the probability density function of the random variable \(X\).
  3. Give a reason why the random variable \(X\) might be unsuitable as a model for the time, in thousands of hours, that these electrical components will last.
  4. Sketch a probability density function of a more realistic model.
Edexcel S2 2017 January Q5
14 marks Standard +0.8
  1. In the manufacture of cloth in a factory, defects occur randomly in the production process at a rate of 2 per \(5 \mathrm {~m} ^ { 2 }\)
The quality control manager randomly selects 12 pieces of cloth each of area \(15 \mathrm {~m} ^ { 2 }\).
  1. Find the probability that exactly half of these 12 pieces of cloth will contain at most 7 defects. The factory introduces a new procedure to manufacture the cloth. After the introduction of this new procedure, the manager takes a random sample of \(25 \mathrm {~m} ^ { 2 }\) of cloth from the next batch produced to test if there has been any change in the rate of defects.
    1. Write down suitable hypotheses for this test.
    2. Describe a suitable test statistic that the manager should use.
    3. Explain what is meant by the critical region for this test.
  3. Using a 5\% level of significance, find the critical region for this test. You should choose the largest critical region for which the probability in each tail is less than 2.5\%
  4. Find the actual significance level for this test.
Edexcel S2 2017 January Q6
7 marks Standard +0.3
  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
14 marks Standard +0.3
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
9 marks Standard +0.3
  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
8 marks Standard +0.8
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
9 marks Moderate -0.3
  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
11 marks Standard +0.3
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
15 marks Standard +0.3
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
8 marks Standard +0.3
  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
15 marks Standard +0.3
  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
7 marks Moderate -0.3
  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
12 marks Standard +0.3
  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
8 marks Moderate -0.3
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
9 marks Standard +0.3
  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
13 marks Standard +0.3
  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
12 marks Moderate -0.3
  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.