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Edexcel S2 2013 January Q4
14 marks Standard +0.3
4. The continuous random variable \(X\) is uniformly distributed over the interval \([ - 4,6 ]\).
  1. Write down the mean of \(X\).
  2. Find \(\mathrm { P } ( X \leqslant 2.4 )\)
  3. Find \(\mathrm { P } ( - 3 < X - 5 < 3 )\) The continuous random variable \(Y\) is uniformly distributed over the interval \([ a , 4 a ]\).
  4. Use integration to show that \(\mathrm { E } \left( Y ^ { 2 } \right) = 7 a ^ { 2 }\)
  5. Find \(\operatorname { Var } ( Y )\).
  6. Given that \(\mathrm { P } \left( X < \frac { 8 } { 3 } \right) = \mathrm { P } \left( Y < \frac { 8 } { 3 } \right)\), find the value of \(a\).
Edexcel S2 2013 January Q5
10 marks Moderate -0.8
5. The continuous random variable \(T\) is used to model the number of days, \(t\), a mosquito survives after hatching. The probability that the mosquito survives for more than \(t\) days is $$\frac { 225 } { ( t + 15 ) ^ { 2 } } , \quad t \geqslant 0$$
  1. Show that the cumulative distribution function of \(T\) is given by $$\mathrm { F } ( t ) = \begin{cases} 1 - \frac { 225 } { ( t + 15 ) ^ { 2 } } & t \geqslant 0 \\ 0 & \text { otherwise } \end{cases}$$
  2. Find the probability that a randomly selected mosquito will die within 3 days of hatching.
  3. Given that a mosquito survives for 3 days, find the probability that it will survive for at least 5 more days. A large number of mosquitoes hatch on the same day.
  4. Find the number of days after which only \(10 \%\) of these mosquitoes are expected to survive.
Edexcel S2 2013 January Q6
10 marks Standard +0.3
6. (a) Explain what you understand by a hypothesis.
(b) Explain what you understand by a critical region. Mrs George claims that 45\% of voters would vote for her.
In an opinion poll of 20 randomly selected voters it was found that 5 would vote for her.
(c) Test at the \(5 \%\) level of significance whether or not the opinion poll provides evidence to support Mrs George's claim. In a second opinion poll of \(n\) randomly selected people it was found that no one would vote for Mrs George.
(d) Using a \(1 \%\) level of significance, find the smallest value of \(n\) for which the hypothesis \(\mathrm { H } _ { 0 } : p = 0.45\) will be rejected in favour of \(\mathrm { H } _ { 1 } : p < 0.45\)
Edexcel S2 2013 January Q7
15 marks Standard +0.3
7. The continuous random variable \(X\) has the following probability density function $$f ( x ) = \begin{cases} a + b x & 0 \leqslant x \leqslant 5 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are constants.
  1. Show that \(10 a + 25 b = 2\) Given that \(\mathrm { E } ( X ) = \frac { 35 } { 12 }\)
  2. find a second equation in \(a\) and \(b\),
  3. hence find the value of \(a\) and the value of \(b\).
  4. Find, to 3 significant figures, the median of \(X\).
  5. Comment on the skewness. Give a reason for your answer.
Edexcel S2 2001 June Q1
6 marks Easy -1.8
  1. The small village of Tornep has a preservation society which is campaigning for a new by-pass to be built. The society needs to measure
    1. the strength of opinion amongst the residents of Tornep for the scheme and
    2. the flow of traffic through the village on weekdays.
    The society wants to know whether to use a census or a sample survey for each of these measures.
    (a) In each case suggest which they should use and specify a suitable sampling frame. For the measurement of traffic flow through Tornep,
    (b) suggest a suitable statistic and a possible statistical model for this statistic.
Edexcel S2 2001 June Q2
7 marks Moderate -0.3
2. On a stretch of motorway accidents occur at a rate of 0.9 per month.
  1. Show that the probability of no accidents in the next month is 0.407 , to 3 significant figures. Find the probability of
  2. exactly 2 accidents in the next 6 month period,
  3. no accidents in exactly 2 of the next 4 months.
Edexcel S2 2001 June Q3
7 marks Moderate -0.3
3. In a sack containing a large number of beads \(\frac { 1 } { 4 }\) are coloured gold and the remainder are of different colours. A group of children use some of the beads in a craft lesson and do not replace them. Afterwards the teacher wishes to know whether or not the proportion of gold beads left in the sack has changed. He selects a random sample of 20 beads and finds that 2 of them are coloured gold. Stating your hypotheses clearly test, at the \(10 \%\) level of significance, whether or not there is evidence that the proportion of gold beads has changed.
Edexcel S2 2001 June Q4
12 marks Moderate -0.3
4. A company always sends letters by second class post unless they are marked first class. Over a long period of time it has been established that \(20 \%\) of letters to be posted are marked first class. In a random selection of 10 letters to be posted, find the probability that the number marked first class is
  1. at least 3,
  2. fewer than 2 . One Monday morning there are only 12 first class stamps. Given that there are 70 letters to be posted that day,
  3. use a suitable approximation to find the probability that there are enough first class stamps.
  4. State an assumption about these 70 letters that is required in order to make the calculation in part (c) valid.
Edexcel S2 2001 June Q5
12 marks Standard +0.3
5. The maintenance department of a college receives requests for replacement light bulbs at a rate of 2 per week. Find the probability that in a randomly chosen week the number of requests for replacement light bulbs is
  1. exactly 4,
  2. more than 5 . Three weeks before the end of term the maintenance department discovers that there are only 5 light bulbs left.
  3. Find the probability that the department can meet all requests for replacement light bulbs before the end of term. The following term the principal of the college announces a package of new measures to reduce the amount of damage to college property. In the first 4 weeks following this announcement, 3 requests for replacement light bulbs are received.
  4. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not there is evidence that the rate of requests for replacement light bulbs has decreased.
Edexcel S2 2001 June Q6
14 marks Standard +0.3
6. The continuous random variable X has cumulative distribution function \(\mathrm { F } ( x )\) given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 , & x < 1 \\ \frac { 1 } { 27 } \left( - x ^ { 3 } + 6 x ^ { 2 } - 5 \right) , & 1 \leq x \leq 4 \\ 1 , & x > 4 \end{array} \right.$$
  1. Find the probability density function \(\mathrm { f } ( x )\).
  2. Find the mode of \(X\).
  3. Sketch \(\mathrm { f } ( x )\) for all values of \(x\).
  4. Find the mean \(\mu\) of X .
  5. Show that \(\mathrm { F } ( \mu ) > 0.5\).
  6. Show that the median of \(X\) lies between the mode and the mean.
Edexcel S2 2001 June Q7
17 marks Moderate -0.3
7. In a computer game, a star moves across the screen, with constant speed, taking 1 s to travel from one side to the other. The player can stop the star by pressing a key. The object of the game is to stop the star in the middle of the screen by pressing the key exactly 0.5 s after the star first appears. Given that the player actually presses the key \(T\) s after the star first appears, a simple model of the game assumes that \(T\) is a continuous uniform random variable defined over the interval \([ 0,1 ]\).
  1. Write down \(\mathrm { P } ( \mathrm { T } < 0.2 )\).
  2. Write down E(T).
  3. Use integration to find \(\operatorname { Var } ( T )\). A group of 20 children each play this game once.
  4. Find the probability that no more than 4 children stop the star in less than 0.2 s . The children are allowed to practise.this game so that this continuous uniform model is no longer applicable.
  5. Explain how you would expect the mean and variance of T to change. It is found that a more appropriate model of the game when played by experienced players assumes that \(T\) has a probability density function \(\mathrm { g } ( t )\) given by $$g ( t ) = \begin{cases} 4 t , & 0 \leq t \leq 0.5 \\ 4 - 4 t , & 0.5 \leq t \leq 1 , \\ 0 , & \text { otherwise } . \end{cases}$$
  6. Using this model show that \(\mathrm { P } ( T < 0.2 ) = 0.08\). A group of 75 experienced players each played this game once.
  7. Using a suitable approximation, find the probability that more than 7 of them stop the star in less than 0.2 s .
    (4) END
Edexcel S2 2002 June Q1
4 marks Easy -2.0
\begin{enumerate} \item The manager of a leisure club is considering a change to the club rules. The club has a large membership and the manager wants to take the views of the members into consideration before deciding whether or not to make the change.
  1. Explain briefly why the manager might prefer to use a sample survey rather than a census to obtain the views.
  2. Suggest a suitable sampling frame.
  3. Identify the sampling units. \item A random sample \(X _ { 1 } , X _ { 2 } , \ldots , X _ { n }\) is taken from a finite population. A statistic \(Y\) is based on this sample.
Edexcel S2 2002 June Q5
13 marks Standard +0.3
5. A garden centre sells canes of nominal length 150 cm . The canes are bought from a supplier who uses a machine to cut canes of length \(L\) where \(L \sim \mathrm {~N} \left( \mu , 0.3 ^ { 2 } \right)\).
  1. Find the value of \(\mu\), to the nearest 0.1 cm , such that there is only a \(5 \%\) chance that a cane supplied to the garden centre will have length less than 150 cm . A customer buys 10 of these canes from the garden centre.
  2. Find the probability that at most 2 of the canes have length less than 150 cm . Another customer buys 500 canes.
  3. Using a suitable approximation, find the probability that fewer than 35 of the canes will have length less than 150 cm .
    (6)
Edexcel S2 2002 June Q6
14 marks Standard +0.3
6. From past records, a manufacturer of twine knows that faults occur in the twine at random and at a rate of 1.5 per 25 m .
  1. Find the probability that in a randomly chosen 25 m length of twine there will be exactly 4 faults. The twine is usually sold in balls of length 100 m . A customer buys three balls of twine.
  2. Find the probability that only one of them will have fewer than 6 faults. As a special order a ball of twine containing 500 m is produced.
  3. Using a suitable approximation, find the probability that it will contain between 23 and 33 faults inclusive.
    (6)
Edexcel S2 2002 June Q7
17 marks Standard +0.3
7. The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { x } { 15 } , & 0 \leq x \leq 2 \\ \frac { 2 } { 15 } , & 2 < x < 7 \\ \frac { 4 } { 9 } - \frac { 2 x } { 45 } , & 7 \leq x \leq 10 \\ 0 , & \text { otherwise } \end{cases}$$
  1. Sketch \(\mathrm { f } ( x )\) for all values of \(x\).
    1. Find expressions for the cumulative distribution function, \(\mathrm { F } ( x )\), for \(0 \leq x \leq 2\) and for \(7 \leq x \leq 10\).
    2. Show that for \(2 < x < 7 , \mathrm {~F} ( x ) = \frac { 2 x } { 15 } - \frac { 2 } { 15 }\).
    3. Specify \(\mathrm { F } ( x )\) for \(x < 0\) and for \(x > 10\).
  3. Find \(\mathrm { P } ( X \leq 8.2 )\).
  4. Find, to 3 significant figures, \(\mathrm { E } ( X )\).
Edexcel S2 2003 June Q1
4 marks Easy -1.2
  1. Explain briefly what you understand by
    1. a statistic,
    2. a sampling distribution.
    3. (a) Write down the condition needed to approximate a Poisson distribution by a Normal distribution.
    The random variable \(Y \sim \operatorname { Po } ( 30 )\).
  2. Estimate \(\mathrm { P } ( Y > 28 )\).
Edexcel S2 2003 June Q3
9 marks Easy -1.2
3. In a town, \(30 \%\) of residents listen to the local radio station. Four residents are chosen at random.
  1. State the distribution of the random variable \(X\), the number of these four residents that listen to local radio.
  2. On graph paper, draw the probability distribution of \(X\).
  3. Write down the most likely number of these four residents that listen to the local radio station.
  4. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
Edexcel S2 2003 June Q4
12 marks Moderate -0.3
4. (a) Write down the conditions under which the binomial distribution may be a suitable model to use in statistical work. A six-sided die is biased. When the die is thrown the number 5 is twice as likely to appear as any other number. All the other faces are equally likely to appear. The die is thrown repeatedly. Find the probability that
(b) (i) the first 5 will occur on the sixth throw,
(ii) in the first eight throws there will be exactly three 5 s .
Edexcel S2 2003 June Q5
13 marks Moderate -0.8
5. A drinks machine dispenses lemonade into cups. It is electronically controlled to cut off the flow of lemonade randomly between 180 ml and 200 ml . The random variable \(X\) is the volume of lemonade dispensed into a cup.
  1. Specify the probability density function of \(X\) and sketch its graph.
  2. Find the probability that the machine dispenses
    1. less than 183 ml ,
    2. exactly 183 ml .
  3. Calculate the inter-quartile range of \(X\).
  4. Determine the value of \(x\) such that \(\mathrm { P } ( X \geq x ) = 2 \mathrm { P } ( X \leq x )\).
  5. Interpret in words your value of \(x\).
Edexcel S2 2003 June Q6
15 marks Standard +0.3
6. A doctor expects to see, on average, 1 patient per week with a particular disease.
  1. Suggest a suitable model for the distribution of the number of times per week that the doctor sees a patient with the disease. Give a reason for your answer.
  2. Using your model, find the probability that the doctor sees more than 3 patients with the disease in a 4 week period. The doctor decides to send information to his patients to try to reduce the number of patients he sees with the disease. In the first 6 weeks after the information is sent out, the doctor sees 2 patients with the disease.
  3. Test, at the \(5 \%\) level of significance, whether or not there is reason to believe that sending the information has reduced the number of times the doctor sees patients with the disease. State your hypotheses clearly. Medical research into the nature of the disease discovers that it can be passed from one patient to another.
  4. Explain whether or not this research supports your choice of model. Give a reason for your answer.
Edexcel S2 2003 June Q7
15 marks Moderate -0.3
7. A continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \begin{cases} k \left( x ^ { 2 } + 2 x + 1 \right) & - 1 \leq x \leq 0 \\ 0 , & \text { otherwise } \end{cases}$$ where \(k\) is a positive integer.
  1. Show that \(k = 3\). Find
  2. \(\mathrm { E } ( X )\),
  3. the cumulative distribution function \(\mathrm { F } ( x )\),
  4. \(\mathrm { P } ( - 0.3 < X < 0.3 )\). END
Edexcel S2 2004 June Q1
3 marks Easy -1.2
  1. Explain briefly what you understand by
    1. a sampling frame,
    2. a statistic.
    3. The continuous random variable \(X\) is uniformly distributed over the interval \([ - 1,4 ]\).
    Find
  2. \(\mathrm { P } ( X < 2.7 )\),
  3. \(\mathrm { E } ( X )\),
  4. \(\operatorname { Var } ( X )\).
Edexcel S2 2004 June Q3
7 marks Moderate -0.3
3. Brad planted 25 seeds in his greenhouse. He has read in a gardening book that the probability of one of these seeds germinating is 0.25 . Ten of Brad's seeds germinated. He claimed that the gardening book had underestimated this probability. Test, at the \(5 \%\) level of significance, Brad's claim. State your hypotheses clearly.
Edexcel S2 2004 June Q4
13 marks Moderate -0.8
4. (a) State two conditions under which a random variable can be modelled by a binomial distribution. In the production of a certain electronic component it is found that \(10 \%\) are defective.
The component is produced in batches of 20 .
(b) Write down a suitable model for the distribution of defective components in a batch. Find the probability that a batch contains
(c) no defective components,
(d) more than 6 defective components.
(e) Find the mean and the variance of the defective components in a batch. A supplier buys 100 components. The supplier will receive a refund if there are more than 15 defective components.
(f) Using a suitable approximation, find the probability that the supplier will receive a refund.
Edexcel S2 2004 June Q5
15 marks Standard +0.3
5. (a) Explain what you understand by a critical region of a test statistic. The number of breakdowns per day in a large fleet of hire cars has a Poisson distribution with mean \(\frac { 1 } { 7 }\).
(b) Find the probability that on a particular day there are fewer than 2 breakdowns.
(c) Find the probability that during a 14-day period there are at most 4 breakdowns. The cars are maintained at a garage. The garage introduced a weekly check to try to decrease the number of cars that break down. In a randomly selected 28-day period after the checks are introduced, only 1 hire car broke down.
(d) Test, at the \(5 \%\) level of significance, whether or not the mean number of breakdowns has decreased. State your hypotheses clearly.