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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
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.
  1. In each case suggest which they should use and specify a suitable sampling frame. For the measurement of traffic flow through Tornep,
  2. 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 2005 June Q1
7 marks Easy -1.2
It is estimated that \(4 \%\) of people have green eyes. In a random sample of size \(n\), the expected number of people with green eyes is 5 .
  1. Calculate the value of \(n\). The expected number of people with green eyes in a second random sample is 3 .
  2. Find the standard deviation of the number of people with green eyes in this second sample. expected number of people with green eyes is 5 .
  1. Calculate the value of \(n\) - The expected number of people with green eyes in a second random sample is 3 .
  2. sample. C) T. " D
Edexcel S2 2005 June Q2
11 marks Easy -1.3
2. The continuous random variable \(X\) is uniformly distributed over the interval \([ 2,6 ]\).
  1. Write down the probability density function \(\mathrm { f } ( x )\). Find
  2. \(\mathrm { E } ( X )\),
  3. \(\operatorname { Var } ( X )\),
  4. the cumulative distribution function of \(X\), for all \(x\),
  5. \(\mathrm { P } ( 2.3 < X < 3.4 )\).
Edexcel S2 2005 June Q3
14 marks Easy -1.2
3. The random variable \(X\) is the number of misprints per page in the first draft of a novel.
  1. State two conditions under which a Poisson distribution is a suitable model for \(X\). The number of misprints per page has a Poisson distribution with mean 2.5. Find the probability that
  2. a randomly chosen page has no misprints,
  3. the total number of misprints on 2 randomly chosen pages is more than 7 . The first chapter contains 20 pages.
  4. Using a suitable approximation find, to 2 decimal places, the probability that the chapter will contain less than 40 misprints.
Edexcel S2 2005 June Q4
4 marks Easy -1.8
4. Explain what you understand by
  1. a sampling unit,
  2. a sampling frame,
  3. a sampling distribution.
Edexcel S2 2005 June Q5
7 marks Moderate -0.8
5. In a manufacturing process, \(2 \%\) of the articles produced are defective. A batch of 200 articles is selected.
  1. Giving a justification for your choice, use a suitable approximation to estimate the probability that there are exactly 5 defective articles.
  2. Estimate the probability there are less than 5 defective articles.
Edexcel S2 2005 June Q6
18 marks Standard +0.3
6. A continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) where $$f ( x ) = \begin{cases} k \left( 4 x - x ^ { 3 } \right) , & 0 \leqslant x \leqslant 2 \\ 0 , & \text { otherwise } \end{cases}$$ where \(k\) is a positive integer.
  1. Show that \(k = \frac { 1 } { 4 }\). Find
  2. \(\mathrm { E } ( X )\),
  3. the mode of \(X\),
  4. the median of \(X\).
  5. Comment on the skewness of the distribution.
  6. Sketch f(x).
Edexcel S2 2005 June Q7
14 marks Standard +0.3
7. A drugs company claims that \(75 \%\) of patients suffering from depression recover when treated with a new drug. A random sample of 10 patients with depression is taken from a doctor's records.
  1. Write down a suitable distribution to model the number of patients in this sample who recover when treated with the new drug. Given that the claim is correct,
  2. find the probability that the treatment will be successful for exactly 6 patients. The doctor believes that the claim is incorrect and the percentage who will recover is lower. From her records she took a random sample of 20 patients who had been treated with the new drug. She found that 13 had recovered.
  3. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, the doctor's belief.
  4. From a sample of size 20, find the greatest number of patients who need to recover for the test in part (c) to be significant at the \(1 \%\) level.
Edexcel S2 2009 June Q1
5 marks Standard +0.3
A bag contains a large number of counters of which \(15 \%\) are coloured red. A random sample of 30 counters is selected and the number of red counters is recorded.
  1. Find the probability of no more than 6 red counters in this sample. A second random sample of 30 counters is selected and the number of red counters is recorded.
  2. Using a Poisson approximation, estimate the probability that the total number of red counters in the combined sample of size 60 is less than 13.
Edexcel S2 2009 June Q2
6 marks Standard +0.3
2. An effect of a certain disease is that a small number of the red blood cells are deformed. Emily has this disease and the deformed blood cells occur randomly at a rate of 2.5 per ml of her blood. Following a course of treatment, a random sample of 2 ml of Emily's blood is found to contain only 1 deformed red blood cell. Stating your hypotheses clearly and using a \(5 \%\) level of significance, test whether or not there has been a decrease in the number of deformed red blood cells in Emily's blood.
Edexcel S2 2009 June Q3
5 marks Moderate -0.8
3. A random sample \(X _ { 1 } , X _ { 2 } , \ldots X _ { n }\) is taken from a population with unknown mean \(\mu\) and unknown variance \(\sigma ^ { 2 }\). A statistic \(Y\) is based on this sample.
  1. Explain what you understand by the statistic \(Y\).
  2. Explain what you understand by the sampling distribution of \(Y\).
  3. State, giving a reason which of the following is not a statistic based on this sample.
    1. \(\sum _ { i = 1 } ^ { n } \frac { \left( X _ { i } - \bar { X } \right) ^ { 2 } } { n }\)
    2. \(\sum _ { i = 1 } ^ { n } \left( \frac { X _ { i } - \mu } { \sigma } \right) ^ { 2 }\)
    3. \(\sum _ { i = 1 } ^ { n } X _ { i } ^ { 2 }\)
Edexcel S2 2009 June Q4
8 marks Standard +0.3
4. Past records suggest that \(30 \%\) of customers who buy baked beans from a large supermarket buy them in single tins. A new manager questions whether or not there has been a change in the proportion of customers who buy baked beans in single tins. A random sample of 20 customers who had bought baked beans was taken.
  1. Using a \(10 \%\) level of significance, find the critical region for a two-tailed test to answer the manager's question. You should state the probability of rejection in each tail which should be less than 0.05 .
  2. Write down the actual significance level of a test based on your critical region from part (a). The manager found that 11 customers from the sample of 20 had bought baked beans in single tins.
  3. Comment on this finding in the light of your critical region found in part (a).
Edexcel S2 2009 June Q5
10 marks Standard +0.3
An administrator makes errors in her typing randomly at a rate of 3 errors every 1000 words.
  1. In a document of 2000 words find the probability that the administrator makes 4 or more errors. The administrator is given an 8000 word report to type and she is told that the report will only be accepted if there are 20 or fewer errors.
  2. Use a suitable approximation to calculate the probability that the report is accepted.
Edexcel S2 2009 June Q6
13 marks Standard +0.3
6. The three independent random variables \(A , B\) and \(C\) each has a continuous uniform distribution over the interval \([ 0,5 ]\).
  1. Find \(\mathrm { P } ( A > 3 )\).
  2. Find the probability that \(A , B\) and \(C\) are all greater than 3 . The random variable \(Y\) represents the maximum value of \(A , B\) and \(C\). The cumulative distribution function of \(Y\) is $$\mathrm { F } ( y ) = \begin{cases} 0 & y < 0 \\ \frac { y ^ { 3 } } { 125 } & 0 \leqslant y \leqslant 5 \\ 1 & y > 5 \end{cases}$$
  3. Find the probability density function of \(Y\).
  4. Sketch the probability density function of \(Y\).
  5. Write down the mode of \(Y\).
  6. Find \(\mathrm { E } ( Y )\).
  7. Find \(\mathrm { P } ( Y > 3 )\).
Edexcel S2 2009 June Q7
15 marks Moderate -0.8
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f3fdcd3c-c1c8-4205-a730-eb0bab8607d4-11_471_816_233_548} \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 \(X\). The part of the sketch from \(x = 0\) to \(x = 4\) consists of an isosceles triangle with maximum at ( \(2,0.5\) ).
  1. Write down \(\mathrm { E } ( X )\). The probability density function \(\mathrm { f } ( x )\) can be written in the following form. $$f ( x ) = \begin{cases} a x & 0 \leqslant x < 2 \\ b - a x & 2 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$
  2. Find the values of the constants \(a\) and \(b\).
  3. Show that \(\sigma\), the standard deviation of \(X\), is 0.816 to 3 decimal places.
  4. Find the lower quartile of \(X\).
  5. State, giving a reason, whether \(\mathrm { P } ( 2 - \sigma < X < 2 + \sigma )\) is more or less than 0.5
Edexcel S2 2009 June Q8
13 marks Standard +0.3
8. A cloth manufacturer knows that faults occur randomly in the production process at a rate of 2 every 15 metres.
  1. Find the probability of exactly 4 faults in a 15 metre length of cloth.
  2. Find the probability of more than 10 faults in 60 metres of cloth. A retailer buys a large amount of this cloth and sells it in pieces of length \(x\) metres. He chooses \(x\) so that the probability of no faults in a piece is 0.80
  3. Write down an equation for \(x\) and show that \(x = 1.7\) to 2 significant figures. The retailer sells 1200 of these pieces of cloth. He makes a profit of 60p on each piece of cloth that does not contain a fault but a loss of \(\pounds 1.50\) on any pieces that do contain faults.
  4. Find the retailer's expected profit.
Edexcel S2 2011 June Q1
3 marks Easy -1.8
A factory produces components. Each component has a unique identity number and it is assumed that \(2 \%\) of the components are faulty. On a particular day, a quality control manager wishes to take a random sample of 50 components.
  1. Identify a sampling frame. The statistic \(F\) represents the number of faulty components in the random sample of size 50.
  2. Specify the sampling distribution of \(F\).
Edexcel S2 2011 June Q2
10 marks Standard +0.3
2. A traffic officer monitors the rate at which vehicles pass a fixed point on a motorway. When the rate exceeds 36 vehicles per minute he must switch on some speed restrictions to improve traffic flow.
  1. Suggest a suitable model to describe the number of vehicles passing the fixed point in a 15 s interval. The traffic officer records 12 vehicles passing the fixed point in a 15 s interval.
  2. Stating your hypotheses clearly, and using a \(5 \%\) level of significance, test whether or not the traffic officer has sufficient evidence to switch on the speed restrictions.
  3. Using a \(5 \%\) level of significance, determine the smallest number of vehicles the traffic officer must observe in a 10 s interval in order to have sufficient evidence to switch on the speed restrictions.