Questions — Edexcel S2 (562 questions)

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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 The children are allowed to practise.this game so that this continuous uniform model is no longer applicable.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
  1. 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 .
  3. Calculate the value of \(n\) - The expected number of people with green eyes in a second random sample is 3 .
  4. 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
  1. 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
  1. 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
  1. 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.
Edexcel S2 2011 June Q3
10 marks Challenging +1.2
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6e4af10e-ee8d-493f-bd72-34b231003d97-05_455_1026_242_484} \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 \(0 \leqslant x \leqslant 3 , \mathrm { f } ( x )\) is represented by a curve \(O B\) with equation \(\mathrm { f } ( x ) = k x ^ { 2 }\), where \(k\) is a constant. For \(3 \leqslant x \leqslant a\), where \(a\) is a constant, \(\mathrm { f } ( x )\) is represented by a straight line passing through \(B\) and the point ( \(a , 0\) ). For all other values of \(x , \mathrm { f } ( x ) = 0\).
Given that the mode of \(X =\) the median of \(X\), find
  1. the mode,
  2. the value of \(k\),
  3. the value of \(a\). Without calculating \(\mathrm { E } ( X )\) and with reference to the skewness of the distribution
  4. state, giving your reason, whether \(\mathrm { E } ( X ) < 3 , \mathrm { E } ( X ) = 3\) or \(\mathrm { E } ( X ) > 3\).
Edexcel S2 2011 June Q4
8 marks Standard +0.3
  1. In a game, players select sticks at random from a box containing a large number of sticks of different lengths. The length, in cm , of a randomly chosen stick has a continuous uniform distribution over the interval [7,10].
A stick is selected at random from the box.
  1. Find the probability that the stick is shorter than 9.5 cm . To win a bag of sweets, a player must select 3 sticks and wins if the length of the longest stick is more than 9.5 cm .
  2. Find the probability of winning a bag of sweets. To win a soft toy, a player must select 6 sticks and wins the toy if more than four of the sticks are shorter than 7.6 cm .
  3. Find the probability of winning a soft toy.
Edexcel S2 2011 June Q5
13 marks Standard +0.3
5. Defects occur at random in planks of wood with a constant rate of 0.5 per 10 cm length. Jim buys a plank of length 100 cm .
  1. Find the probability that Jim's plank contains at most 3 defects. Shivani buys 6 planks each of length 100 cm .
  2. Find the probability that fewer than 2 of Shivani's planks contain at most 3 defects.
  3. Using a suitable approximation, estimate the probability that the total number of defects on Shivani's 6 planks is less than 18.
Edexcel S2 2011 June Q6
14 marks Standard +0.3
  1. A shopkeeper knows, from past records, that \(15 \%\) of customers buy an item from the display next to the till. After a refurbishment of the shop, he takes a random sample of 30 customers and finds that only 1 customer has bought an item from the display next to the till.
    1. Stating your hypotheses clearly, and using a \(5 \%\) level of significance, test whether or not there has been a change in the proportion of customers buying an item from the display next to the till.
    During the refurbishment a new sandwich display was installed. Before the refurbishment \(20 \%\) of customers bought sandwiches. The shopkeeper claims that the proportion of customers buying sandwiches has now increased. He selects a random sample of 120 customers and finds that 31 of them have bought sandwiches.
  2. Using a suitable approximation and stating your hypotheses clearly, test the shopkeeper's claim. Use a \(10 \%\) level of significance.
Edexcel S2 2011 June Q7
17 marks Standard +0.3
  1. The continuous random variable \(X\) has probability density function given by
$$f ( x ) = \left\{ \begin{array} { c c } \frac { 3 } { 32 } ( x - 1 ) ( 5 - x ) & 1 \leqslant x \leqslant 5 \\ 0 & \text { otherwise } \end{array} \right.$$
  1. Sketch \(\mathrm { f } ( x )\) showing clearly the points where it meets the \(x\)-axis.
  2. Write down the value of the mean, \(\mu\), of \(X\).
  3. Show that \(\mathrm { E } \left( X ^ { 2 } \right) = 9.8\)
  4. Find the standard deviation, \(\sigma\), of \(X\). The cumulative distribution function of \(X\) is given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 1 \\ \frac { 1 } { 32 } \left( a - 15 x + 9 x ^ { 2 } - x ^ { 3 } \right) & 1 \leqslant x \leqslant 5 \\ 1 & x > 5 \end{array} \right.$$ where \(a\) is a constant.
  5. Find the value of \(a\).
  6. Show that the lower quartile of \(X , q _ { 1 }\), lies between 2.29 and 2.31
  7. Hence find the upper quartile of \(X\), giving your answer to 1 decimal place.
  8. Find, to 2 decimal places, the value of \(k\) so that $$\mathrm { P } ( \mu - k \sigma < X < \mu + k \sigma ) = 0.5$$
Edexcel S2 2012 June Q1
7 marks Moderate -0.3
  1. A manufacturer produces sweets of length \(L \mathrm {~mm}\) where \(L\) has a continuous uniform distribution with range [15, 30].
    1. Find the probability that a randomly selected sweet has a length greater than 24 mm .
    These sweets are randomly packed in bags of 20 sweets.
  2. Find the probability that a randomly selected bag will contain at least 8 sweets with length greater than 24 mm .
  3. Find the probability that 2 randomly selected bags will both contain at least 8 sweets with length greater than 24 mm .