Questions — Edexcel S2 (494 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Edexcel S2 2006 June Q3
3. An estate agent sells properties at a mean rate of 7 per week.
  1. Suggest a suitable model to represent the number of properties sold in a randomly chosen week. Give two reasons to support your model.
  2. Find the probability that in any randomly chosen week the estate agent sells exactly 5 properties.
  3. Using a suitable approximation find the probability that during a 24 week period the estate agent sells more than 181 properties.
Edexcel S2 2006 June Q4
  1. Breakdowns occur on a particular machine at random at a mean rate of 1.25 per week.
    1. Find the probability that fewer than 3 breakdowns occurred in a randomly chosen week.
    Over a 4 week period the machine was monitored. During this time there were 11 breakdowns.
  2. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the rate of breakdowns has changed over this period. State your hypotheses clearly.
Edexcel S2 2006 June Q5
  1. A manufacturer produces large quantities of coloured mugs. It is known from previous records that \(6 \%\) of the production will be green.
A random sample of 10 mugs was taken from the production line.
  1. Define a suitable distribution to model the number of green mugs in this sample.
  2. Find the probability that there were exactly 3 green mugs in the sample. A random sample of 125 mugs was taken.
  3. Find the probability that there were between 10 and 13 (inclusive) green mugs in this sample, using
    1. a Poisson approximation,
    2. a Normal approximation.
Edexcel S2 2006 June Q6
6. The continuous random variable \(X\) has probability density function $$f ( x ) = \left\{ \begin{array} { c c } \frac { 1 + x } { k } , & 1 \leqslant x \leqslant 4
0 , & \text { otherwise } \end{array} \right.$$
  1. Show that \(k = \frac { 21 } { 2 }\).
  2. Specify fully the cumulative distribution function of \(X\).
  3. Calculate \(\mathrm { E } ( X )\).
  4. Find the value of the median.
  5. Write down the mode.
  6. Explain why the distribution is negatively skewed.
Edexcel S2 2006 June Q7
  1. It is known from past records that 1 in 5 bowls produced in a pottery have minor defects. To monitor production a random sample of 25 bowls was taken and the number of such bowls with defects was recorded.
    1. Using a 5\% level of significance, find critical regions for a two-tailed test of the hypothesis that 1 in 5 bowls have defects. The probability of rejecting, in either tail, should be as close to \(2.5 \%\) as possible.
    2. State the actual significance level of the above test.
    At a later date, a random sample of 20 bowls was taken and 2 of them were found to have defects.
  2. Test, at the \(10 \%\) level of significance, whether or not there is evidence that the proportion of bowls with defects has decreased. State your hypotheses clearly.
Edexcel S2 2009 June Q1
  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
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
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
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
  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
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
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
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 2010 June Q3
3. A rectangle has a perimeter of 20 cm . The length, \(X \mathrm {~cm}\), of one side of this rectangle is uniformly distributed between 1 cm and 7 cm . Find the probability that the length of the longer side of the rectangle is more than 6 cm long.
Edexcel S2 2010 June Q5
  1. A company has a large number of regular users logging onto its website. On average 4 users every hour fail to connect to the company's website at their first attempt.
    1. Explain why the Poisson distribution may be a suitable model in this case.
    Find the probability that, in a randomly chosen 2 hour period,
    1. all users connect at their first attempt,
    2. at least 4 users fail to connect at their first attempt. The company suffered from a virus infecting its computer system. During this infection it was found that the number of users failing to connect at their first attempt, over a 12 hour period, was 60 .
  3. Using a suitable approximation, test whether or not the mean number of users per hour who failed to connect at their first attempt had increased. Use a \(5 \%\) level of significance and state your hypotheses clearly.
Edexcel S2 2010 June Q6
  1. A company claims that a quarter of the bolts sent to them are faulty. To test this claim the number of faulty bolts in a random sample of 50 is recorded.
    1. Give two reasons why a binomial distribution may be a suitable model for the number of faulty bolts in the sample.
    2. Using a 5\% significance level, find the critical region for a two-tailed test of the hypothesis that the probability of a bolt being faulty is \(\frac { 1 } { 4 }\). The probability of rejection in either tail should be as close as possible to 0.025
    3. Find the actual significance level of this test.
    In the sample of 50 the actual number of faulty bolts was 8 .
  2. Comment on the company's claim in the light of this value. Justify your answer. The machine making the bolts was reset and another sample of 50 bolts was taken. Only 5 were found to be faulty.
  3. Test at the \(1 \%\) level of significance whether or not the probability of a faulty bolt has decreased. State your hypotheses clearly.
Edexcel S2 2011 June Q1
  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
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
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
  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
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
  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
  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
  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 .
Edexcel S2 2012 June Q2
2. A test statistic has a distribution \(\mathrm { B } ( 25 , p )\). Given that $$\mathrm { H } _ { 0 } : p = 0.5 \quad \mathrm { H } _ { 1 } : p \neq 0.5$$
  1. find the critical region for the test statistic such that the probability in each tail is as close as possible to \(2.5 \%\).
  2. State the probability of incorrectly rejecting \(\mathrm { H } _ { 0 }\) using this critical region.