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 2011 January Q2
2. A student takes a multiple choice test. The test is made up of 10 questions each with 5 possible answers. The student gets 4 questions correct. Her teacher claims she was guessing the answers. Using a one tailed test, at the \(5 \%\) level of significance, test whether or not there is evidence to reject the teacher's claim. State your hypotheses clearly.
Edexcel S2 2011 January Q3
3. The continuous random variable \(X\) is uniformly distributed over the interval \([ - 1,3 ]\). Find
  1. \(\mathrm { E } ( X )\)
  2. \(\operatorname { Var } ( X )\)
  3. \(\mathrm { E } \left( X ^ { 2 } \right)\)
  4. \(\mathrm { P } ( X < 1.4 )\) A total of 40 observations of \(X\) are made.
  5. Find the probability that at least 10 of these observations are negative.
Edexcel S2 2011 January Q4
  1. Richard regularly travels to work on a ferry. Over a long period of time, Richard has found that the ferry is late on average 2 times every week. The company buys a new ferry to improve the service. In the 4-week period after the new ferry is launched, Richard finds the ferry is late 3 times and claims the service has improved. Assuming that the number of times the ferry is late has a Poisson distribution, test Richard's claim at the \(5 \%\) level of significance. State your hypotheses clearly.
    (6)
  2. A continuous random variable \(X\) has the probability density function \(\mathrm { f } ( x )\) shown in Figure 1.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{58e5aa9e-f177-48ad-8bb8-54c0e2c21e6d-07_591_689_358_630} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure}
  1. Show that \(\mathrm { f } ( x ) = 4 - 8 x\) for \(0 \leqslant x \leqslant 0.5\) and specify \(\mathrm { f } ( x )\) for all real values of \(x\).
  2. Find the cumulative distribution function \(\mathrm { F } ( x )\).
  3. Find the median of \(X\).
  4. Write down the mode of \(X\).
  5. State, with a reason, the skewness of \(X\).
Edexcel S2 2011 January Q6
6. Cars arrive at a motorway toll booth at an average rate of 150 per hour.
  1. Suggest a suitable distribution to model the number of cars arriving at the toll booth, \(X\), per minute.
  2. State clearly any assumptions you have made by suggesting this model. Using your model,
  3. find the probability that in any given minute
    1. no cars arrive,
    2. more than 3 cars arrive.
  4. In any given 4 minute period, find \(m\) such that \(\mathrm { P } ( X > m ) = 0.0487\)
  5. Using a suitable approximation find the probability that fewer than 15 cars arrive in any given 10 minute period.
    January 2011
Edexcel S2 2011 January Q7
7. The queuing time in minutes, \(X\), of a customer at a post office is modelled by the probability density function $$f ( x ) = \begin{cases} k x \left( 81 - x ^ { 2 } \right) & 0 \leqslant x \leqslant 9
0 & \text { otherwise } \end{cases}$$
  1. Show that \(k = \frac { 4 } { 6561 }\). Using integration, find
  2. the mean queuing time of a customer,
  3. the probability that a customer will queue for more than 5 minutes. Three independent customers shop at the post office.
  4. Find the probability that at least 2 of the customers queue for more than 5 minutes.
Edexcel S2 2012 January Q1
  1. The time in minutes that Elaine takes to checkout at her local supermarket follows a continuous uniform distribution defined over the interval [3,9].
Find
  1. Elaine's expected checkout time,
  2. the variance of the time taken to checkout at the supermarket,
  3. the probability that Elaine will take more than 7 minutes to checkout. Given that Elaine has already spent 4 minutes at the checkout,
  4. find the probability that she will take a total of less than 6 minutes to checkout.
Edexcel S2 2012 January Q2
2. David claims that the weather forecasts produced by local radio are no better than those achieved by tossing a fair coin and predicting rain if a head is obtained or no rain if a tail is obtained. He records the weather for 30 randomly selected days. The local radio forecast is correct on 21 of these days. Test David's claim at the \(5 \%\) level of significance. State your hypotheses clearly.
Edexcel S2 2012 January Q3
3. The probability of a telesales representative making a sale on a customer call is 0.15 Find the probability that
  1. no sales are made in 10 calls,
  2. more than 3 sales are made in 20 calls. Representatives are required to achieve a mean of at least 5 sales each day.
  3. Find the least number of calls each day a representative should make to achieve this requirement.
  4. Calculate the least number of calls that need to be made by a representative for the probability of at least 1 sale to exceed 0.95
Edexcel S2 2012 January Q4
4. A website receives hits at a rate of 300 per hour.
  1. State a distribution that is suitable to model the number of hits obtained during a 1 minute interval.
  2. State two reasons for your answer to part (a). Find the probability of
  3. 10 hits in a given minute,
  4. at least 15 hits in 2 minutes. The website will go down if there are more than 70 hits in 10 minutes.
  5. Using a suitable approximation, find the probability that the website will go down in a particular 10 minute interval.
Edexcel S2 2012 January Q5
  1. The probability of an electrical component being defective is 0.075 The component is supplied in boxes of 120
    1. Using a suitable approximation, estimate the probability that there are more than 3 defective components in a box.
    A retailer buys 2 boxes of components.
  2. Estimate the probability that there are at least 4 defective components in each box.
Edexcel S2 2012 January Q6
6. A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 2 } & 0 \leqslant x < 1
x - \frac { 1 } { 2 } & 1 \leqslant x \leqslant k
0 & \text { otherwise } \end{cases}$$ where \(k\) is a positive constant.
  1. Sketch the graph of \(\mathrm { f } ( x )\).
  2. Show that \(k = \frac { 1 } { 2 } ( 1 + \sqrt { 5 } )\).
  3. Define fully the cumulative distribution function \(\mathrm { F } ( x )\).
  4. Find \(\mathrm { P } ( 0.5 < X < 1.5 )\).
  5. Write down the median of \(X\) and the mode of \(X\).
  6. Describe the skewness of the distribution of \(X\). Give a reason for your answer.
Edexcel S2 2012 January Q7
7. (a) Explain briefly what you understand by
  1. a critical region of a test statistic,
  2. the level of significance of a hypothesis test.
    (b) An estate agent has been selling houses at a rate of 8 per month. She believes that the rate of sales will decrease in the next month.
  3. Using a \(5 \%\) level of significance, find the critical region for a one tailed test of the hypothesis that the rate of sales will decrease from 8 per month.
  4. Write down the actual significance level of the test in part (b)(i). The estate agent is surprised to find that she actually sold 13 houses in the next month. She now claims that this is evidence of an increase in the rate of sales per month.
    (c) Test the estate agent's claim at the \(5 \%\) level of significance. State your hypotheses clearly.
Edexcel S2 2013 January Q1
  1. (a) Write down the conditions under which the Poisson distribution can be used as an approximation to the binomial distribution.
The probability of any one letter being delivered to the wrong house is 0.01 On a randomly selected day Peter delivers 1000 letters.
(b) Using a Poisson approximation, find the probability that Peter delivers at least 4 letters to the wrong house. Give your answer to 4 decimal places.
Edexcel S2 2013 January Q2
2. In a village, power cuts occur randomly at a rate of 3 per year.
  1. Find the probability that in any given year there will be
    1. exactly 7 power cuts,
    2. at least 4 power cuts.
  2. Use a suitable approximation to find the probability that in the next 10 years the number of power cuts will be less than 20
Edexcel S2 2013 January Q3
  1. A random variable \(X\) has the distribution \(\mathrm { B } ( 12 , p )\).
    1. Given that \(p = 0.25\) find
      1. \(\mathrm { P } ( X < 5 )\)
      2. \(\mathrm { P } ( X \geqslant 7 )\)
    2. Given that \(\mathrm { P } ( X = 0 ) = 0.05\), find the value of \(p\) to 3 decimal places.
    3. Given that the variance of \(X\) is 1.92 , find the possible values of \(p\).
Edexcel S2 2013 January Q4
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
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
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
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
  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
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
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
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
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
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.