Questions — Edexcel S2 (494 questions)

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Edexcel S2 2014 January Q1
  1. The probability of a leaf cutting successfully taking root is 0.05
Find the probability that, in a batch of 10 randomly selected leaf cuttings, the number taking root will be
    1. exactly 1
    2. more than 2 A second random sample of 160 leaf cuttings is selected.
  2. Using a suitable approximation, estimate the probability of at least 10 leaf cuttings taking root.
Edexcel S2 2014 January Q2
2. Bill owns a restaurant. Over the next four weeks Bill decides to carry out a sample survey to obtain the customers’ opinions.
  1. Suggest a suitable sampling frame for the sample survey.
  2. Identify the sampling units.
  3. Give one advantage and one disadvantage of taking a census rather than a sample survey. Bill believes that only \(30 \%\) of customers would like a greater choice on the menu. He takes a random sample of 50 customers and finds that 20 of them would like a greater choice on the menu.
  4. Test, at the \(5 \%\) significance level, whether or not the percentage of customers who would like a greater choice on the menu is more than Bill believes. State your hypotheses clearly.
Edexcel S2 2014 January Q3
  1. The continuous random variable \(X\) has cumulative distribution function given by
$$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 0
\frac { 1 } { 6 } x ( x + 1 ) & 0 \leqslant x \leqslant 2
1 & x > 2 \end{array} \right.$$
  1. Find the value of \(a\) such that \(\mathrm { P } ( X > a ) = 0.4\) Give your answer to 3 significant figures.
  2. Use calculus to find (i) \(\mathrm { E } ( X )\)
    (ii) \(\operatorname { Var } ( X )\).
Edexcel S2 2014 January Q4
  1. The number of telephone calls per hour received by a business is a random variable with distribution \(\operatorname { Po } ( \lambda )\).
Charlotte records the number of calls, \(C\), received in 4 hours. A test of the null hypothesis \(\mathrm { H } _ { 0 } : \lambda = 1.5\) is carried out.
\(\mathrm { H } _ { 0 }\) is rejected if \(C > 10\)
  1. Write down the alternative hypothesis.
  2. Find the significance level of the test. Given that \(\mathrm { P } ( C > 10 ) < 0.1\)
  3. find the largest possible value of \(\lambda\) that can be found by using the tables.
Edexcel S2 2014 January Q5
5. A school photocopier breaks down randomly at a rate of 15 times per year.
  1. Find the probability that there will be exactly 3 breakdowns in the next month.
  2. Show that the probability that there will be at least 2 breakdowns in the next month is 0.355 to 3 decimal places.
  3. Find the probability of at least 2 breakdowns in each of the next 4 months. The teachers would like a new photocopier. The head teacher agrees to monitor the situation for the next 12 months. The head teacher decides he will buy a new photocopier if there is more than 1 month when the photocopier has at least 2 breakdowns.
  4. Find the probability that the head teacher will buy a new photocopier.
Edexcel S2 2014 January Q6
  1. The continuous random variable \(X\) has probability density function given by
$$f ( x ) = \left\{ \begin{array} { c c } k ( x + 1 ) ^ { 2 } & - 1 \leqslant x \leqslant 1
k ( 6 - 2 x ) & 1 < x \leqslant 3
0 & \text { otherwise } \end{array} \right.$$ where \(k\) is a positive constant.
  1. Sketch the graph of \(\mathrm { f } ( x )\).
  2. Show that the value of \(k\) is \(\frac { 3 } { 20 }\)
  3. Define fully the cumulative distribution function \(\mathrm { F } ( x )\).
  4. Find the median of \(X\), giving your answer to 3 significant figures.
Edexcel S2 2014 January Q7
  1. The random variable \(Y \sim \mathrm {~B} ( n , p )\).
Using a normal approximation the probability that \(Y\) is at least 65 is 0.2266 and the probability that \(Y\) is more than 52 is 0.8944 Find the value of \(n\) and the value of \(p\).
Edexcel S2 2015 January Q1
  1. The number of cars caught speeding per day, by a particular camera, has a Poisson distribution with mean 0.8
    1. Find the probability that in a given 4 day period exactly 3 cars will be caught speeding by this camera.
    A car has been caught speeding by this camera.
  2. Find the probability that the period of time that elapses before the next car is caught speeding by this camera is less than 48 hours. Given that 4 cars were caught speeding by this camera in a two day period,
  3. find the probability that 1 was caught on the first day and 3 were caught on the second day. Each car that is caught speeding by this camera is fined \(\pounds 60\)
  4. Using a suitable approximation, find the probability that, in 90 days, the total amount of fines issued will be more than \(\pounds 5000\)
Edexcel S2 2015 January Q2
2. A continuous random variable \(X\) has cumulative distribution function $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 1
\frac { 1 } { 5 } ( x - 1 ) & 1 \leqslant x \leqslant 6
1 & x > 6 \end{array} \right.$$
  1. Find \(\mathrm { P } ( X > 4 )\)
  2. Write down the value of \(\mathrm { P } ( X \neq 4 )\)
  3. Find the probability density function of \(X\), specifying it for all values of \(X\)
  4. Write down the value of \(\mathrm { E } ( X )\)
  5. Find \(\operatorname { Var } ( X )\)
  6. Hence or otherwise find \(\mathrm { E } \left( 3 X ^ { 2 } + 1 \right)\)
Edexcel S2 2015 January Q3
3. Explain what you understand by
  1. a statistic,
  2. a sampling distribution. A factory stores screws in packets. A small packet contains 100 screws and a large packet contains 200 screws. The factory keeps small and large packets in the ratio 4:3 respectively.
  3. Find the mean and the variance of the number of screws in the packets stored at the factory. A random sample of 3 packets is taken from the factory and \(Y _ { 1 } , Y _ { 2 }\) and \(Y _ { 3 }\) denote the number of screws in each of these packets.
  4. List all the possible samples.
  5. Find the sampling distribution of \(\bar { Y }\)
Edexcel S2 2015 January Q4
4. Accidents occur randomly at a crossroads at a rate of 0.5 per month. A researcher records the number of accidents, \(X\), which occur at the crossroads in a year.
  1. Find \(\mathrm { P } ( 5 \leqslant X < 7 )\) A new system is introduced at the crossroads. In the first 18 months, 4 accidents occur at the crossroads.
  2. Test, at the \(5 \%\) level of significance, whether or not there is reason to believe that the new system has led to a reduction in the mean number of accidents per month. State your hypotheses clearly.
Edexcel S2 2015 January Q5
5. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by $$f ( x ) = \left\{ \begin{array} { c c } k \left( x ^ { 2 } + a \right) & - 1 < x \leqslant 2
3 k & 2 < x \leqslant 3
0 & \text { otherwise } \end{array} \right.$$ where \(k\) and \(a\) are constants.
Given that \(\mathrm { E } ( X ) = \frac { 17 } { 12 }\)
  1. find the value of \(k\) and the value of \(a\)
  2. Write down the mode of \(X\)
Edexcel S2 2015 January Q6
6. The Headteacher of a school claims that \(30 \%\) of parents do not support a new curriculum. In a survey of 20 randomly selected parents, the number, \(X\), who do not support the new curriculum is recorded. Assuming that the Headteacher's claim is correct, find
  1. the probability that \(X = 5\)
  2. the mean and the standard deviation of \(X\) The Director of Studies believes that the proportion of parents who do not support the new curriculum is greater than \(30 \%\). Given that in the survey of 20 parents 8 do not support the new curriculum,
  3. test, at the \(5 \%\) level of significance, the Director of Studies' belief. State your hypotheses clearly. The teachers believe that the sample in the original survey was biased and claim that only \(25 \%\) of the parents are in support of the new curriculum. A second random sample, of size \(2 n\), is taken and exactly half of this sample supports the new curriculum. A test is carried out at a 10\% level of significance of the teachers' belief using this sample of size \(2 n\) Using the hypotheses \(\mathrm { H } _ { 0 } : p = 0.25\) and \(\mathrm { H } _ { 1 } : p > 0.25\)
  4. find the minimum value of \(n\) for which the outcome of the test is that the teachers' belief is rejected.
Edexcel S2 2015 January Q7
7. A multiple choice examination paper has \(n\) questions where \(n > 30\) Each question has 5 answers of which only 1 is correct. A pass on the paper is obtained by answering 30 or more questions correctly. The probability of obtaining a pass by randomly guessing the answer to each question should not exceed 0.0228 Use a normal approximation to work out the greatest number of questions that could be used.
Edexcel S2 2016 January Q1
  1. The manager of a clothing shop wishes to investigate how satisfied customers are with the quality of service they receive. A database of the shop's customers is used as a sampling frame for this investigation.
    1. Identify one potential problem with this sampling frame.
    Customers are asked to complete a survey about the quality of service they receive. Past information shows that \(35 \%\) of customers complete the survey. A random sample of 20 customers is taken.
  2. Write down a suitable distribution to model the number of customers in this sample that complete the survey.
  3. Find the probability that more than half of the customers in the sample complete the survey.
Edexcel S2 2016 January Q2
  1. The continuous random variable \(X\) is uniformly distributed over the interval \([ a , b ]\) Given that \(\mathrm { P } ( 3 < X < 5 ) = \frac { 1 } { 8 }\) and \(\mathrm { E } ( X ) = 4\)
    1. find the value of \(a\) and the value of \(b\)
    2. find the value of the constant, \(c\), such that \(\mathrm { E } ( c X - 2 ) = 0\)
    3. find the exact value of \(\mathrm { E } \left( X ^ { 2 } \right)\)
    4. find \(\mathrm { P } ( 2 X - b > a )\)
    5. Left-handed people make up \(10 \%\) of a population. A random sample of 60 people is taken from this population. The discrete random variable \(Y\) represents the number of left-handed people in the sample.
      1. Write down an expression for the exact value of \(\mathrm { P } ( Y \leqslant 1 )\)
      2. Evaluate your expression, giving your answer to 3 significant figures.
    7. Using a Poisson approximation, estimate \(\mathrm { P } ( Y \leqslant 1 )\)
    8. Using a normal approximation, estimate \(\mathrm { P } ( Y \leqslant 1 )\)
    9. Give a reason why the Poisson approximation is a more suitable estimate of \(\mathrm { P } ( Y \leqslant 1 )\)
Edexcel S2 2016 January Q4
4. A continuous random variable \(X\) has cumulative distribution function $$F ( x ) = \left\{ \begin{array} { l r } 0 & x < 0
\frac { 1 } { 4 } x & 0 \leqslant x \leqslant 1
\frac { 1 } { 20 } x ^ { 4 } + \frac { 1 } { 5 } & 1 < x \leqslant d
1 & x > d \end{array} \right.$$
  1. Show that \(d = 2\)
  2. Find \(\mathrm { P } ( X < 1.5 )\)
  3. Write down the value of the lower quartile of \(X\)
  4. Find the median of \(X\)
  5. Find, to 3 significant figures, the value of \(k\) such that \(\mathrm { P } ( X > 1.9 ) = \mathrm { P } ( X < k )\)
Edexcel S2 2016 January Q5
5. The number of eruptions of a volcano in a 10 year period is modelled by a Poisson distribution with mean 1
  1. Find the probability that this volcano erupts at least once in each of 2 randomly selected 10 year periods.
  2. Find the probability that this volcano does not erupt in a randomly selected 20 year period. The probability that this volcano erupts exactly 4 times in a randomly selected \(w\) year period is 0.0443 to 3 significant figures.
  3. Use the tables to find the value of \(w\) A scientist claims that the mean number of eruptions of this volcano in a 10 year period is more than 1 She selects a 100 year period at random in order to test her claim.
  4. State the null hypothesis for this test.
  5. Determine the critical region for the test at the \(5 \%\) level of significance.
Edexcel S2 2016 January Q6
6. A continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} a x ^ { 2 } + b x & 1 \leqslant x \leqslant 7
0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are constants.
  1. Show that \(114 a + 24 b = 1\) Given that \(a = \frac { 1 } { 90 }\)
  2. use algebraic integration to find \(\mathrm { E } ( X )\)
  3. find the cumulative distribution function of \(X\), specifying it for all values of \(x\)
  4. find \(\mathrm { P } ( X > \mathrm { E } ( X ) )\)
  5. use your answer to part (d) to describe the skewness of the distribution.
Edexcel S2 2016 January Q7
  1. A fisherman is known to catch fish at a mean rate of 4 per hour. The number of fish caught by the fisherman in an hour follows a Poisson distribution.
The fisherman takes 5 fishing trips each lasting 1 hour.
  1. Find the probability that this fisherman catches at least 6 fish on exactly 3 of these trips. The fisherman buys some new equipment and wants to test whether or not there is a change in the mean number of fish caught per hour. Given that the fisherman caught 14 fish in a 2 hour period using the new equipment,
  2. carry out the test at the \(5 \%\) level of significance. State your hypotheses clearly.
    (6) by the fisherman in an hour follows a Poisson distribution.
    The fisherman takes 5 fishing trips each lasting 1 hour.
  3. Find the probability that this fisherman catches at least 6 fish on exactly 3 of these trips.
    7.
    \includegraphics[max width=\textwidth, alt={}]{00d4f7c7-6ad5-43c0-8512-16c83cde107a-13_2632_1826_121_123}
Edexcel S2 2017 January Q1
  1. The continuous random variable \(W\) has the normal distribution \(\mathrm { N } \left( 32,4 { } ^ { 2 } \right)\)
    1. Write down the value of \(\mathrm { P } ( W = 36 )\)
    The discrete random variable \(X\) has the binomial distribution \(\mathrm { B } ( 20,0.45 )\)
  2. Find \(\mathrm { P } ( X = 8 )\)
  3. Find the probability that \(X\) lies within one standard deviation of its mean.
Edexcel S2 2017 January Q2
2. The continuous random variable \(X\) is uniformly distributed over the interval \([ \alpha , \beta ]\) where \(\beta > \alpha\) Given that \(\mathrm { E } ( X ) = 8\)
  1. write down an equation involving \(\alpha\) and \(\beta\) Given also that \(\mathrm { P } ( X \leqslant 13 ) = 0.7\)
  2. find the value of \(\alpha\) and the value of \(\beta\)
  3. find \(\operatorname { Var } ( X )\)
  4. find \(\mathrm { P } ( 5 \leqslant X \leqslant 35 )\)
Edexcel S2 2017 January Q3
3. (a) State the condition under which the normal distribution may be used as an approximation to the Poisson distribution. The number of reported first aid incidents per week at an airport terminal has a Poisson distribution with mean 3.5
(b) Find the modal number of reported first aid incidents in a randomly selected week. Justify your answer. The random variable \(X\) represents the number of reported first aid incidents at this airport terminal in the next 2 weeks.
(c) Find \(\mathrm { P } ( X > 5 )\)
(d) Given that there were exactly 6 reported first aid incidents in a 2 week period, find the probability that exactly 4 were reported in the first week.
(e) Using a suitable approximation, find the probability that in the next 40 weeks there will be at least 120 reported first aid incidents.
Edexcel S2 2017 January Q4
  1. The time, in thousands of hours, that a certain electrical component will last is modelled by the random variable \(X\), with probability density function
$$f ( x ) = \begin{cases} \frac { 3 } { 64 } x ^ { 2 } ( 4 - x ) & 0 \leqslant x \leqslant 4
0 & \text { otherwise } \end{cases}$$ Using this model, find, by algebraic integration,
  1. the mean number of hours that a component will last,
  2. the standard deviation of \(X\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{ce1f9aa7-cf16-4293-98b1-157eed35b761-06_478_974_1069_479} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows a sketch of the probability density function of the random variable \(X\).
  3. Give a reason why the random variable \(X\) might be unsuitable as a model for the time, in thousands of hours, that these electrical components will last.
  4. Sketch a probability density function of a more realistic model.
Edexcel S2 2017 January Q5
  1. In the manufacture of cloth in a factory, defects occur randomly in the production process at a rate of 2 per \(5 \mathrm {~m} ^ { 2 }\)
The quality control manager randomly selects 12 pieces of cloth each of area \(15 \mathrm {~m} ^ { 2 }\).
  1. Find the probability that exactly half of these 12 pieces of cloth will contain at most 7 defects. The factory introduces a new procedure to manufacture the cloth. After the introduction of this new procedure, the manager takes a random sample of \(25 \mathrm {~m} ^ { 2 }\) of cloth from the next batch produced to test if there has been any change in the rate of defects.
    1. Write down suitable hypotheses for this test.
    2. Describe a suitable test statistic that the manager should use.
    3. Explain what is meant by the critical region for this test.
  3. Using a 5\% level of significance, find the critical region for this test. You should choose the largest critical region for which the probability in each tail is less than 2.5\%
  4. Find the actual significance level for this test.