Questions — Edexcel S2 (562 questions)

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Edexcel S2 2018 June Q4
10 marks Moderate -0.3
  1. David aims to catch the train to work each morning. The scheduled departure time of the train is 0830
The number of minutes after 0830 that the train departs may be modelled by the random variable \(X\). Given that \(X\) has a continuous uniform distribution over \([ \alpha , \beta ]\) and that \(\mathrm { E } ( X ) = 4\) and \(\operatorname { Var } ( X ) = 12\)
  1. find the value of \(\alpha\) and the value of \(\beta\). Each morning, the probability that David oversleeps is 0.05 If David oversleeps he will be late for work. If he does not oversleep he will be in time to catch the train, but will be late for work if the train departs after 0835
  2. Find the probability that David will be late for work. Given that David is late for work,
  3. find the probability that he overslept.
Edexcel S2 2018 June Q5
16 marks Standard +0.3
5. Past records show that the proportion of customers buying organic vegetables from Tesson supermarket is 0.35 During a particular day, a random sample of 40 customers from Tesson supermarket was taken and 18 of them bought organic vegetables.
  1. Test, at the \(5 \%\) level of significance, whether or not this provides evidence that the proportion of customers who bought organic vegetables has increased. State your hypotheses clearly. The manager of Tesson supermarket claims that the proportion of customers buying organic eggs is different from the proportion of those buying organic vegetables. To test this claim the manager decides to take a random sample of 50 customers.
  2. Using a \(5 \%\) level of significance, find the critical region to enable the Tesson supermarket manager to test her claim. The probability for each tail of the region should be as close as possible to \(2.5 \%\) During a particular day, a random sample of 50 customers from Tesson supermarket is taken and 8 of them bought organic eggs.
  3. Using your answer to part (b), state whether or not this sample supports the manager's claim. Use a \(5 \%\) level of significance.
  4. State the actual significance level of this test. The proportion of customers who buy organic fruit from Tesson supermarket is 0.2 During a particular day, a random sample of 200 customers from Tesson supermarket is taken. Using a suitable approximation, the probability that fewer than \(n\) of these customers bought organic fruit is 0.0465 correct to 4 decimal places.
  5. Find the value of \(n\).
Edexcel S2 2018 June Q6
10 marks Challenging +1.2
  1. The continuous random variable \(X\) has the following cumulative distribution function
$$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x \leqslant 1 \\ \frac { 4 } { 15 } ( x - 1 ) & 1 < x \leqslant 2 \\ k \left( \frac { a x ^ { 3 } } { 3 } - \frac { x ^ { 4 } } { 4 } \right) + b & 2 < x \leqslant 4 \\ 1 & x > 4 \end{array} \right.$$ where \(k , a\) and \(b\) are constants.
Given that the mode of \(X\) is \(\frac { 8 } { 3 }\)
  1. show that \(a = 4\)
  2. Find \(\mathrm { P } ( X < 2.5 )\) giving your answer to 3 significant figures.
Edexcel S2 Q1
9 marks Standard +0.3
  1. The lifetime, in tens of hours, of a certain delicate electrical component can be modelled by the random variable \(X\) with probability density function
$$f ( x ) = \begin{cases} \frac { 1 } { 42 } x , & 0 \leq x < 6 \\ \frac { 1 } { 7 } & 6 \leq x \leq 10 \\ 0 , & \text { otherwise } \end{cases}$$
  1. Sketch \(\mathrm { f } ( x )\) for all values of \(x\).
  2. Find the probability that a component lasts at least 50 hours. A particular device requires two of these components and it will not operate if one or more of the components fail. The device has just been fitted with two new components and the lifetimes of these two components are independent.
  3. Find the probability that the device breaks down within the next 50 hours.
Edexcel S2 Q2
11 marks Standard +0.3
2. The continuous random variable \(X\) represents the error, in mm, made when a machine cuts piping to a target length. The distribution of \(X\) is rectangular over the interval \([ - 5.0,5.0 ]\). Find
  1. \(\mathrm { P } ( X < - 4.2 )\),
  2. \(\mathrm { P } ( | X | < 1.5 )\). A supervisor checks a random sample of 10 lengths of piping cut by the machine.
  3. Find the probability that more than half of them are within 1.5 cm of the target length.
    (3 marks)
    If \(X < - 4.2\), the length of piping cannot be used. At the end of each day the supervisor checks a random sample of 60 lengths of piping.
  4. Use a suitable approximation to estimate the probability that no more than 2 of these lengths of piping cannot be used.
    (5 marks)
Edexcel S2 Q3
11 marks Moderate -0.8
3. An athletics teacher has kept careful records over the past 20 years of results from school sports days. There are always 10 competitors in the javelin competition. Each competitor is allowed 3 attempts and the teacher has a record of the distances thrown by each competitor at each attempt. The random variable \(D\) represents the greatest distance thrown by each competitor and the random variable \(A\) represents the number of the attempt in which the competitor achieved their greatest distance.
  1. State which of the two random variables \(D\) or \(A\) is continuous. A new athletics coach wishes to take a random sample of the records of 36 javelin competitors.
  2. Specify a suitable sampling frame and explain how such a sample could be taken.
    (2 marks)
    The coach assumes that \(\mathrm { P } ( A = 2 ) = \frac { 1 } { 3 }\), and is therefore surprised to find that 20 of the 36 competitors in the sample achieved their greatest distance on their second attempt. Using a suitable approximation, and assuming that \(\mathrm { P } ( A = 2 ) = \frac { 1 } { 3 }\),
  3. find the probability that at least 20 of the competitors achieved their greatest distance on their second attempt.
    (6 marks)
  4. Comment on the assumption that \(\mathrm { P } ( A = 2 ) = \frac { 1 } { 3 }\).
Edexcel S2 Q4
12 marks Standard +0.3
4. From past records a manufacturer of glass vases knows that \(15 \%\) of the production have slight defects. To monitor the production, a random sample of 20 vases is checked each day and the number of vases with slight defects is recorded.
  1. Using a 5\% significance level, find the critical regions for a two-tailed test of the hypothesis that the probability of a vase with slight defects is 0.15 . The probability of rejecting, in either tail, should be as close as possible to \(2.5 \%\).
  2. State the actual significance level of the test described in part (a). A shop sells these vases at a rate of 2.5 per week. In the 4 weeks of December the shop sold 15 vases.
  3. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not there is evidence that the rate of sales per week had increased in December.
    (6 marks)
Edexcel S2 Q5
14 marks Standard +0.3
5. The continuous random variable \(T\) represents the time in hours that students spend on homework. The cumulative distribution function of \(T\) is $$\mathrm { F } ( t ) = \begin{cases} 0 , & t < 0 \\ k \left( 2 t ^ { 3 } - t ^ { 4 } \right) & 0 \leq t \leq 1.5 \\ 1 , & t > 1.5 \end{cases}$$ where \(k\) is a positive constant.
  1. Show that \(k = \frac { 16 } { 27 }\).
  2. Find the proportion of students who spend more than 1 hour on homework.
  3. Find the probability density function \(\mathrm { f } ( t )\) of \(T\).
  4. Show that \(\mathrm { E } ( T ) = 0.9\).
  5. Show that \(\mathrm { F } ( \mathrm { E } ( T ) ) = 0.4752\). A student is selected at random. Given that the student spent more than the mean amount of time on homework,
  6. find the probability that this student spent more than 1 hour on homework.
Edexcel S2 Q6
18 marks Standard +0.8
6. On a typical weekday morning customers arrive at a village post office independently and at a rate of 3 per 10 minute period. Find the probability that
  1. at least 4 customers arrive in the next 10 minutes,
  2. no more than 7 customers arrive between 11.00 a.m. and 11.30 a.m. The period from 11.00 a.m. to 11.30 a.m. next Tuesday morning will be divided into 6 periods of 5 minutes each.
  3. Find the probability that no customers arrive in at most one of these periods. The post office is open for \(3 \frac { 1 } { 2 }\) hours on Wednesday mornings.
  4. Using a suitable approximation, estimate the probability that more than 49 customers arrive at the post office next Wednesday morning. END
Edexcel S2 Q1
4 marks Easy -1.8
  1. (a) Briefly describe the difference between a census and a sample survey.
    (b) Illustrate the difference by considering the case of a village council which has to decide whether or not to build a new village hall.
Given that the council decides to use a sample survey,
(c) suggest suitable sampling units.
Edexcel S2 Q2
6 marks Moderate -0.3
2. The number of copies of The Statistician that a newsagent sells each week is modelled by a Poisson distribution. On average, he sells 1.5 copies per week.
  1. Find the probability that he sells no copies in a particular week.
  2. If he stocks 5 copies each week, find the probability he will not have enough copies to meet that week's demand.
  3. Find the minimum number of copies that he should stock in order to have at least a \(95 \%\) probability of being able to satisfy the week's demand.
Edexcel S2 Q3
10 marks Standard +0.3
3. A die is rolled 60 times, and results in 16 sixes.
  1. Use a suitable approximation to test, at the \(5 \%\) significance level, whether the probability of scoring a six is \(\frac { 1 } { 6 }\) or not. State your hypotheses clearly.
  2. Describe how you would change the test if you wished to investigate whether the probability of scoring a six is greater than \(\frac { 1 } { 6 }\). Carry out this modified test.
Edexcel S2 Q4
12 marks Standard +0.3
4. A continuous random variable \(X\) has the cumulative distribution function $$\mathrm { F } ( x ) = \left\{ \begin{array} { l } 0 \\ \frac { 1 } { 84 } \left( x ^ { 2 } - 16 \right) \\ 1 \end{array} \right.$$ $$\begin{aligned} & x < 4 , \\ & 4 \leq x \leq 10 , \\ & x > 10 . \end{aligned}$$
  1. Find the median value of \(X\).
  2. Find the interquartile range for \(X\).
  3. Find the probability density function \(\mathrm { f } ( x )\) of \(X\).
  4. Sketch the graph of \(\mathrm { f } ( x )\) and hence write down the mode of \(X\), explaining how you obtain your answer from the graph. \section*{STATISTICS 2 (A) TEST PAPER 1 Page 2}
Edexcel S2 Q5
12 marks Standard +0.3
  1. Lupin seeds are sold in packets of 15 . On average, 9 seeds in a packet are green and 6 are red. Find, to 2 decimal places, the probability that in any particular packet there are
    1. less than 2 red seeds,
    2. more red than green seeds.
    The seeds from 10 packets are then combined together.
  2. Use a suitable approximation to find the probability that the total number of green seeds is more than 100 .
Edexcel S2 Q6
14 marks Standard +0.3
6. Patients suffering from 'flu are treated with a drug. The number of days, \(t\), that it then takes for them to recover is modelled by the continuous random variable \(T\) with the probability density function $$\begin{array} { l l } \mathrm { f } ( t ) = \frac { 3 t ^ { 2 } ( 4 - t ) } { 64 } & 0 \leq t \leq 4 \\ \mathrm { f } ( t ) = 0 & \text { otherwise. } \end{array}$$
  1. Find the mean and standard deviation of \(T\).
  2. Find the probability that a patient takes more than 3 days to recover.
  3. Two patients are selected at random. Find the probability that they both recover within three days.
  4. Comment on the suitability of the model.
Edexcel S2 Q7
17 marks Standard +0.8
7. In a certain field, daisies are randomly distributed, at an average density of 0.8 daisies per \(\mathrm { cm } ^ { 2 }\). One particular patch, of area \(1 \mathrm {~cm} ^ { 2 }\), is selected at random. Assuming that the number of daisies per \(\mathrm { cm } ^ { 2 }\) has a Poisson distribution,
  1. find the probability that the chosen patch contains
    1. no daisies,
    2. one daisy. Ten such patches are chosen. Using your answers to part (a),
  2. find the probability that the total number of daisies is less than two.
  3. By considering the distribution of daisies over patches of \(10 \mathrm {~cm} ^ { 2 }\), use the Poisson distribution to find the probability that a particular area of \(10 \mathrm {~cm} ^ { 2 }\) contains no more than one daisy.
  4. Compare your answers to parts (b) and (c).
  5. Use a suitable approximation to find the probability that a patch of area \(1 \mathrm {~m} ^ { 2 }\) contains more than 8100 daisies.
Edexcel S2 Q1
4 marks Easy -1.8
  1. Explain what is meant by
    1. a population,
    2. a sampling unit.
    Suggest suitable sampling frames for surveys of
  2. families who have holidays in Greece,
  3. mothers with children under two years old.
Edexcel S2 Q2
6 marks Easy -1.3
2. A continuous random variable \(X\) has the probability density function $$\begin{array} { l l } \mathrm { f } ( x ) = k & 5 \leq x \leq 15 , \\ \mathrm { f } ( x ) = 0 & \text { otherwise. } \end{array}$$
  1. Find \(k\) and specify the cumulative density function \(\mathrm { F } ( x )\).
  2. Write down the value of \(\mathrm { P } ( X < 8 )\).
Edexcel S2 Q3
8 marks Moderate -0.3
3. A coin is tossed 20 times, giving 16 heads.
  1. Test at the \(1 \%\) significance level whether the coin is fair, stating your hypotheses clearly.
  2. Find the critical region for the same test at the \(0.1 \%\) significance level.
Edexcel S2 Q4
9 marks Standard +0.8
4. Alison and Gemma play table tennis. Alison starts by serving for the first five points. The probability that she wins a point when serving is \(p\).
  1. Show that the probability that Alison is ahead at the end of her five serves is given by $$p ^ { 3 } \left( 6 p ^ { 2 } - 15 p + 10 \right) .$$
  2. Evaluate this probability when \(p = 0.6\).
Edexcel S2 Q5
15 marks Moderate -0.3
5. In a certain school, \(32 \%\) of Year 9 pupils are left-handed. A random sample of 10 Year 9 pupils is chosen.
  1. Find the probability that none are left-handed.
  2. Find the probability that at least two are left-handed.
  3. Use a suitable approximation to find the probability of getting more than 5 but less than 15 left-handed pupils in a group of 35 randomly selected Year 9 pupils.
    Explain what adjustment is necessary when using this approximation. \section*{STATISTICS 2 (A) TEST PAPER 3 Page 2}
Edexcel S2 Q6
15 marks Standard +0.3
  1. A sample of radioactive material decays randomly, with an approximate mean of 1.5 counts per minute.
    1. Name a distribution that would be suitable for modelling the number of counts per minute.
    Give any parameters required for the model.
  2. Find the probability of at least 4 counts in a randomly chosen minute.
  3. Find the probability of 3 counts or fewer in a random interval lasting 5 minutes. More careful measurements, over 50 one-minute intervals, give the following data for \(x\), the number of counts per minute: $$\sum x = 84 , \quad \sum x ^ { 2 } = 226$$
  4. Decide whether these data support your answer to part (a).
  5. Use the improved data to find probability of exactly two counts in a given one-minute interval.
Edexcel S2 Q7
18 marks Standard +0.3
7. Each day on the way to work, a commuter encounters a similar traffic jam. The length of time, in 10-minute units, spent waiting in the traffic jam is modelled by the random variable \(T\) with the cumulative distribution function: $$\begin{array} { l l } \mathrm { F } ( t ) = 0 & t < 0 , \\ \mathrm {~F} ( t ) = \frac { t ^ { 2 } \left( 3 t ^ { 2 } - 16 t + 24 \right) } { 16 } & 0 \leq t \leq 2 , \\ \mathrm {~F} ( t ) = 1 & t > 2 . \end{array}$$
  1. Show that 0.77 is approximately the median value of \(T\).
  2. Given that he has already waited for 12 minutes, find the probability that he will have to wait another 3 minutes.
  3. Find, and sketch, the probability density function of \(T\).
  4. Hence find the modal value of \(T\).
  5. Comment on the validity of this model.
Edexcel S2 Q1
4 marks Easy -1.8
  1. A random sample is to be taken from the A-level results obtained by the final-year students in a Sixth Form College. Suggest
    1. suitable sampling units,
    2. a suitable sampling frame.
    3. Would it be advisable simply to use the results of all those doing A-level Maths?
    Explain your answer.
Edexcel S2 Q2
5 marks Moderate -0.8
2. The random variable \(X\), which can take any value in the interval \(1 \leq X \leq n\), is modelled by the continuous uniform distribution with mean 12.
  1. Show that \(n = 23\) and find the variance of \(X\).
  2. Find \(\mathrm { P } ( 10 < X < 14 )\).