Questions — Edexcel (9685 questions)

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Edexcel S1 Q6
12 marks
6. At the start of a gameshow there are 10 contestants of which 6 are female. In each round of the game, one contestant is eliminated. All of the contestants have the same chance of progressing to the next round each time.
  1. Show that the probability that the first two contestants to be eliminated are both male is \(\frac { 2 } { 15 }\).
  2. Find the probability that more females than males are eliminated in the first three rounds of the game.
  3. Given that the first contestant to be eliminated is male, find the probability that the next two contestants to be eliminated are both female.
    (3 marks)
Edexcel S1 Q7
15 marks Moderate -0.8
7. A cyber-cafe recorded how long each user stayed during one day giving the following results.
Length of stay
(minutes)
\(0 -\)\(30 -\)\(60 -\)\(90 -\)\(120 -\)\(240 -\)\(360 -\)
Number of users153132231720
  1. Use linear interpolation to estimate the median and quartiles of these data. The results of a previous study had led to the suggestion that the length of time each user stays can be modelled by a normal distribution with a mean of 72 minutes and a standard deviation of 48 minutes.
  2. Find the median and quartiles that this model would predict.
  3. Comment on the suitability of the suggested model in the light of the new results.
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 Moderate -0.8
  1. (a) Explain why it is often useful to take samples as a means of obtaining information.
    (b) Briefly define the term sampling frame.
    (c) Suggest a suitable sampling frame for a sample survey on a proposal to install speed humps on a road.
  2. An insurance company conducts its business by using a Call Centre. The average number of calls per minute is \(3 \cdot 5\). In the first minute after a TV advertisement is shown, the number of calls received is 7 .
    (a) Stating your hypotheses carefully, and working at the \(5 \%\) significance level, test whether the advertisement has had an effect.
    (b) Find the number of calls that would be required in the first minute for the null hypothesis to be rejected at the \(0.1 \%\) significance level.
  3. On average, \(35 \%\) of the candidates in a certain subject get an A or B grade in their exam. In a class of 20 students, find the probability that
    (a) less than 5 get A or B grades,
    (b) exactly 8 get A or B grades.
Five such classes of 20 students are combined to sit the exam.
(c) Use a suitable approximation to find the probability that less than a quarter of the total get A or B grades.
Edexcel S2 Q4
12 marks Standard +0.3
4. Light bulbs produced in a certain factory have lifetimes, in 100 s of hours, whose distribution is modelled by the random variable \(X\) with probability density function $$\begin{array} { l l } \mathrm { f } ( x ) = \frac { 2 x ( 3 - x ) } { 9 } , & 0 \leq x \leq 3 \\ \mathrm { f } ( x ) = 0 & \text { otherwise } \end{array}$$
  1. Sketch \(\mathrm { f } ( x )\).
  2. Write down the mean lifetime of a bulb.
  3. Show that ten times as many bulbs fail before 200 hours as survive beyond 250 hours.
  4. Given that a bulb lasts for 200 hours, find the probability that it will then last for at least another 50 hours.
  5. State, with a reason, whether you consider that the density function \(f\) is a realistic model for the lifetimes of light bulbs. \section*{STATISTICS 2 (A) TEST PAPER 2 Page 2}
Edexcel S2 Q5
11 marks Standard +0.3
  1. In a packet of 40 biscuits, the number of currants in each biscuit is as follows
Number of currants, \(x\)0123456
Number of biscuits49118431
  1. Find the mean and variance of the random variable \(X\) representing the number of currants per biscuit.
  2. State an appropriate model for the distribution of \(X\), giving two reasons for your answer. Another machine produces biscuits with a mean of 1.9 currants per biscuit.
  3. Determine which machine is more likely to produce a biscuit with at least two currants.
Edexcel S2 Q6
12 marks Standard +0.3
6. A greengrocer sells apples from a barrel in his shop. He claims that no more than \(5 \%\) of the apples are of poor quality. When he takes 10 apples out for a customer, 2 of them are bad.
  1. Stating your hypotheses clearly, test his claim at the \(1 \%\) significance level.
  2. State an assumption that has been made about the selection of the apples.
  3. When five other customers also buy 10 apples each, the numbers of bad apples they get are \(1,3,1,2\) and 1 respectively. By combining all six customers' results, and using a suitable approximation, test at the \(1 \%\) significance level whether the combined results provide evidence that the proportion of bad apples in the barrel is greater than \(5 \%\).
  4. Comment briefly on your results in parts (a) and (c).
Edexcel S2 Q7
18 marks Standard +0.3
7. Some children are asked to mark the centre of a scale 10 cm long. The position they choose is indicated by the variable \(X\), where \(0 \leq X \leq 10\). Initially, \(X\) is modelled as a random variable with a continuous uniform distribution.
  1. Find the mean and the standard deviation of \(X\). It is suggested that a better model would be the distribution with probability density function $$f ( x ) = c x , 0 \leq x \leq 5 , \quad f ( x ) = c ( 10 - x ) , 5 < x \leq 10 , \quad f ( x ) = 0 \text { otherwise. }$$
  2. Write down the mean of \(X\).
  3. Find \(c\), and hence find the standard deviation of \(X\) in this model.
  4. Find \(\mathrm { P } ( 4 < X < 6 )\). It is then proposed that an even better model for \(X\) would be a Normal distribution with the mean and standard deviation found in parts (b) and (c).
  5. Use these results to find \(\mathrm { P } ( 4 < X < 6 )\) in the third model.
  6. Compare your answer with (d). Which model do you think is most appropriate? (1 mark)
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 \(\mathrm { P } ( 10 < X < 14 )\).
Edexcel S2 Q3
10 marks Moderate -0.8
3. The Driving Theory Test includes 30 questions which require one answer to be selected from four options.
  1. Phil ticks answers at random. Find how many of the 30 he should expect to get right.
  2. If he gets 15 correct, decide whether this is evidence that he has actually done some revision. Use a \(5 \%\) significance level. Another candidate, Sarah, has revised and has a 0.9 probability of getting each question right.
  3. Determine the expected number of answers that Sarah will get right.
  4. Find the probability that Sarah gets more than 25 correct answers out of 30.
Edexcel S2 Q4
10 marks Moderate -0.3
4. A continuous random variable \(X\) has probability density function $$\begin{array} { l l } \mathrm { f } ( x ) = 0 & x < 1 , \\ \mathrm { f } ( x ) = k x & 1 \leq x \leq 4 , \\ \mathrm { f } ( x ) = 0 & x > 4 . \end{array}$$
  1. Sketch a graph of \(\mathrm { f } ( x )\), and hence find the value of \(k\).
  2. Calculate the mean and the variance of \(X\). \section*{STATISTICS 2 (A)TEST PAPER 4 Page 2}